# Excess molar quantity

Excess molar quantities are properties of mixtures which characterize the nonideal behaviour of real mixtures. They are the difference between the partial molar property of a component in a real mixture and that of the component in an ideal mixture. The most frequently used excess molar quantities are the excess molar volume, excess molar enthalpies and heat capacities, excess chemical potential. For volumes, internal energies and enthalpies the excess quantities are identical to the mixing quantities. They can be expressed as functions of derivatives of the activity coefficients.

## Definition

By definition, excess properties of a mixture are related to those of the pure substances in an ideal mixture by:

${\displaystyle z^{E}=z-\sum _{i}x_{i}z_{i}^{\text{id}}.}$

Here ${\displaystyle *}$[clarification needed] denotes the pure substance, ${\displaystyle E}$ the excess molar property, and ${\displaystyle z}$ corresponds to the specific property under consideration. From the definition of partial molar properties,

${\displaystyle z=\sum _{i}x_{i}{\overline {Z}}_{i},}$

substitution yields:

${\displaystyle z^{E}=\sum _{i}x_{i}\left({\overline {Z}}_{i}-z_{i}^{\text{id}}\right).}$

For volumes, internal energies and enthalpies the excess quantities are identical to the mixing quantities. They can be expressed as functions of derivatives of the activity coefficients.

## Examples and properties

{\displaystyle {\begin{aligned}{\overline {V^{E}}}_{i}&={\overline {V}}_{i}-{\overline {V^{\text{id}}}}_{i}\\{\overline {H^{E}}}_{i}&={\overline {H}}_{i}-{\overline {H^{\text{id}}}}_{i}\\{\overline {S^{E}}}_{i}&={\overline {S}}_{i}-{\overline {S^{\text{id}}}}_{i}\\{\overline {G^{E}}}_{i}&={\overline {G}}_{i}-{\overline {G^{\text{id}}}}_{i}\end{aligned}}}

The volume of a mixture from the sum of the excess volumes of the components of a mixture is given by the formula:

${\displaystyle {V}=\sum _{i}V_{i}+\sum _{i}V_{i}^{E}}$

### Relation to activity coefficients

The excess molar volume of the component i is connected to its activity coefficient.

${\displaystyle {\overline {V^{E}}}_{i}=RT{\frac {\partial (ln(\gamma _{i}))}{\partial P}}}$

This expression can be further processed by taking the activity coefficient derivative out of the logarithm by logarithmic derivative.

${\displaystyle {\overline {V^{E}}}_{i}=RT{\frac {\frac {\partial (\gamma _{i})}{\partial P}}{\gamma _{i}}}}$

This formula can be substituted in the definition of the excess volume.

## Derivatives to state parameters

### Thermal expansivities

By taking the derivative in respect to temperature the thermal expansivities of the components in a mixture can be related to the expansivity of the mixture:

${\displaystyle {\frac {\partial V}{\partial T}}=\sum _{i}{\frac {\partial V_{i}}{\partial T}}+\sum _{i}{\frac {\partial V_{i}^{E}}{\partial T}}}$

Equivalently:

${\displaystyle \alpha _{V}V=\sum _{i}\alpha _{V,i}V_{i}+\sum _{i}{\frac {\partial V_{i}^{E}}{\partial T}}}$

Substituting the temperature derivative of the excess molar volume

${\displaystyle {\frac {\partial {\overline {V^{E}}}_{i}}{\partial T}}=R{\frac {\partial (ln(\gamma _{i}))}{\partial P}}+RT{\partial ^{2} \over \partial T\partial P}ln(\gamma _{i})}$

one can relate activity coefficients to thermal expansivity.