In chemistry, molality is a measure of the amount of solute in a solution relative to a given mass of solvent. This contrasts with the definition of molarity which is based on a given volume of solution. A commonly used unit for molality is moles of solute per kilogram solvent (mol/kg). A solution of concentration 1 mol/kg is often written as 1 molal (or 1 m).
Definition The molality (b), of a solution is defined as the amount of substance (in moles) of solute, nsolute, divided by the mass (in kg) of the solvent, msolvent:
b = n s o l u t e m s o l v e n t {\displaystyle b={\frac {n_{\mathrm {solute} }}{m_{\mathrm {solvent} }}}} . In the case of solutions with more than one solvent, molality can be defined for the mixed solvent considered as a pure pseudo-solvent. Instead of mole solute per kilogram solvent as in the binary case, units are defined as mole solute per kilogram mixed solvent.
Origin The term molality is formed in analogy to molarity which is the molar concentration of a solution. The earliest known use of the intensive property molality and of its adjectival unit, the now-deprecated molal, appears to have been published by G. N. Lewis and M. Randall in the 1923 publication of Thermodynamics and the Free Energies of Chemical Substances. Though the two terms are subject to being confused with one another, the molality and molarity of a dilute aqueous solution are nearly the same, as one kilogram of water (solvent) occupies the volume of 1 liter at room temperature and a small amount of solute has little effect on the volume.
Unit The SI unit for molality is moles per kilogram of solvent. A solution with a molality of 3 mol/kg is often described as "3 molal", "3 m" 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.
Usage considerations
Advantages The primary advantage of using molality as a measure of concentration is that molality only depends on the masses of solute and solvent, which are unaffected by variations in temperature and pressure. In contrast, solutions prepared volumetrically (e.g. molar concentration or mass concentration) are likely to change as temperature and pressure change. In many applications, this is a significant advantage because the mass, or the amount, of a substance is often more important than its volume (e.g. in a limiting reagent problem). Another advantage of molality is the fact that the molality of one solute in a solution is independent of the presence or absence of other solutes.
Problem areas 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.
Relation to other compositional quantities 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 b0, is found to be nothing more than the reciprocal of its molar mass, M0 (expressed in the unit kg/mol):
b 0 = n 0 n 0 M 0 = 1 M 0 {\displaystyle b_{0}={\frac {n_{0}}{n_{0}M_{0}}}={\frac {1}{M_{0}}}} . For the solutes the expression of molalities is similar:
b i = n i n 0 M 0 = x i x 0 M 0 = c i c 0 M 0 {\displaystyle b_{i}={\frac {n_{i}}{n_{0}M_{0}}}={\frac {x_{i}}{x_{0}M_{0}}}={\frac {c_{i}}{c_{0}M_{0}}}} . The expressions linking molalities to mass fractions and mass concentrations contain the molar masses of the solutes Mi:
b i = n i n 0 M 0 = w i w 0 M i = ρ i ρ 0 M i {\displaystyle b_{i}={\frac {n_{i}}{n_{0}M_{0}}}={\frac {w_{i}}{w_{0}M_{i}}}={\frac {\rho _{i}}{\rho _{0}M_{i}}}} . Similarly the equalities below are obtained from the definitions of the molalities and of the other compositional quantities. The mole fraction of solvent can be obtained from the definition by dividing the numerator and denominator to the amount of solvent n0:
x 0 = n 0 n 0 + n 1 + n 2 + ∑ j = 3 n n j = 1 1 + n 1 n 0 + n 2 n 0 + ∑ j = 3 n n j n 0 {\displaystyle x_{0}={\frac {n_{0}}{n_{0}+n_{1}+n_{2}+\displaystyle \sum _{j=3}^{n}{n_{j}}}}={\frac {1}{1+{\frac {n_{1}}{n_{0}}}+{\frac {n_{2}}{n_{0}}}+\displaystyle \sum _{j=3}^{n}{\frac {n_{j}}{n_{0}}}}}} . Then the sum of ratios of the other mole amounts to the amount of solvent is substituted with expressions from below containing molalities:
n i n 0 = b i M 0 {\displaystyle {\frac {n_{i}}{n_{0}}}=b_{i}M_{0}}
∑ i n n i n 0 = M 0 ∑ i n b i {\displaystyle \sum _{i}^{n}{\frac {n_{i}}{n_{0}}}=M_{0}\sum _{i}^{n}b_{i}}
giving the result
x 0 = 1 1 + M 0 b 1 + M 0 b 2 + M 0 ∑ i = 3 n b i = 1 1 + M 0 ∑ i = 1 n b i {\displaystyle x_{0}={\frac {1}{1+\displaystyle M_{0}b_{1}+M_{0}b_{2}+M_{0}\displaystyle \sum _{i=3}^{n}b_{i}}}={\frac {1}{1+M_{0}\displaystyle \sum _{i=1}^{n}b_{i}}}} .
