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{{merge|Saturation vapor density|date=August 2012|discuss=talk: Vapor pressure #Four articles for one phenomenon}}
'''Vapor–liquid equilibrium''' ('''VLE''') is a condition where a [[liquid]] and its [[vapor]] (gas phase) are in [[Chemical equilibrium|equilibrium]] with each other, a condition or state where the rate of [[evaporation]] (liquid changing to vapor) equals the rate of [[condensation]] (vapor changing to liquid) on a molecular level such that there is no net (overall) vapor–liquid interconversion. A substance at vapor–liquid equilibrium is generally referred to as a '''saturated fluid'''. For a pure chemical substance this implies that it is at its [[boiling point]].<ref>"Thermodynamics: An Engineering Approach" by Yunus A. Çengel, Michael A. Boles, p.&nbsp;65, ISBN 0-07-121688-X</ref> The notion of "saturated fluid" includes saturated liquid (about to vaporize), saturated liquid–vapor mixture, and [[saturated vapor]] (about to condense).
 
Although in theory equilibrium takes forever to reach, such an equilibrium is practically reached in a relatively closed location if a liquid and its vapor are allowed to stand in contact with each other long enough with no interference or only gradual interference from the outside. This is not the case for intensive heat exchange or rapid pressure change, though.
 
==VLE data introduction==
 
The concentration of a vapor in contact with its liquid, especially at equilibrium, is often in terms of [[vapor pressure]], which could be a [[partial pressure]] (part of the total gas pressure) if any other gas(es) are present with the vapor.  The equilibrium vapor pressure of a liquid is usually very dependent on temperature.  At vapor–liquid equilibrium, a liquid with individual components (compounds) in certain concentrations will have an equilibrium vapor in which the concentrations or partial pressures of the vapor components will have certain set values depending on all of the liquid component concentrations and the temperature.  The converse is also true: if a vapor with components at certain concentrations or partial pressures is in vapor–liquid equilibrium with its liquid, then the component concentrations in the liquid will be set dependent on the vapor concentrations, again also depending on the temperature.  The equilibrium concentration of each component in the liquid phase is often different from its concentration (or vapor pressure) in the vapor phase, but there is a correlation.  Such VLE concentration data is often known or can be determined experimentally for vapor–liquid mixtures with various components. In certain cases such VLE data can be determined or approximated with the help of certain theories such as [[Raoult's law]], [[Dalton's law]], and/or [[Henry's law]]. 
 
Such VLE information is useful in designing [[Fractionating column|columns]] for [[distillation]], especially [[fractional distillation]], which is a particular specialty of [[chemical engineer]]s.<ref name=Kister>{{cite book|author=Kister, Henry Z.|title=[[Distillation Design]]|edition=1st Edition|publisher=McGraw-hill|year=1992|isbn=0-07-034909-6}}</ref><ref name=Perry>{{cite book|author=Perry, R.H. and Green, D.W. (Editors)|title=[[Perry's Chemical Engineers' Handbook]]|edition=7th Edition|publisher=McGraw-hill|year=1997|isbn=0-07-049841-5}}</ref><ref  name=SeaderHenley>{{cite book|author= Seader, J. D., and Henley, Ernest J.|title=Separation Process Principles|publisher=Wiley| location=New York|year=1998|isbn=0-471-58626-9}}</ref> Distillation is a process used to separate or partially separate components in a mixture by boiling (vaporization) followed by [[condensation]].  Distillation takes advantage of differences in concentrations of components in the liquid and vapor phases. 
 
In mixtures containing two or more components where their concentrations are compared in the vapor and liquid phases, concentrations of each component are often expressed as [[mole fraction]]s.  A mole fraction is number of [[Mole (unit)|moles]] of a given component in an amount of mixture in a [[Phase (matter)|phase]] (either vapor or liquid phase) divided by the total number of moles of all components in that amount of mixture in that phase. 
 
Binary mixtures are those having two components.  Three-component mixtures could be called ternary mixtures.  There can be VLE data for mixtures with even more components, but such data becomes copious and is often hard to show graphically.  VLE data is often shown at a certain overall pressure, such as 1&nbsp;[[Atmosphere|atm]] or whatever pressure a process of interest is conducted at.  When at a certain temperature, the total of partial pressures of all the components becomes equal to the overall pressure of the system such that vapors generated from the liquid displace any air or other gas which maintained the overall pressure, the mixture is said to [[Boiling|boil]] and the corresponding temperature is the boiling point (This assumes excess pressure is relieved by letting out gases to maintain a desired total pressure).  A boiling point at an overall pressure of 1&nbsp;atm is called the ''[[normal boiling point]]''.
 
==Thermodynamic description of vapor–liquid equilibrium==
The field of [[thermodynamics]] describes when vapor–liquid equilibrium is possible, and its properties. Much of the analysis depends on whether the vapor and liquid consist of a single component, or if they are mixtures.
 
