Molality

The conversions to and from the mass fraction, w, of the solute in a single-solute solution are

${\displaystyle w={\frac {1}{1+{\dfrac {1}{bM}}}},\quad b={\frac {w}{(1-w)M}},}$

where b is the molality and M is the molar mass of the solute.

More generally, for an n-solute/one-solvent solution, letting b_i and w_i be, respectively, the molality and mass fraction of the i-th solute,

${\displaystyle w_{i}=w_{0}b_{i}M_{i},\quad b_{i}={\frac {w_{i}}{w_{0}M_{i}}},}$

where M_i 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,

${\displaystyle w_{0}={\frac {1}{1+\displaystyle \sum _{j=1}^{n}{b_{j}M_{j}}}}=1-\sum _{j=1}^{n}{w_{j}}.}$

The conversions to and from the mole fraction, x, of the solute in a single-solute solution are

${\displaystyle x={\frac {1}{1+{\dfrac {1}{M_{0}b}}}},\quad b={\frac {x}{M_{0}(1-x)}},}$

where M₀ is the molar mass of the solvent.

More generally, for an n-solute/one-solvent solution, letting x_i be the mole fraction of the ith solute,

${\displaystyle x_{i}=x_{0}M_{0}b_{i},\quad b_{i}={\frac {b_{0}x_{i}}{x_{0}}},}$

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:

${\displaystyle x_{0}={\frac {1}{1+M_{0}\displaystyle \sum _{j=1}^{n}{b_{j}}}}=1-\sum _{j=1}^{n}{x_{j}}.}$

The conversions to and from the molar concentration, c, for one-solute solutions are

${\displaystyle c={\frac {\rho b}{1+bM}},\quad b={\frac {c}{\rho -cM}},}$

where ρ is the mass density of the solution, b is the molality, and M is the molar mass (in kg/mol) of the solute.

For solutions with n solutes, the conversions are

${\displaystyle c_{i}=c_{0}M_{0}b_{i},\quad b_{i}={\frac {b_{0}c_{i}}{c_{0}}},}$

where the molar concentration of the solvent c₀ is expressible both as a function of the molalities as well as a function of the molarities:

${\displaystyle c_{0}={\frac {\rho b_{0}}{1+\displaystyle \sum _{j=1}^{n}{b_{j}M_{j}}}}={\frac {\rho -\displaystyle \sum _{j=1}^{n}{c_{i}M_{i}}}{M_{0}}}.}$

The conversions to and from the mass concentration, ρ_solute, of a single-solute solution are

${\displaystyle \rho _{\mathrm {solute} }={\frac {\rho bM}{1+bM}},\quad b={\frac {\rho _{\mathrm {solute} }}{M\left(\rho -\rho _{\mathrm {solute} }\right)}},}$

where ρ is the mass density of the solution, b is the molality, and M is the molar mass of the solute.

For the general n-solute solution, the mass concentration of the ith solute, ρ_i, is related to its molality, b_i, as follows:

${\displaystyle \rho _{i}=\rho _{0}b_{i}M_{i},\quad b_{i}={\frac {\rho _{i}}{\rho _{0}M_{i}}},}$

where the mass concentration of the solvent, ρ₀, is expressible both as a function of the molalities as well as a function of the mass concentrations:

${\displaystyle \rho _{0}={\frac {\rho }{1+\displaystyle \sum _{j=1}^{n}b_{j}M_{j}}}=\rho -\sum _{j=1}^{n}{\rho _{i}}.}$

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:

${\displaystyle {\frac {b_{i}}{b_{j}}}={\frac {x_{i}}{x_{j}}}={\frac {c_{i}}{c_{j}}}={\frac {\rho _{i}M_{j}}{\rho _{j}M_{i}}}={\frac {w_{i}M_{j}}{w_{j}M_{i}}},}$

where i and j are subscripts representing all the constituents, the n solutes plus the solvent.

An acid mixture consists of 0.76, 0.04, and 0.20 mass fractions of 70% HNO₃, 49% HF, and H₂O, where the percentages refer to mass fractions of the bottled acids carrying a balance of H₂O. The first step is determining the mass fractions of the constituents:

${\displaystyle {\begin{aligned}w_{\mathrm {HNO_{3}} }&=0.70\times 0.76=0.532\\w_{\mathrm {HF} }&=0.49\times 0.04=0.0196\\w_{\mathrm {H_{2}O} }&=1-w_{\mathrm {HNO_{3}} }-w_{\mathrm {HF} }=0.448\\\end{aligned}}}$

The approximate molar masses in kg/mol are

${\displaystyle M_{\mathrm {HNO_{3}} }=0.063\ \mathrm {kg/mol} ,\quad M_{\mathrm {HF} }=0.020\ \mathrm {kg/mol} ,\ M_{\mathrm {H_{2}O} }=0.018\ \mathrm {kg/mol} .}$

First derive the molality of the solvent, in mol/kg,

${\displaystyle b_{\mathrm {H_{2}O} }={\frac {1}{M_{\mathrm {H_{2}O} }}}={\frac {1}{0.018}}\ \mathrm {mol/kg} ,}$

and use that to derive all the others by use of the equal ratios:

${\displaystyle {\frac {b_{\mathrm {HNO_{3}} }}{b_{\mathrm {H_{2}O} }}}={\frac {w_{\mathrm {HNO_{3}} }M_{\mathrm {H_{2}O} }}{w_{\mathrm {H_{2}O} }M_{\mathrm {HNO_{3}} }}}\quad \therefore b_{\mathrm {HNO_{3}} }=18.83\ \mathrm {mol/kg} .}$

Actually, b_H2O 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:

${\displaystyle b_{\mathrm {HF} }={\frac {w_{\mathrm {HF} }}{w_{\mathrm {H_{2}O} }M_{\mathrm {HF} }}}=2.19\ \mathrm {mol/kg} .}$

The mole fractions may be derived from this result:

${\displaystyle x_{\mathrm {H_{2}O} }={\frac {1}{1+M_{\mathrm {H_{2}O} }\left(b_{\mathrm {HNO_{3}} }+b_{\mathrm {HF} }\right)}}=0.726,}$

${\displaystyle {\frac {x_{\mathrm {HNO_{3}} }}{x_{\mathrm {H_{2}O} }}}={\frac {b_{\mathrm {HNO_{3}} }}{b_{\mathrm {H_{2}O} }}}\quad \therefore x_{\mathrm {HNO_{3}} }=0.246,}$

${\displaystyle x_{\mathrm {HF} }=1-x_{\mathrm {HNO_{3}} }-x_{\mathrm {H_{2}O} }=0.029.}$

Osmolality is a variation of molality that takes into account only solutes that contribute to a solution's osmotic pressure. It is measured in osmoles 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).

For concentrated ionic solutions the activity coefficient of the electrolyte is split into electric and statistical components.

The statistical part includes molality b, hydration index number h, the number of ions from the dissociation and the ratio r_a between the apparent molar volume of the electrolyte and the molar volume of water.

Concentrated solution statistical part of the activity coefficient is:

${\displaystyle \ln \gamma _{s}={\frac {h-\nu }{\nu }}\ln \left(1+{\frac {br_{a}}{55.5}}\right)-{\frac {h}{\nu }}\ln \left(1-{\frac {br_{a}}{55.5}}\right)+{\frac {br_{a}\left(r_{a}+h-\nu \right)}{55.5\left(1+{\frac {br_{a}}{55.5}}\right)}}}$,[6][7][8]