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| {{Distinguish|Molarity}}
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| In [[chemistry]], the '''molality''', ''b'' (or ''m''), of a [[solution]] is defined as the [[amount of substance]] (in [[mole (unit)|mol]]) of [[solute]], <math>n_{solute}</math>, divided by the [[mass]] (in [[kg]]) of the [[solvent]], <math>m_{solvent}</math> ('''not''' the mass of the solution):<ref>{{GoldBookRef | file = M03970 | title = molality}}</ref>
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| :<math>b = \frac{n_{solute}}{m_{solvent}}</math>
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| == Origin ==
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| The earliest such definition of the [[intensive property]] '''molality''' and of its adjectival unit, the now-deprecated '''molal''' (formerly, a variant of '''molar''', describing a solution of unit [[molar concentration]]), appear to have been coined by [[G. N. Lewis]] and [[Merle Randall|M. Randall]] in their 1923 publication of ''Thermodynamics and the Free Energies of Chemical Substances.''<ref name=OED>{{cite book|title=www.OED.com|year=2011|publisher=Oxford University Press}}</ref> Though the two words are subject to being confused with one another, the '''molality''' and [[molarity]] of a ''weak'' [[aqueous solution]] happen to be nearly the same, as one kilogram of water (the solvent) occupies 1 liter of volume at room temperature and the small amount of solute would have little effect on the volume.
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| == Unit ==
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| The [[SI]] unit for molality is mol/kg.
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| A solution with a molality of 3 mol/kg is often described as "3 molal" or "3 m". However, following the SI system of units, the [[National Institute of Standards and Technology]], the [[United States]] authority on [[measurement]], considers the term "molal" and the unit symbol "m" to be obsolete, and suggests mol/kg or a related unit of the SI.<ref>{{cite web|url=http://physics.nist.gov/Pubs/SP811/sec11.html |title=NIST Guide to SI Units |accessdate=2007-12-17 }}</ref> This recommendation has not been universally implemented in academia yet.
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| == Usage considerations ==
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| '''Advantages:'''
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| Compared to [[molar concentration]] or [[mass concentration (chemistry)|mass concentration]], the preparation of a solution of a given molality requires only a good scale: both solvent and solute need to be weighed, as opposed to measured volumetrically, which would be subject to variations in [[density]] due to the ambient conditions of [[temperature]] and [[pressure]]; this is an advantage because, in ''chemical'' compositions, the mass, or the amount, of a pure known substance is more relevant than its volume: a ''contained'' measured amount of substance may change in volume with ambient conditions, but its amount and mass are unvarying, and chemical reactions occur in proportions of mass, not volume. The mass-based nature of molality implies that it can be readily converted into a mass ratio (or [[mass fraction]], "''w''," ratio),
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| :<math>b\,M_{solute}=\frac{m_{solute}}{m_{solvent}}=\frac{w_{solute}}{w_{solvent}},</math>
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| where the symbol ''M'' stands for [[molar mass]], or into a mole ratio (or [[mole fraction]], "''x''," ratio)
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| :<math>b\,M_{solvent}=\frac{n_{solute}}{n_{solvent}}=\frac{x_{solute}}{x_{solvent}}.</math>
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| The advantage of molality over other mass-based fractions is the fact that the molality of one solute in a single-solvent solution is independent of the presence or absence of other solutes.
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| '''Problem areas:'''
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| Unlike all the other compositional properties listed in "Relation" section (below), molality ''depends'' on the choice of the substance to be called “solvent” in an arbitrary mixture. If there is only one pure liquid substance in a mixture, the choice is clear, but not all solutions are this clear-cut: in an alcohol-water solution, either one could be called the solvent; in an alloy, or [[solid solution]], there is no clear choice and all constituents may be treated alike. In such situations, mass or mole fraction is the preferred compositional specification.
