## Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

### 11. Mixing Liquids: Molecular Solutions

### 11.7. The Activity Coefficient for Other Composition Measures*

So far, we have dealt with activity coefficients using the mole fraction *x* as a measure for composition. In practice, people use also molality *m* (*m* moles of solute per 1 kg of solvent) and molarity *c* (*c* moles per liter solution). Whichever measure is used, the difference *μ* − *μ*^{∞}, where *μ*^{∞} is the infinite dilution reference value, should be the same. Denoting with *a _{x}* and

*a*the activities using the mole fraction and molality scale we must have

_{m}(11.111)

So we obtain

(11.112)

with *γ* the activity coefficient. For the molality scale we have (see __Chapter 2__)

(11.113)

where *M*_{1} is the molar mass of the solvent. For infinite dilution

Moreover, for infinite dilution,

In total, we thus obtain

(11.114)

A similar calculation for the molarity scale yields *γ _{c}*/

*γ*=

_{x}*x*

_{2}

*ρ*/

*c*with

_{2}M_{1}*ρ*the mass density and

*c*

_{2}the molarity of the solute.

Problem 11.17

Calculate the relationship between the activity coefficients on the mole fraction scale and the molarity scale.