===Pure (single-component) systems===
If the liquid and vapor are pure, in that they consist of only one molecular component and no impurities, then the equilibrium state between the two phases is described by the following equations:
 
:<math>P^{liq} = P^{vap}\,</math>;
:<math>T^{liq} = T^{vap}\,</math>; and
:<math>\tilde{G}^{liq} = \tilde{G}^{vap}</math>
 
where <math>P^{liq}\,</math> and <math>P^{vap}\,</math> are the [[pressure]]s within the liquid and vapor, <math>T^{liq}\,</math> and <math>T^{vap}\,</math> are the [[temperature]]s within the liquid and vapor, and <math>\tilde{G}^{liq}</math> and <math>\tilde{G}^{vap}</math> are the molar [[Gibbs free energy|Gibbs free energies]] (units of energy per [[amount of substance]]) within the liquid and vapor, respectively.<ref>Balzhiser et al. (1972), ''Chemical Engineering Thermodynamics'', p.&nbsp;215.</ref> In other words, the temperature, pressure and molar Gibbs free energy are the same between the two phases when they are at equilibrium.
 
An equivalent, more common way to express the vapor–liquid equilibrium condition in a pure system is by using the concept of [[fugacity]]. Under this view, equilibrium is described by the following equation:
 
:<math>f^{\,liq}(T_s,P_s) = f^{\,vap}(T_s,P_s)</math>
 
where <math>f^{\,liq}(T_s,P_s)</math> and <math>f^{\,vap}(T_s,P_s)</math> are the [[fugacity|fugacities]] of the liquid and vapor, respectively, at the system temperature {{mvar|T<sub>s</sub>}} and pressure {{mvar|P<sub>s</sub>}}.<ref>Balzhiser et al. (1972), ''Chemical Engineering Thermodynamics'', p.&nbsp;216.</ref> Using fugacity is often more convenient for calculation, given that the fugacity of the liquid is, to a good approximation, pressure-independent,<ref>Balzhiser et al. (1972), ''Chemical Engineering Thermodynamics'', p.&nbsp;218.</ref> and it is often convenient to use the quantity <math>\phi=f/P\,</math>, the dimensionless ''fugacity coefficient'', which is 1 for an [[ideal gas]].
 
===Multicomponent systems===
 
In a multicomponent system, where the vapor and liquid consist of more than one type of molecule, describing the equilibrium state is more complicated. For all components {{mvar|i}} in the system, the equilibrium state between the two phases is described by the following equations:
 
:<math>P^{liq} = P^{vap}\,</math>;
:<math>T^{liq} = T^{vap}\,</math>; and
:<math>\bar{G}_i^{liq} = \bar{G}_i^{vap}</math>
 
where {{mvar|P}} and {{mvar|T}} are the temperature and pressure for each phase, and <math>\bar{G}_i^{liq}</math> and <math>\bar{G}_i^{vap}</math> are the [[partial molar Gibbs free energy]] also called [[chemical potential]] (units of energy per [[amount of substance]]) within the liquid and vapor, respectively, for each phase. The partial molar Gibbs free energy is defined by:
 
:<math>\bar{G}_i \ \stackrel{\mathrm{def}}{=}\  \frac{\partial G}{\partial n_i}</math>
 
where {{mvar|G}} is the ([[extensive property|extensive]]) Gibbs free energy, and {{mvar|n<sub>i</sub>}} is the [[amount of substance]] of component&nbsp;{{mvar|i}}.
 
==Boiling-point diagrams==
 
Binary mixture VLE data at a certain overall pressure, such as 1&nbsp;atm, showing mole fraction vapor and liquid concentrations when boiling at various temperatures can be shown as a two-dimensional [[graphics|graph]] called a '''boiling-point diagram'''. The mole fraction of component&nbsp;1 in the mixture can be represented by the symbol {{math|''x''<sub>1</sub>}}.  The mole fraction of component&nbsp;2, represented by {{math|''x''<sub>2</sub>}}, is related to {{math|''x''<sub>1</sub>}} in a binary mixture as follows:
:{{math|1=''x''<sub>1</sub> +  ''x''<sub>2</sub> = 1}}
 
In multi-component mixtures in general with n components, this becomes:
:{{math|1=''x''<sub>1</sub> +  ''x''<sub>2</sub> + ... +  ''x''<sub>''n''</sub>  = 1}}
 
[[Image:Binary Boiling Point Diagram new.svg|frame|Boiling-point diagram]] 
The preceding equations are typically applied for each phase (liquid or vapor) individually. In a binary boiling-point diagram, temperature&nbsp;({{mvar|T }}) is graphed vs. {{math|''x''<sub>1</sub>}}.  At any given temperature where boiling is present, vapor with a certain mole fraction is in equilibrium with liquid with a certain mole fraction, often differing from the vapor.  These vapor and liquid mole fractions are both on a horizontal isotherm (constant {{mvar|T }}) line.  When an entire range of boiling temperatures vs. vapor and liquid mole fractions is graphed, two (usually curved) lines are made.  The lower one, representing boiling liquid mole fraction at various temperatures, is called a ''[[bubble point]] curve''.  The upper one, representing vapor mole fraction at corresponding temperatures, is called a ''dew point curve''.<ref name=Kister/>
 