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| == Relation to other compositional properties ==
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| In what follows, the solvent may be given the same treatment as the other constituents of the solution, such that the molality of the solvent of an ''n''-solute solution, say ''b''<sub>0</sub>, is found to be nothing more than the reciprocal of its molar mass, ''M''<sub>0</sub>:
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| :<math>b_0=\frac{{n_0}}{{n_0 M_0}}=M_0^{-1}.</math> | |
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| === Mass fraction===
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| The conversions to and from the [[mass fraction (chemistry)|mass fraction]], <math>w</math>, of the solute in a single-solute solution are
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| :<math>w=(1+(b\,M)^{-1})^{-1},\ b=\frac{w}{(1-w)M} ,</math>
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| where ''b'' is the molality and ''M'' is the [[molar mass]] of the solute.
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| More generally, for an ''n''-solute/one-solvent solution, letting ''b<sub>i</sub>'' and ''w<sub>i</sub>'' be, respectively, the molality and mass fraction of the ''i''-th solute,
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| :<math>w_i=w_0\,b_i M_i,\ b_i=\frac{w_i}{w_0 M_i},</math>
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| where ''M<sub>i</sub>'' is the molar mass of the ''i''-th solute, and ''w<sub>''0</sub> is the mass fraction of the solvent, which is expressible both as a function of the molalities as well as a function of the other mass fractions,
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| :<math>w_0=(1+\textstyle\sum_{j=1}^{n}{b_j M_j})^{-1}=1-\sum_{j=1}^{n}{w_j}.</math>
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| === Mole fraction ===
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| The conversions to and from the [[mole fraction]], ''x'', of the solute in a single-solute solution are
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| :<math>x=(1+(M_0\,b)^{-1})^{-1},\ b=\frac{x}{M_0(1-x)},</math>
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| where ''M''<sub>0</sub> is the molar mass of the solvent.
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| More generally, for an ''n''-solute/one-solvent solution, letting ''x<sub>i</sub>'' be the mole fraction of the ''i''-th solute,
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| :<math>x_i=x_0 M_0\,b_i,\ b_i=\frac{b_0 x_i}{x_0},</math>
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| where ''x''<sub>0</sub> is the mole fraction of the solvent, expressible both as a function of the molalities as well as a function of the other mole fractions:
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| :<math>x_0=(1+M_0\textstyle \sum_{j=1}^{n}{b_j})^{-1}=1-\sum_{j=1}^{n}{x_j}.</math>
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| === Molar concentration (Molarity)===
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| The conversions to and from the [[molar concentration]], ''c'', for one-solute solutions are
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| :<math>c = \frac{\rho\, b}{1+ b M},\ b=\frac{c}{\rho-cM},</math>
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| where ''ρ'' is the [[mass density]] of the solution, ''b'' is the molality, and ''M'' is the molar mass of the solute.
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| For solutions with ''n'' solutes, the conversions are
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| :<math>c_i =c_0 M_0\,b_i,\ b_i=\frac{b_0 c_i}{c_0},</math>
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| where the molar concentration of the solvent ''c''<sub>0</sub> is expressible both as a function of the molalities as well as a function of the molarities:
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| :<math>c_0=\frac{\rho\,b_0}{1+ \sum_{j=1}^{n}{b_j M_j}}=\frac{\rho-\sum_{j=1}^{n}{c_i M_i}}{M_0}.</math>
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| === Mass concentration ===
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| The conversions to and from the [[mass concentration (chemistry)|mass concentration]], ''ρ<sub>solute</sub>'', of an single-solute solution are
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| :<math>\rho_{solute} = \frac{\rho\,b\,M}{1+b M},\ b=\frac{\rho_{solute}}{M (\rho-\rho_{solute})},</math>
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| where ''ρ'' is the mass density of the solution, ''b'' is the molality, and ''M'' is the molar mass of the solute.