Mass fraction The conversions to and from the mass fraction, w1, of the solute in a single-solute solution are
w 1 = 1 1 + 1 b 1 M 1 , b 1 = w 1 ( 1 − w 1 ) M 1 , {\displaystyle w_{1}={\frac {1}{1+{\dfrac {1}{b_{1}M_{1}}}}},\quad b_{1}={\frac {w_{1}}{(1-w_{1})M_{1}}},}
where b1 is the molality and M1 is the molar mass of the solute. More generally, for an n-solute/one-solvent solution, letting bi and wi be, respectively, the molality and mass fraction of the i-th solute,
w i = w 0 b i M i , b i = w i w 0 M i {\displaystyle w_{i}=w_{0}b_{i}M_{i},\quad b_{i}={\frac {w_{i}}{w_{0}M_{i}}}} , where Mi is the molar mass of the ith solute, and w0 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,
w 0 = 1 1 + ∑ j = 1 n b j M j = 1 − ∑ j = 1 n w j {\displaystyle w_{0}={\frac {1}{1+\displaystyle \sum _{j=1}^{n}{b_{j}M_{j}}}}=1-\sum _{j=1}^{n}{w_{j}}} . Substitution gives:
w i = b i M i 1 + ∑ j = 1 n b j M j , b i = w i ( 1 − ∑ j = 1 n w j ) M i {\displaystyle w_{i}={\frac {b_{i}M_{i}}{1+\displaystyle \sum _{j=1}^{n}b_{j}M_{j}}},\quad b_{i}={\frac {w_{i}}{\left(1-\displaystyle \sum _{j=1}^{n}w_{j}\right)M_{i}}}} .
Mole fraction The conversions to and from the mole fraction, x1 mole fraction of the solute in a single-solute solution are
x 1 = 1 1 + 1 M 0 b 1 , b 1 = x 1 M 0 ( 1 − x 1 ) {\displaystyle x_{1}={\frac {1}{1+{\dfrac {1}{M_{0}b_{1}}}}},\quad b_{1}={\frac {x_{1}}{M_{0}(1-x_{1})}}} , where M0 is the molar mass of the solvent. More generally, for an n-solute/one-solvent solution, letting xi be the mole fraction of the ith solute,
x i = x 0 M 0 b i , b i = b 0 x i x 0 = x i M 0 x 0 {\displaystyle x_{i}=x_{0}M_{0}b_{i},\quad b_{i}={\frac {b_{0}x_{i}}{x_{0}}}={\frac {x_{i}}{M_{0}x_{0}}}} , where x0 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:
x 0 = 1 1 + M 0 ∑ i = 1 n b i = 1 − ∑ i = 1 n x i {\displaystyle x_{0}={\frac {1}{1+M_{0}\displaystyle \sum _{i=1}^{n}{b_{i}}}}=1-\sum _{i=1}^{n}{x_{i}}} . Substitution gives:
x i = M 0 b i 1 + M 0 ∑ j = 1 n b j , b i = x i M 0 ( 1 − ∑ j = 1 n x j ) {\displaystyle x_{i}={\frac {M_{0}b_{i}}{1+M_{0}\displaystyle \sum _{j=1}^{n}b_{j}}},\quad b_{i}={\frac {x_{i}}{M_{0}\left(1-\displaystyle \sum _{j=1}^{n}x_{j}\right)}}} .