These two lines (or curves) meet where the mixture becomes purely one component, where {{math|1=''x''<sub>1</sub> = 0}} (and {{math|1=''x''<sub>2</sub> = 1}}, pure component&nbsp;2) or {{math|1=''x''<sub>1</sub> = 1}} (and {{math|1=''x''<sub>2</sub> = 0}}, pure component&nbsp;1).  The temperatures at those two points correspond to the boiling points of the two pure components.  In certain combinations of components, the two curves may also meet at a point somewhere in between {{math|1=''x''<sub>1</sub> = 0}} and {{math|1=''x''<sub>1</sub> = 1}}.  That point represents an '''[[azeotrope]]''' in that particular combination of components.  That point has an azeotrope temperature and an azeotropic composition often represented as a mole fraction.  There can be '''maximum-boiling azeotropes''', where the azeotrope temperature is at a maximum in the boiling curves, or '''minimum-boiling azeotropes''', where the azeotrope temperature is at a minimum in the boiling curves. 
 
If one wants to represent a VLE data for a three-component mixture as a boiling point "diagram", a [[three-dimensional graph]] can be used.  Two of the dimensions would be used to represent the composition mole fractions, and the third dimension would be the temperature.  Using two dimensions, the composition can be represented as an equilateral triangle in which each corner representing one of the pure components.  The edges of the triangle represent a mixture of the two components at each end of the edge. Any point inside the triangle represent the composition of a mixture of all three components. The mole fraction of each component would correspond to where a point lies along a line starting at that component's corner and perpendicular to the opposite edge. The [[bubble point]] and [[dew point]] data would become curved surfaces inside a triangular prism, which connect the three boiling points on the vertical temperature "axes". Each face of this triangular prism would represent a two-dimensional boiling-point diagram for the corresponding binary mixture. Due to their three-dimensional complexity, such boiling-point diagrams are rarely seen. Alternatively, the three-dimensional curved surfaces can be represented on a two-dimensional graph by the use of curved isotherm lines at graduated intervals, similar to iso-altitude lines on a map. Two sets of such isotherm lines are needed on such a two-dimensional graph: one set for the bubble point surface and another set for the dew point surface.
 
===''K'' values and relative volatility values ===
{{main|Relative volatility}}
[[File:Vapor-Liquid Equilibrium K Values Chloroform + Methanol.png||thumb|K Values Diagram (with [[UNIQUAC]] Best-Fit Curve), Mixture of [[Chloroform]]/[[Methanol]]]]
The tendency of a given chemical species to partition itself preferentially between liquid and vapor phases is the equilibrium ratio <math>K_i</math>.<ref>Smith, J. M., Van Ness, H. C., & Abbott, M. M. (2005)."Introduction to Chemical Engineering Thermodynamics (Seventh ed.)". Boston: McGraw Hill</ref>
There can be VLE data for mixtures of four or more components, but such a boiling-point diagram is hard to show in either tabular or graphical form. For such multi-component mixtures, as well as binary
mixtures, the vapor–liquid equilibrium data are represented in terms of {{mvar|K}} values ([[Relative volatility|vapor–liquid distribution ratio]]s)<ref name=Kister/><ref name=Perry/> defined by
 
:<math>K_i=\frac {y_i}x_i</math>
 
where {{mvar|y<sub>i</sup>}} and {{mvar|x<sub>i</sub>}} are the mole fractions of component&nbsp;{{mvar|i}} in the phases {{mvar|y}} and {{mvar|x}} respectively.
 
For [[Raoult's law]] 
                                   
:<math>K_i=\frac {P_i^{\star}}P</math> 
 
                 
For modified [[Raoult's law]]
:<math>K_i=\frac {\gamma_iP_i^{\star}}P</math>
 
where <math>\gamma_i</math> is the [[activity coefficient]], {{mvar|P<sub>i</sup>}} is the [[partial pressure]] and {{mvar|P</sup>}} is the [[pressure]]
 
The values of the ratio {{mvar|K<sub>i</sub>}} are correlated empirically or theoretically in terms of temperature, pressure and phase compositions in the form of equations, tables or graph such as the DePriester charts (Shown on the right).<ref>DePriester, C.L., ''Chem. Eng. Prog. Symposium Series'', 7, '''49''', pages 1–43</ref>
[[File:DePriester Chart 1.JPG|thumb|K-Values for systems of light hydrocarbons Low Temperature Range]]
[[File:DePriester Chart 2.JPG|thumb|K-Values for systems of light hydrocarbons High Temperature Range]]
 
