Duhem–Margules equation


The Duhem–Margules equation, named for Pierre Duhem and Max Margules, is a thermodynamic statement of the relationship between the two components of a single liquid where the vapour mixture is regarded as an ideal gas:
where PA and PB are the partial vapour pressures of the two constituents and xA and xB are the mole fractions of the liquid.
Derivation
Duhem - Margulus equation give the relation between change of mole fraction with partial pressure of a component in a liquid mixture.
Let consider a binary liquid mixture of two component in equilibrium with their vapour at constant temperature and pressure. Then from Gibbs - Duhem equation is
Where nA and nB are number of moles of the component A and B while μA and μB is their chemical potential.
Dividing equation by nA + nB, then
Or
Now the chemical potential of any component in mixture is depend upon temperature, pressure and composition of mixture. Hence if temperature and pressure taking constant then chemical potential
Putting these values in equation, then
Because the sum of mole fraction of all component in the mixture is unity i.e.,
Hence
so equation can be re-written:
Now the chemical potential of any component in mixture is such that
where P is partial pressure of component. By differentiating this equation with respect to the mole fraction of a component:
So we have for components A and B
Substituting these value in equation, then
or
this is the final equation of Duhem- Margules equation.