Molar concentration (molarity) The conversions to and from the molar concentration, c1, for one-solute solutions are
c 1 = ρ b 1 1 + b 1 M 1 , b 1 = c 1 ρ − c 1 M 1 {\displaystyle c_{1}={\frac {\rho b_{1}}{1+b_{1}M_{1}}},\quad b_{1}={\frac {c_{1}}{\rho -c_{1}M_{1}}}} , where ρ is the mass density of the solution, b1 is the molality, and M1 is the molar mass (in kg/mol) of the solute. For solutions with n solutes, the conversions are
c i = c 0 M 0 b i , b i = b 0 c i c 0 {\displaystyle c_{i}=c_{0}M_{0}b_{i},\quad b_{i}={\frac {b_{0}c_{i}}{c_{0}}}} , where the molar concentration of the solvent c0 is expressible both as a function of the molalities as well as a function of the other molarities:
c 0 = ρ b 0 1 + ∑ j = 1 n b j M j = ρ − ∑ j = 1 n c i M i M 0 {\displaystyle c_{0}={\frac {\rho b_{0}}{1+\displaystyle \sum _{j=1}^{n}{b_{j}M_{j}}}}={\frac {\rho -\displaystyle \sum _{j=1}^{n}{c_{i}M_{i}}}{M_{0}}}} . Substitution gives:
c i = ρ b i 1 + ∑ j = 1 n b j M j , b i = c i ρ − ∑ j = 1 n c j M j {\displaystyle c_{i}={\frac {\rho b_{i}}{1+\displaystyle \sum _{j=1}^{n}{b_{j}M_{j}}}},\quad b_{i}={\frac {c_{i}}{\rho -\displaystyle \sum _{j=1}^{n}{c_{j}M_{j}}}}} ,
Mass concentration The conversions to and from the mass concentration, ρsolute, of a single-solute solution are
ρ s o l u t e = ρ b M 1 + b M , b = ρ s o l u t e M ( ρ − ρ s o l u t e ) {\displaystyle \rho _{\mathrm {solute} }={\frac {\rho bM}{1+bM}},\quad b={\frac {\rho _{\mathrm {solute} }}{M\left(\rho -\rho _{\mathrm {solute} }\right)}}} , or
ρ 1 = ρ b 1 M 1 1 + b 1 M 1 , b 1 = ρ 1 M 1 ( ρ − ρ 1 ) {\displaystyle \rho _{1}={\frac {\rho b_{1}M_{1}}{1+b_{1}M_{1}}},\quad b_{1}={\frac {\rho _{1}}{M_{1}\left(\rho -\rho _{1}\right)}}} , where ρ is the mass density of the solution, b1 is the molality, and M1 is the molar mass of the solute. For the general n-solute solution, the mass concentration of the ith solute, ρi, is related to its molality, bi, as follows:
ρ i = ρ 0 b i M i , b i = ρ i ρ 0 M i {\displaystyle \rho _{i}=\rho _{0}b_{i}M_{i},\quad b_{i}={\frac {\rho _{i}}{\rho _{0}M_{i}}}} , where the mass concentration of the solvent, ρ0, is expressible both as a function of the molalities as well as a function of the other mass concentrations:
ρ 0 = ρ 1 + ∑ j = 1 n b j M j = ρ − ∑ i = 1 n ρ i {\displaystyle \rho _{0}={\frac {\rho }{1+\displaystyle \sum _{j=1}^{n}b_{j}M_{j}}}=\rho -\sum _{i=1}^{n}{\rho _{i}}} . Substitution gives:
ρ i = ρ b i M i 1 + ∑ j = 1 n b j M j b i = ρ i M i ( ρ − ∑ j = 1 n ρ j ) {\displaystyle \rho _{i}={\frac {\rho b_{i}M_{i}}{1+\displaystyle \sum _{j=1}^{n}b_{j}M_{j}}}\quad b_{i}={\frac {\rho _{i}}{M_{i}\left(\rho -\displaystyle \sum _{j=1}^{n}\rho _{j}\right)}}} .
Equal ratios Alternatively, one 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:
b i b j = x i x j = c i