For binary mixtures, the ratio of the {{mvar|K}} values for the two components is called the [[relative volatility]] denoted by {{mvar|α}}
 
:<math>\alpha=\frac {K_i}K_j=\frac {(y_i/x_i)}{(y_j/x_j)}</math>
 
which is a measure of the relative ease or difficulty of separating the two components. Large-scale industrial distillation is rarely undertaken if the relative volatility is less than 1.05 with the volatile component being {{mvar|i}} and the less volatile component being {{mvar|j}}.<ref name=Perry/>
 
{{mvar|K}} values are widely used in the design calculations of [[continuous distillation]] columns for distilling multicomponent mixtures.
 
==Vapor–liquid equilibrium diagrams==
 
[[Image:Vapor-Liquid Equilibrium Diagram.PNG|frame|Vapor-Liquid Equilibrium Diagram]] 
For each component in a binary mixture, one could make a vapor–liquid equilibrium diagram.  Such a diagram would graph liquid mole fraction on a horizontal axis and vapor mole fraction on a vertical axis.  In such VLE diagrams, liquid mole fractions for components 1 and 2 can be represented as {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} respectively, and vapor mole fractions of the corresponding components are commonly represented as {{math|''y''<sub>1</sub>}} and {{math|''y''<sub>2</sub>}}.<ref name=Perry/>  Similarly for binary mixtures in these VLE diagrams:
:{{math|1=''x''<sub>1</sub> +  ''x''<sub>2</sub> = 1 }} and&nbsp; {{math|1=  ''y''<sub>1</sub> +  ''y''<sub>2</sub> = 1}}
 
Such VLE diagrams are square with a diagonal line running from the ({{math|1=''x''<sub>1</sub> = 0, ''y''<sub>1</sub> = 0}}) corner to the ({{math|1=''x''<sub>1</sub> = 1, ''y''<sub>1</sub> = 1}}) corner for reference. 
 
These types of VLE diagrams are used in the [[McCabe–Thiele method]] to determine the number of equilibrium stages (or [[theoretical plate]]s) needed to distill a given composition binary feed mixture into one distillate fraction and one bottoms fraction.  Corrections can also be made to take into account the incomplete efficiency of each tray in a distillation column when compared to a theoretical plate.
 
==Raoult's law==
{{main|Raoult's law}}
At boiling and higher temperatures the sum of the individual component partial pressures becomes equal to the overall pressure, which can symbolized as ''P''<sub>tot</sub>. 
 
Under such conditions, [[Dalton's law]] would be in effect as follows: 
:''P''<sub>tot</sub> = ''P''<sub>1</sub> + ''P''<sub>2</sub> + ...
 
Then for each component in the vapor phase: 
:''y''<sub>1</sub> = ''P''<sub>1</sub>&thinsp;/&thinsp;''P''<sub>tot</sub>,&nbsp;&nbsp; ''y''<sub>2</sub> = ''P''<sub>2</sub>&thinsp;/&thinsp;''P''<sub>tot</sub>, ... etc.   
 
where ''P''<sub>1</sub> = partial pressure of component 1, ''P''<sub>2</sub> = partial pressure of component 2, etc. 
 
[[Raoult's law]] is approximately valid for mixtures of components between which there is very little interaction other than the effect of dilution by the other components.  Examples of such mixtures includes mixtures of [[alkane]]s, which are non-[[Chemical polarity|polar]], relatively inert [[Chemical compound|compounds]] in many ways, so there is little attraction or repulsion between the molecules.  Raoult's law states that for components 1, 2, etc. in a mixture: 
 
:''P''<sub>1</sub> = ''x''<sub>1</sub> ''P''<sup>&thinsp;o</sup><sub>1</sub>,&nbsp;&nbsp; ''P''<sub>2</sub> = ''x''<sub>2</sub> ''P''<sup>&thinsp;o</sup><sub>2</sub>, ...  etc.
 
where ''P''<sup>&thinsp;o</sup><sub>1</sub>, P<sup>&thinsp;o</sup><sub>2</sub>, etc. are the vapor pressures of components 1, 2, etc. when they are pure, and ''x''<sub>1</sub>, ''x''<sub>2</sub>, etc. are mole fractions of the corresponding component in the liquid. 
 
Recall from the first section that vapor pressures of liquids are very dependent on temperature.  Thus the ''P''<sup>o</sup> pure vapor pressures for each component are a function of temperature (''T''):  For example, commonly for a pure liquid component, the [[Clausius–Clapeyron relation]] may be used to approximate how the vapor pressure varies as a function of temperature.  This makes each of the partial pressures dependent on temperature also regardless of whether Raoult's law applies or not.  When Raoult's law is valid these expressions become: 
 
:''P''<sub>1</sub>''T'' = ''x''<sub>1</sub> ''P''<sup>&thinsp;o</sup><sub>1</sub>''T'',&nbsp;&nbsp; ''P''<sub>2</sub>''T'' = ''x''<sub>2</sub> ''P''<sup>&thinsp;o</sup><sub>2</sub>''T'', ...  etc.
 
