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The '''Antoine equation'''<ref>{{Citation | |||
|last= Antoine | |||
|first= C. | |||
|title= Tensions des vapeurs; nouvelle relation entre les tensions et les températures | |||
|journal= Comptes Rendus des Séances de l'Académie des Sciences | |||
|volume= 107 | |||
|pages= 681–684, 778–780, 836–837 | |||
|year= 1888 | |||
|doi= }} | |||
</ref> | |||
is a vapor pressure equation and describes the relation between [[vapor pressure]] and temperature for pure components. The Antoine equation is derived from the [[Clausius-Clapeyron relation]]. | |||
==The equation== | |||
:<math>\log_{10} p = A-\frac{B}{C+T}.</math> | |||
where ''p'' is the vapor pressure, ''T'' is [[temperature]] and ''A'', ''B'' and ''C'' are component-specific constants. | |||
The simplified form with ''C'' set to zero: | |||
:<math>\log_{10} p = A-\frac{B}{T}</math> | |||
is named the '''August equation''', after the German physicist [[Ernst Ferdinand August]] (1795–1870). The August equation describes a linear relation between the logarithm of the pressure and the reciprocal temperature. This assumes a temperature-independent [[heat of vaporization]]. The Antoine equation allows an improved, but still inexact description of the change of the heat of vaporization with the temperature. | |||
The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations: | |||
:<math>T = \frac{B}{A-\log_{10}\, p} - C</math> | |||
==Validity range== | |||
Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the [[triple point]] to the [[critical point (thermodynamics)|critical point]], because it is not flexible enough. Therefore, multiple parameter sets for a single component are commonly used. A low-pressure parameter set is used to describe the vapour pressure curve up to the [[normal boiling point]] and the second set of parameters is used for the range from the normal boiling point to the critical point. | |||
<gallery caption="Typical deviations of a parameter fit over the entire range (experimental data for Benzene)" widths="150px" > | |||
Image:VaporPressureFitAugust.png | Deviations of a ''August'' equation fit (2 parameters) | |||
Image:VaporPressureFitAntoine.png | Deviations of a ''Antoine'' equation fit (3 parameters) | |||
Image:VaporPressureFitDIPPR101.png | Deviations of a ''DIPPR 101'' equation fit (4 parameters) | |||
</gallery> | |||
==Example parameters== | |||
{| cellpadding="4" rules="all" style="margin: 1em 0em; background: #ffffff; border: 2px solid #aaa;" | |||
|- align="center" bgcolor="#f0f0f0" | |||
! | |||
!''A'' | |||
!''B'' | |||
!''C'' | |||
!''T'' min. | |||
°C | |||
!''T'' max | |||
°C | |||
|- align="center" | |||
| bgcolor="#f0f0f0" |Water||8.07131||1730.63||233.426||1||100 | |||
|- align="center" | |||
| bgcolor="#f0f0f0" |Water||8.14019||1810.94||244.485||99||374 | |||
|- align="center" | |||
| bgcolor="#f0f0f0"|Ethanol||8.20417||1642.89||230.300||-57||80 | |||
|- align="center" | |||
| bgcolor="#f0f0f0"|Ethanol||7.68117||1332.04||199.200||77||243 | |||
|} | |||
The constants are given in °C and [[mmHg]]. | |||
===Example calculation=== | |||
The normal boiling point of ethanol is ''T''<sub>B</sub> = 78.32 °C. | |||
:<math>P = 10^{(8{.}20417 - \frac{1642{.}89}{78{.}32 + 230{.}300})} = 760{.}0\ \mathrm{mmHg}</math> | |||
:<math>P = 10^{(7{.}68117 - \frac{1332{.}04}{78{.}32 + 199{.}200})} = 761{.}0\ \mathrm{mmHg}</math> | |||
(760 mmHg = 101.325 kPa = 1.000 atm = normal pressure) | |||
This example shows a severe problem caused by using two different sets of coefficients. The described vapor pressure is not [[Continuous function|continuous]]—at the normal boiling point the two sets give different results. This causes severe problems for computational techniques which rely on a continuous vapor pressure curve. | |||
Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. A variant of this single set approach is using a special parameter set fitted for the examined temperature range. The second solution is switching to another vapor pressure equation with more than three parameters. Commonly used are simple extensions of the Antoine equation (see below) and the equations of DIPPR or Wagner.<ref>{{Citation|last= Wagner|first= W.|title= New vapour pressure measurements for argon and nitrogen and an new method for establishing rational vapour pressure equations|journal= Cryogenics|volume= 13|issue= 8|pages= 470–482 |year= 1973|doi= }}</ref><ref>{{Citation|last= Reid|first= R. C.|title= Properties of Gases and Liquids|year= |edition= 3rd |isbn= |doi= }}</ref> | |||
