Duhem–Margules equation

${\displaystyle {\left(\frac{\mathrm{d}\ln P_A}{\mathrm{d}\ln x_A}\right)}_{T,P}={\left(\frac{\mathrm{d}\ln P_B}{\mathrm{d}\ln x_B}\right)}_{T,P}}$

where P_A and P_B are the partial vapour pressures of the two constituents and x_A and x_B are the mole fractions of the liquid. The equation gives the relation between changes in mole fraction and partial pressure of the components.

Derivation

Let us consider a binary liquid mixture of two component in equilibrium with their vapor at constant temperature and pressure. Then from the Gibbs–Duhem equation, we have

Where n_A and n_B are number of moles of the component A and B while μ_A and μ_B are their chemical potentials.

Dividing equation (1) by n_A + n_B, then

${\displaystyle \frac{n_A}{n_A+n_B}\mathrm{d}{\mu }_A+\frac{n_B}{n_A+n_B}\mathrm{d}{\mu }_B=0}$

Or

Now the chemical potential of any component in mixture is dependent upon temperature, pressure and the composition of the mixture. Hence if temperature and pressure are taken to be constant, the chemical potentials must satisfy

Putting these values in equation (2), then

Because the sum of mole fractions of all components in the mixture is unity, i.e.,

${\displaystyle x_1+x_2=1}$

we have

${\displaystyle \mathrm{d}{x}_1+\mathrm{d}{x}_2=0}$

so equation (5) can be re-written:

Now the chemical potential of any component in mixture is such that

${\displaystyle \mu ={\mu }_0+RT\ln P}$

where P is the partial pressure of that component. By differentiating this equation with respect to the mole fraction of a component:

${\displaystyle \frac{\mathrm{d}\mu }{\mathrm{d}x}=RT\frac{\mathrm{d}\ln P}{\mathrm{d}x}}$

we have for components A and B

Substituting these value in equation (6), then

${\displaystyle x_A\frac{\mathrm{d}\ln P_A}{\mathrm{d}{x}_A}=x_B\frac{\mathrm{d}\ln P_B}{\mathrm{d}{x}_B}}$

or

${\displaystyle {\left(\frac{\mathrm{d}\ln P_A}{\mathrm{d}\ln x_A}\right)}_{T,P}={\left(\frac{\mathrm{d}\ln P_B}{\mathrm{d}\ln x_B}\right)}_{T,P}}$

This final equation is the Duhem–Margules equation.