At boiling temperatures if Raoult's law applies, the total pressure becomes:
:''P''<sub>tot</sub> = ''x''<sub>1</sub> ''P''<sup>&thinsp;o</sup><sub>1</sub>''T'' + ''x''<sub>2</sub> ''P''<sup>&thinsp;o</sup><sub>2</sub>''T'' + ... etc.
 
At a given ''P''<sub>tot</sub> such as 1&nbsp;atm and a given liquid composition, ''T'' can be solved for to give the liquid mixture's boiling point or bubble point, although the solution for ''T'' may not be mathematically analytical (i.e., may require a numerical solution or approximation).  For a binary mixture at a given ''P''<sub>tot</sub>, the bubble point&nbsp;''T'' can become a function of ''x''<sub>1</sub> (or ''x''<sub>2</sub>) and this function can be shown on a two-dimensional graph like a binary boiling point diagram. 
 
At boiling temperatures if Raoult's law applies, a number of the preceding equations in this section can be combined to give the following expressions for vapor mole fractions as a function of liquid mole fractions and temperature: 
:''y''<sub>1</sub> = ''x''<sub>1</sub> ''P''<sup>&thinsp;o</sup><sub>1</sub>''T''&thinsp;/&thinsp;''P''<sub>tot</sub>,&nbsp;&nbsp; ''y''<sub>2</sub> = ''x''<sub>2</sub> ''P''<sup>&thinsp;o</sup><sub>2</sub>''T''&thinsp;/&thinsp;''P''<sub>tot</sub>, ... etc.
 
Once the bubble point ''T'''s as a function of liquid composition in terms of mole fractions have been determined, these values can be inserted into the above equations to obtain corresponding vapor compositions in terms of mole fractions.  When this is finished over a complete range of liquid mole fractions and their corresponding temperatures, one effectively obtains a temperature&nbsp; ''T'' function of vapor composition mole fractions.  This function effectively acts as the dew point ''T'' function of vapor composition. 
 
In the case of a binary mixture, ''x''<sub>2</sub> = 1 − ''x''<sub>1</sub>  and the above equations can be expressed as: 
:''y''<sub>1</sub> = ''x''<sub>1</sub> ''P''<sup>&thinsp;o</sup><sub>1</sub>''T''&thinsp;/&thinsp;''P''<sub>tot</sub>, and   
:''y''<sub>2</sub> = (1  −  ''x''<sub>1</sub>) ''P''<sup>&thinsp;o</sup><sub>2</sub>''T''&thinsp;/&thinsp;''P''<sub>tot</sub>
 
For many kinds of mixtures, particularly where there is interaction between components beyond simply the effects of dilution, Raoult's law does not work well for determining the shapes of the curves in the boiling point or VLE diagrams.  Even in such mixtures, there are usually still differences in the vapor and liquid equilibrium concentrations at most points, and distillation is often still useful for separating components at least partially.  For such mixtures, empirical data is typically used in determining such boiling point and VLE diagrams.  [[Chemical engineer]]s have done a significant amount of research trying to develop equations for correlating and/or predicting VLE data for various kinds of mixtures which do not obey Raoult's law well.
 
==See also==
 
*[[Continuous distillation]]
*[[Dortmund Data Bank]] (includes a collection of VLE data)
*[[Fenske equation]]
*[[Flash evaporation]]
*[[DECHEMA model]]
*[[Hand boiler]]
*[[Van Laar equation]]
*[[Margules activity model]]
*[[Pervaporation]]
*[[Supercooling]]
*[[Superheated steam]]
 
==External links==
* [http://lorien.ncl.ac.uk/ming/distil/distilpri.htm Introduction to Distillation: Distillation Principles]
* [http://lorien.ncl.ac.uk/ming/distil/distilvle.htm Introduction to Distillation: Vapor Liquid Equilibria]
* [http://www.et.byu.edu:8080/~rowley/VLEfinal/background2.htm VLE Thermodynamics] (Chemical Engineering Dept., Prof. Richard Rowley, Brigham Young University)
* [http://www.nist.gov/srd/nist103b.htm NIST Standard Reference Database 103b] (Describes the extensive VLE database available from [[NIST]])
* [http://www.ddbst.com/en/EED/VLE/VLEindex.php Some VLE data sets and diagrams for mixtures of 30 common components], a small subset of the [[Dortmund Data Bank]]
 
==References==
 
<references/>
 
{{Chemical equilibria}}
{{states of matter}}
 
{{DEFAULTSORT:Vapor-Liquid Equilibrium}}
[[Category:Chemical engineering]]
[[Category:Equilibrium chemistry]]
[[Category:Distillation]]
[[Category:Phases of matter]]
[[Category:Thermodynamics]]
[[Category:Steam engines]]
 
[[be:Насычаная пара]]
[[cs:Sytá pára]]
[[pl:Para nasycona]]

Revision as of 20:46, 29 November 2013



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Although in theory equilibrium takes forever to reach, such an equilibrium is practically reached in a relatively closed location if a liquid and its vapor are allowed to stand in contact with each other long enough with no interference or only gradual interference from the outside. This is not the case for intensive heat exchange or rapid pressure change, though.