==Units== | |||
The coefficients of Antoine's equation are normally given in '''mmHg'''—even today where the [[International System of Units|SI]] is recommended and [[Pascal (unit)|pascal]]s are preferred. The usage of the pre-SI units has only historic reasons and originates directly from Antoine's original publication. | |||
It is however easy to convert the parameters to different pressure and temperature units. For switching from degrees Celsius to kelvin it is sufficient to subtract 273.15 from the C parameter. For switching from millimeters of mercury to pascals it is sufficient to add the common [[logarithm]] of the factor between both units to the A parameter: | |||
<math>A_{\mathrm{Pa}} = A_{\mathrm{mmHg}} + \log_{10}\frac{101325}{760} = A_{\mathrm{mmHg}} + 2.124903.</math> | |||
The parameters for '''°C''' and '''mmHg''' for [[ethanol]] | |||
:{| cellpadding="2" rules="all" style="border: 1px solid #aaa;" | |||
|- | |||
!A | |||
!B | |||
!C | |||
|- | |||
||8.20417||1642.89||230.300 | |||
|} | |||
are converted for '''K''' and '''Pa''' to | |||
:{| cellpadding="2" rules="all" style="border: 1px solid #aaa;" | |||
|- | |||
!A | |||
!B | |||
!C | |||
|- | |||
||10.32907||1642.89||-42.85 | |||
|} | |||
The first example calculation with ''T''<sub>B</sub> = 351.47 K becomes | |||
:<math>\log_{10}(P) = 10{.}3291 - \frac{1642{.}89}{351{.}47 - 42{.}85} = 5{.}005727378 = \log_{10}(101328\ \mathrm{Pa}).</math> | |||
A similarly simple transformation can be used if the common logarithm should be exchanged by the natural logarithm. It is sufficient to multiply the A and B parameters by ln(10) = 2.302585. | |||
The example calculation with the converted parameters (for '''K''' and '''Pa'''): | |||
:{| cellpadding="2" rules="all" style="border: 1px solid #aaa;" | |||
|- | |||
!A | |||
!B | |||
!C | |||
|- | |||
||23.7836||3782.89||-42.85 | |||
|} | |||
becomes | |||
:<math>\ln P = 23{.}7836 - \frac{3782{.}89}{351{.}47 - 42{.}85} = 11{.}52616367 = \ln(101332\,\mathrm{Pa}).</math> | |||
(The small differences in the results are only caused by the used limited precision of the coefficients). | |||
==Extension of the Antoine equations== | |||
To overcome the limits of the Antoine equation some simple extension by additional terms are used: | |||
* <math>P = \exp{\left( A + \frac{B}{C+T} + D \cdot T + E \cdot T^2 + F \cdot \ln \left( T \right) \right)}</math> | |||
* <math>P = \exp\left( A + \frac{B}{C+T} + D \cdot \ln \left( T \right) + E \cdot T^F\right).</math> | |||
The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The extended equation forms can be reduced to the original form by setting the additional parameters ''D'', ''E'' and ''F'' to 0. | |||
A further difference is that the extended equations use the e as base for the exponential function and the natural logarithm. This doesn't affect the equation form. | |||
==Sources for Antoine equation parameters== | |||
* [http://webbook.nist.gov/chemistry/ NIST Chemistry WebBook] | |||
* [[Dortmund Data Bank]] | |||
* [http://www.eqi.ethz.ch/fmi/xsl/eqi/eqi_property_details_en.xsl?node_id=983 Directory of reference books and data banks containing Antoine constants] | |||
* Several reference books and publications, e. g. | |||
** Lange's Handbook of Chemistry, McGraw-Hill Professional | |||
** Wichterle I., Linek J., "Antoine Vapor Pressure Constants of Pure Compounds" | |||
** Yaws C. L., Yang H.-C., "To Estimate Vapor Pressure Easily. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds", Hydrocarbon Processing, 68(10), Pages 65–68, 1989 | |||
==See also== | |||
* [[Lee-Kesler method]] (Estimation of vapor pressure) | |||
==Literature== | |||
<references/> | |||
==External links== | |||
* [http://gallica.bnf.fr/ark:/12148/bpt6k30635 Gallica, scanned original paper] | |||
* [http://webbook.nist.gov/chemistry/ NIST Chemistry Web Book] | |||
* [http://ddbonline.ddbst.de/AntoineCalculation/AntoineCalculationCGI.exe Calculation of vapor pressures with the Antoine equation] | |||
{{DEFAULTSORT:Antoine Equation}} | |||
[[Category:Equations]] | |||
[[Category:Thermodynamic equations]] |
Latest revision as of 15:08, 26 June 2013
The Antoine equation[1] is a vapor pressure equation and describes the relation between vapor pressure and temperature for pure components. The Antoine equation is derived from the Clausius-Clapeyron relation.