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| For the general ''n''-solute solution, the mass concentration of the ''i''-th solute, ''ρ<sub>i</sub>'', is related to its molality, ''b<sub>i</sub>'', as follows:
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| :<math>\rho_i = \rho_0\,b_i M_i,\ b_i=\frac{\rho_i}{\rho_0 M_i},</math>
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| where the mass concentration of the solvent, ''ρ''<sub>0</sub>, is expressible both as a function of the molalities as well as a function of the mass concentrations:
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| :<math>\rho_0=\frac{\rho}{1+\sum_{j=1}^n b_j M_j}=\rho-\sum_{j=1}^{n}{\rho_i}.</math>
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| === Equal ratios ===
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| Alternatively, we may use just the last two equations given for the compositional property of the solvent in each of the preceding sections, together with the relationships given below, to derive the remainder of properties in that set:
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| :<math>\frac{b_i}{b_j}=\frac{x_i}{x_j}=\frac{c_i}{c_j}=\frac{\rho_i\,M_j}{\rho_j\,M_i}=\frac{w_i\,M_j}{w_j\,M_i},</math>
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| where ''i'' and ''j'' are subscripts representing ''all'' the constituents, the ''n'' solutes plus the solvent.
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| === Example of conversion ===
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| An acid mixture consists of 0.76/0.04/0.20 mass fractions of (70% HNO<sub>3</sub>)/(49% HF)/(H<sub>2</sub>O), where the percentages refer to mass fractions of the bottled acids carrying a balance of H<sub>2</sub>O. The first step is determining the mass fractions of the constituents:
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| :<math>\begin{align}
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| w_{HNO_3}&=0.70\times 0.76=0.532\\
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| w_{HF}&=0.49\times 0.04=0.0196\\
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| w_{H_2O}&=1-w_{HNO_3}-w_{HF}=0.448\\
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| \end{align}
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| </math>
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| The approximate molar masses in kg/mol are
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| :<math>M_{HNO_3}=0.063,\ M_{HF}=0.020,\ M_{H_2O}=0.018.</math>
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| First derive the molality of the solvent, in mol/kg,
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| :<math>b_{H_2O}=(M_{H_2O})^{-1}=1/0.018,</math>
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| and use that to derive all the others by use of the equal ratios:
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| :<math>\frac{b_{HNO_3}}{b_{H_2O}}=\frac{w_{HNO_3}M_{H_2O}}{w_{H_2O}M_{HNO_3}}
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| \ \therefore b_{HNO_3}=18.83.</math>
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| Actually, ''b<sub>H<sub>2</sub>O</sub>'' cancels out, because it is not needed. In this case, there is a more direct equation: we use it to derive the molality of HF:
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| :<math>b_{HF}=\frac{w_{HF}}{w_{H_2O}M_{HF}}=2.19.</math>
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| The mole fractions may be derived from this result:
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| :<math>\begin{align}
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| &x_{H_2O}=(1+M_{H_2O}(b_{HNO_3}+b_{HF}))^{-1}=0.726,\\
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| &\frac{x_{HNO_3}}{x_{H_2O}}=\frac{b_{HNO_3}}{b_{H_2O}}\ \therefore x_{HNO_3}=0.246,\\
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| &x_{HF}=1-x_{HNO_3}-x_{H_2O}=0.029.
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| \end{align}</math>
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| === Osmolality ===
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| Osmolality is a variation of molality that only takes into account solutes that contribute to a solution's [[osmotic pressure]]. It is measured in [[osmole]]s of the solute per [[kilogram]] of water. This unit is frequently used in [[medical laboratory]] results in place of [[osmolarity]], because it can be measured simply by depression of the freezing point of a solution, or [[cryoscopy]] (see also: [[osmostat]] and [[colligative properties]]).
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| == References ==
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| {{Reflist}}
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| {{wikt}}
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| {{Chemical solutions}}
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| [[Category:Chemical properties]]
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| [[es:Concentración#Molalidad]]
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