VLE data introduction

The concentration of a vapor in contact with its liquid, especially at equilibrium, is often in terms of vapor pressure, which could be a partial pressure (part of the total gas pressure) if any other gas(es) are present with the vapor. The equilibrium vapor pressure of a liquid is usually very dependent on temperature. At vapor–liquid equilibrium, a liquid with individual components (compounds) in certain concentrations will have an equilibrium vapor in which the concentrations or partial pressures of the vapor components will have certain set values depending on all of the liquid component concentrations and the temperature. The converse is also true: if a vapor with components at certain concentrations or partial pressures is in vapor–liquid equilibrium with its liquid, then the component concentrations in the liquid will be set dependent on the vapor concentrations, again also depending on the temperature. The equilibrium concentration of each component in the liquid phase is often different from its concentration (or vapor pressure) in the vapor phase, but there is a correlation. Such VLE concentration data is often known or can be determined experimentally for vapor–liquid mixtures with various components. In certain cases such VLE data can be determined or approximated with the help of certain theories such as Raoult's law, Dalton's law, and/or Henry's law.

Such VLE information is useful in designing columns for distillation, especially fractional distillation, which is a particular specialty of chemical engineers.[2][3][4] Distillation is a process used to separate or partially separate components in a mixture by boiling (vaporization) followed by condensation. Distillation takes advantage of differences in concentrations of components in the liquid and vapor phases.

In mixtures containing two or more components where their concentrations are compared in the vapor and liquid phases, concentrations of each component are often expressed as mole fractions. A mole fraction is number of moles of a given component in an amount of mixture in a phase (either vapor or liquid phase) divided by the total number of moles of all components in that amount of mixture in that phase.

Binary mixtures are those having two components. Three-component mixtures could be called ternary mixtures. There can be VLE data for mixtures with even more components, but such data becomes copious and is often hard to show graphically. VLE data is often shown at a certain overall pressure, such as 1 atm or whatever pressure a process of interest is conducted at. When at a certain temperature, the total of partial pressures of all the components becomes equal to the overall pressure of the system such that vapors generated from the liquid displace any air or other gas which maintained the overall pressure, the mixture is said to boil and the corresponding temperature is the boiling point (This assumes excess pressure is relieved by letting out gases to maintain a desired total pressure). A boiling point at an overall pressure of 1 atm is called the normal boiling point.

Thermodynamic description of vapor–liquid equilibrium

The field of thermodynamics describes when vapor–liquid equilibrium is possible, and its properties. Much of the analysis depends on whether the vapor and liquid consist of a single component, or if they are mixtures.

Pure (single-component) systems

If the liquid and vapor are pure, in that they consist of only one molecular component and no impurities, then the equilibrium state between the two phases is described by the following equations:

;
; and

where and are the pressures within the liquid and vapor, and are the temperatures within the liquid and vapor, and and are the molar Gibbs free energies (units of energy per amount of substance) within the liquid and vapor, respectively.[5] In other words, the temperature, pressure and molar Gibbs free energy are the same between the two phases when they are at equilibrium.

An equivalent, more common way to express the vapor–liquid equilibrium condition in a pure system is by using the concept of fugacity. Under this view, equilibrium is described by the following equation:

where and are the fugacities of the liquid and vapor, respectively, at the system temperature Template:Mvar and pressure Template:Mvar.[6] Using fugacity is often more convenient for calculation, given that the fugacity of the liquid is, to a good approximation, pressure-independent,[7] and it is often convenient to use the quantity , the dimensionless fugacity coefficient, which is 1 for an ideal gas.

Multicomponent systems

In a multicomponent system, where the vapor and liquid consist of more than one type of molecule, describing the equilibrium state is more complicated. For all components Template:Mvar in the system, the equilibrium state between the two phases is described by the following equations:

;
; and

where Template:Mvar and Template:Mvar are the temperature and pressure for each phase, and and are the partial molar Gibbs free energy also called chemical potential (units of energy per amount of substance) within the liquid and vapor, respectively, for each phase. The partial molar Gibbs free energy is defined by:

where Template:Mvar is the (extensive) Gibbs free energy, and Template:Mvar is the amount of substance of component Template:Mvar.