The equation
where p is the vapor pressure, T is temperature and A, B and C are component-specific constants.
The simplified form with C set to zero:
is named the August equation, after the German physicist Ernst Ferdinand August (1795–1870). The August equation describes a linear relation between the logarithm of the pressure and the reciprocal temperature. This assumes a temperature-independent heat of vaporization. The Antoine equation allows an improved, but still inexact description of the change of the heat of vaporization with the temperature.
The Antoine equation can also be transformed in a temperature-explicit form with simple algebraic manipulations:
Validity range
Usually, the Antoine equation cannot be used to describe the entire saturated vapour pressure curve from the triple point to the critical point, because it is not flexible enough. Therefore, multiple parameter sets for a single component are commonly used. A low-pressure parameter set is used to describe the vapour pressure curve up to the normal boiling point and the second set of parameters is used for the range from the normal boiling point to the critical point.
-
Deviations of a August equation fit (2 parameters)
-
Deviations of a Antoine equation fit (3 parameters)
-
Deviations of a DIPPR 101 equation fit (4 parameters)
Example parameters
A | B | C | T min.
°C |
T max
°C | |
---|---|---|---|---|---|
Water | 8.07131 | 1730.63 | 233.426 | 1 | 100 |
Water | 8.14019 | 1810.94 | 244.485 | 99 | 374 |
Ethanol | 8.20417 | 1642.89 | 230.300 | -57 | 80 |
Ethanol | 7.68117 | 1332.04 | 199.200 | 77 | 243 |
The constants are given in °C and mmHg.
Example calculation
The normal boiling point of ethanol is TB = 78.32 °C.
(760 mmHg = 101.325 kPa = 1.000 atm = normal pressure)
This example shows a severe problem caused by using two different sets of coefficients. The described vapor pressure is not continuous—at the normal boiling point the two sets give different results. This causes severe problems for computational techniques which rely on a continuous vapor pressure curve.
Two solutions are possible: The first approach uses a single Antoine parameter set over a larger temperature range and accepts the increased deviation between calculated and real vapor pressures. A variant of this single set approach is using a special parameter set fitted for the examined temperature range. The second solution is switching to another vapor pressure equation with more than three parameters. Commonly used are simple extensions of the Antoine equation (see below) and the equations of DIPPR or Wagner.[2][3]
Units
The coefficients of Antoine's equation are normally given in mmHg—even today where the SI is recommended and pascals are preferred. The usage of the pre-SI units has only historic reasons and originates directly from Antoine's original publication.
It is however easy to convert the parameters to different pressure and temperature units. For switching from degrees Celsius to kelvin it is sufficient to subtract 273.15 from the C parameter. For switching from millimeters of mercury to pascals it is sufficient to add the common logarithm of the factor between both units to the A parameter:
The parameters for °C and mmHg for ethanol
A B C 8.20417 1642.89 230.300
are converted for K and Pa to
A B C 10.32907 1642.89 -42.85
The first example calculation with TB = 351.47 K becomes
A similarly simple transformation can be used if the common logarithm should be exchanged by the natural logarithm. It is sufficient to multiply the A and B parameters by ln(10) = 2.302585.
The example calculation with the converted parameters (for K and Pa):
A B C 23.7836 3782.89 -42.85
becomes
(The small differences in the results are only caused by the used limited precision of the coefficients).
Extension of the Antoine equations
To overcome the limits of the Antoine equation some simple extension by additional terms are used:
The additional parameters increase the flexibility of the equation and allow the description of the entire vapor pressure curve. The extended equation forms can be reduced to the original form by setting the additional parameters D, E and F to 0.
A further difference is that the extended equations use the e as base for the exponential function and the natural logarithm. This doesn't affect the equation form.
Sources for Antoine equation parameters
- NIST Chemistry WebBook
- Dortmund Data Bank
- Directory of reference books and data banks containing Antoine constants
- Several reference books and publications, e. g.
- Lange's Handbook of Chemistry, McGraw-Hill Professional
- Wichterle I., Linek J., "Antoine Vapor Pressure Constants of Pure Compounds"
- Yaws C. L., Yang H.-C., "To Estimate Vapor Pressure Easily. Antoine Coefficients Relate Vapor Pressure to Temperature for Almost 700 Major Organic Compounds", Hydrocarbon Processing, 68(10), Pages 65–68, 1989
See also
- Lee-Kesler method (Estimation of vapor pressure)
Literature
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010