TheDuhem–Margules equation, named forPierre Duhem andMax Margules, is athermodynamic statement of the relationship between the twocomponents of a singleliquid where thevapour mixture is regarded as anideal gas:

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

whereP_A andP_B are the partialvapour pressures of the two constituents andx_A andx_B are themole fractions of the liquid. The equation gives the relation between changes in mole fraction and partial pressure of the components.

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

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

1

Wheren_A andn_B are number of moles of the component A and B while μ_A and μ_B are their chemical potentials.

Dividing equation (1) byn_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

${\displaystyle x_A\mathrm{d}{\mu }_A+x_B\mathrm{d}{\mu }_B=0}$

2

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

${\displaystyle \mathrm{d}{\mu }_A={\left(\frac{\mathrm{d}{\mu }_A}{\mathrm{d}{x}_A}\right)}_{T,P}\mathrm{d}{x}_A}$

3

${\displaystyle \mathrm{d}{\mu }_B={\left(\frac{\mathrm{d}{\mu }_B}{\mathrm{d}{x}_B}\right)}_{T,P}\mathrm{d}{x}_B}$

4

Putting these values in equation (2), then

${\displaystyle x_A{\left(\frac{\mathrm{d}{\mu }_A}{\mathrm{d}{x}_A}\right)}_{T,P}\mathrm{d}{x}_A+x_B{\left(\frac{\mathrm{d}{\mu }_B}{\mathrm{d}{x}_B}\right)}_{T,P}\mathrm{d}{x}_B=0}$

5

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:

${\displaystyle x_A{\left(\frac{\mathrm{d}{\mu }_A}{\mathrm{d}{x}_A}\right)}_{T,P}=x_B{\left(\frac{\mathrm{d}{\mu }_B}{\mathrm{d}{x}_B}\right)}_{T,P}}$

6

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

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

whereP 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

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

7

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

8

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.

• Atkins, Peter and Julio de Paula. 2002.Physical Chemistry, 7th ed. New York: W. H. Freeman and Co.
• Carter, Ashley H. 2001.Classical and Statistical Thermodynamics. Upper Saddle River: Prentice Hall.

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