Boiling-point diagrams

Binary mixture VLE data at a certain overall pressure, such as 1 atm, showing mole fraction vapor and liquid concentrations when boiling at various temperatures can be shown as a two-dimensional graph called a boiling-point diagram. The mole fraction of component 1 in the mixture can be represented by the symbol Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.. The mole fraction of component 2, represented by Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., is related to Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. in a binary mixture as follows:

Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.

In multi-component mixtures in general with n components, this becomes:

Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.
Boiling-point diagram

The preceding equations are typically applied for each phase (liquid or vapor) individually. In a binary boiling-point diagram, temperature (Template:Mvar) is graphed vs. Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.. At any given temperature where boiling is present, vapor with a certain mole fraction is in equilibrium with liquid with a certain mole fraction, often differing from the vapor. These vapor and liquid mole fractions are both on a horizontal isotherm (constant Template:Mvar) line. When an entire range of boiling temperatures vs. vapor and liquid mole fractions is graphed, two (usually curved) lines are made. The lower one, representing boiling liquid mole fraction at various temperatures, is called a bubble point curve. The upper one, representing vapor mole fraction at corresponding temperatures, is called a dew point curve.[2]

These two lines (or curves) meet where the mixture becomes purely one component, where Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. (and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., pure component 2) or Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. (and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer., pure component 1). The temperatures at those two points correspond to the boiling points of the two pure components. In certain combinations of components, the two curves may also meet at a point somewhere in between Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.. That point represents an azeotrope in that particular combination of components. That point has an azeotrope temperature and an azeotropic composition often represented as a mole fraction. There can be maximum-boiling azeotropes, where the azeotrope temperature is at a maximum in the boiling curves, or minimum-boiling azeotropes, where the azeotrope temperature is at a minimum in the boiling curves.

If one wants to represent a VLE data for a three-component mixture as a boiling point "diagram", a three-dimensional graph can be used. Two of the dimensions would be used to represent the composition mole fractions, and the third dimension would be the temperature. Using two dimensions, the composition can be represented as an equilateral triangle in which each corner representing one of the pure components. The edges of the triangle represent a mixture of the two components at each end of the edge. Any point inside the triangle represent the composition of a mixture of all three components. The mole fraction of each component would correspond to where a point lies along a line starting at that component's corner and perpendicular to the opposite edge. The bubble point and dew point data would become curved surfaces inside a triangular prism, which connect the three boiling points on the vertical temperature "axes". Each face of this triangular prism would represent a two-dimensional boiling-point diagram for the corresponding binary mixture. Due to their three-dimensional complexity, such boiling-point diagrams are rarely seen. Alternatively, the three-dimensional curved surfaces can be represented on a two-dimensional graph by the use of curved isotherm lines at graduated intervals, similar to iso-altitude lines on a map. Two sets of such isotherm lines are needed on such a two-dimensional graph: one set for the bubble point surface and another set for the dew point surface.

K values and relative volatility values

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.

K Values Diagram (with UNIQUAC Best-Fit Curve), Mixture of Chloroform/Methanol

The tendency of a given chemical species to partition itself preferentially between liquid and vapor phases is the equilibrium ratio .[8] There can be VLE data for mixtures of four or more components, but such a boiling-point diagram is hard to show in either tabular or graphical form. For such multi-component mixtures, as well as binary mixtures, the vapor–liquid equilibrium data are represented in terms of Template:Mvar values (vapor–liquid distribution ratios)[2][3] defined by

where Template:Mvar and Template:Mvar are the mole fractions of component Template:Mvar in the phases Template:Mvar and Template:Mvar respectively.

For Raoult's law


For modified Raoult's law

where is the activity coefficient, Template:Mvar is the partial pressure and Template:Mvar is the pressure

The values of the ratio Template:Mvar are correlated empirically or theoretically in terms of temperature, pressure and phase compositions in the form of equations, tables or graph such as the DePriester charts (Shown on the right).[9]

K-Values for systems of light hydrocarbons Low Temperature Range
K-Values for systems of light hydrocarbons High Temperature Range

For binary mixtures, the ratio of the Template:Mvar values for the two components is called the relative volatility denoted by Template:Mvar

which is a measure of the relative ease or difficulty of separating the two components. Large-scale industrial distillation is rarely undertaken if the relative volatility is less than 1.05 with the volatile component being Template:Mvar and the less volatile component being Template:Mvar.[3]

Template:Mvar values are widely used in the design calculations of continuous distillation columns for distilling multicomponent mixtures.

Vapor–liquid equilibrium diagrams

Vapor-Liquid Equilibrium Diagram

For each component in a binary mixture, one could make a vapor–liquid equilibrium diagram. Such a diagram would graph liquid mole fraction on a horizontal axis and vapor mole fraction on a vertical axis. In such VLE diagrams, liquid mole fractions for components 1 and 2 can be represented as Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. respectively, and vapor mole fractions of the corresponding components are commonly represented as Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer..[3] Similarly for binary mixtures in these VLE diagrams:

Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. and  Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.

Such VLE diagrams are square with a diagonal line running from the (Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.) corner to the (Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.) corner for reference.

These types of VLE diagrams are used in the McCabe–Thiele method to determine the number of equilibrium stages (or theoretical plates) needed to distill a given composition binary feed mixture into one distillate fraction and one bottoms fraction. Corrections can also be made to take into account the incomplete efficiency of each tray in a distillation column when compared to a theoretical plate.

Raoult's law

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. At boiling and higher temperatures the sum of the individual component partial pressures becomes equal to the overall pressure, which can symbolized as Ptot.

Under such conditions, Dalton's law would be in effect as follows:

Ptot = P1 + P2 + ...

Then for each component in the vapor phase:

y1 = P1 / Ptot,   y2 = P2 / Ptot, ... etc.

where P1 = partial pressure of component 1, P2 = partial pressure of component 2, etc.

Raoult's law is approximately valid for mixtures of components between which there is very little interaction other than the effect of dilution by the other components. Examples of such mixtures includes mixtures of alkanes, which are non-polar, relatively inert compounds in many ways, so there is little attraction or repulsion between the molecules. Raoult's law states that for components 1, 2, etc. in a mixture:

P1 = x1 P o1,   P2 = x2 P o2, ... etc.

where P o1, P o2, etc. are the vapor pressures of components 1, 2, etc. when they are pure, and x1, x2, etc. are mole fractions of the corresponding component in the liquid.

Recall from the first section that vapor pressures of liquids are very dependent on temperature. Thus the Po pure vapor pressures for each component are a function of temperature (T): For example, commonly for a pure liquid component, the Clausius–Clapeyron relation may be used to approximate how the vapor pressure varies as a function of temperature. This makes each of the partial pressures dependent on temperature also regardless of whether Raoult's law applies or not. When Raoult's law is valid these expressions become:

P1T = x1 P o1T,   P2T = x2 P o2T, ... etc.

At boiling temperatures if Raoult's law applies, the total pressure becomes:

Ptot = x1 P o1T + x2 P o2T + ... etc.

At a given Ptot such as 1 atm and a given liquid composition, T can be solved for to give the liquid mixture's boiling point or bubble point, although the solution for T may not be mathematically analytical (i.e., may require a numerical solution or approximation). For a binary mixture at a given Ptot, the bubble point T can become a function of x1 (or x2) and this function can be shown on a two-dimensional graph like a binary boiling point diagram.

At boiling temperatures if Raoult's law applies, a number of the preceding equations in this section can be combined to give the following expressions for vapor mole fractions as a function of liquid mole fractions and temperature:

y1 = x1 P o1T / Ptot,   y2 = x2 P o2T / Ptot, ... etc.

Once the bubble point T's as a function of liquid composition in terms of mole fractions have been determined, these values can be inserted into the above equations to obtain corresponding vapor compositions in terms of mole fractions. When this is finished over a complete range of liquid mole fractions and their corresponding temperatures, one effectively obtains a temperature  T function of vapor composition mole fractions. This function effectively acts as the dew point T function of vapor composition.

In the case of a binary mixture, x2 = 1 − x1 and the above equations can be expressed as:

y1 = x1 P o1T / Ptot, and
y2 = (1  −  x1) P o2T / Ptot

For many kinds of mixtures, particularly where there is interaction between components beyond simply the effects of dilution, Raoult's law does not work well for determining the shapes of the curves in the boiling point or VLE diagrams. Even in such mixtures, there are usually still differences in the vapor and liquid equilibrium concentrations at most points, and distillation is often still useful for separating components at least partially. For such mixtures, empirical data is typically used in determining such boiling point and VLE diagrams. Chemical engineers have done a significant amount of research trying to develop equations for correlating and/or predicting VLE data for various kinds of mixtures which do not obey Raoult's law well.

See also

External links

References

  1. ↑ "Thermodynamics: An Engineering Approach" by Yunus A. Çengel, Michael A. Boles, p. 65, ISBN 0-07-121688-X
  2. ↑ 2.0 2.1 2.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  3. ↑ 3.0 3.1 3.2 3.3 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  4. ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  5. ↑ Balzhiser et al. (1972), Chemical Engineering Thermodynamics, p. 215.
  6. ↑ Balzhiser et al. (1972), Chemical Engineering Thermodynamics, p. 216.
  7. ↑ Balzhiser et al. (1972), Chemical Engineering Thermodynamics, p. 218.
  8. ↑ Smith, J. M., Van Ness, H. C., & Abbott, M. M. (2005)."Introduction to Chemical Engineering Thermodynamics (Seventh ed.)". Boston: McGraw Hill
  9. ↑ DePriester, C.L., Chem. Eng. Prog. Symposium Series, 7, 49, pages 1–43

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