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:
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1bd0be668886f0e7270ab9b5fbc65aaea175ff6)
where PA and PB are the partial
vapour pressures of the two constituents and xA and xB 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
![{\displaystyle n_{A}\mathrm {d} \mu _{A}+n_{B}\mathrm {d} \mu _{B}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e59a9225319ad4949bb6265235e347213539234d) | | (1) |
Where nA and nB are number of moles of the component A and B while μA and μB are their chemical potentials.
Dividing equation (
1) by nA + nB, 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8307745f69af4679db01e797443ba94c4737e07)
Or
![{\displaystyle x_{A}\mathrm {d} \mu _{A}+x_{B}\mathrm {d} \mu _{B}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dda9558b7efaed6675102295ba01243e3adb804) | | (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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/674414510fffec9fd948ca5b52903fccc7b2515a) | | (3) |
![{\displaystyle \mathrm {d} \mu _{B}=\left({\frac {\mathrm {d} \mu _{B}}{\mathrm {d} x_{B}}}\right)_{T,P}\mathrm {d} x_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5a8dd90edf4707e13be694bbdecca372ffc777) | | (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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eacd5036c070ffc69d0559976396b6efb0500097) | | (5) |
Because the sum of mole fractions of all components in the mixture is unity, i.e.,
![{\displaystyle x_{1}+x_{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d55dfda3a46f225300e83a6bc95b3112da9180c)
we have
![{\displaystyle \mathrm {d} x_{1}+\mathrm {d} x_{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc55e7df9800a9e38032f4d53a218b683391926e)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14bd751b430827ffbb24a52d1d39ab70fc5bac93) | | (6) |
Now the chemical potential of any component in mixture is such that
![{\displaystyle \mu =\mu _{0}+RT\ln P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c87b6fe9b752ba584f45d9c0db2f123cd1ffcf74)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d569e3a886b1b746f2d1c5d31b88113d34f33d9c)
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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b62d35ce5083213b3b9066cd91176dc9f5472d63) | | (7) |
![{\displaystyle {\frac {\mathrm {d} \mu _{B}}{\mathrm {d} x_{B}}}=RT{\frac {\mathrm {d} \ln P_{B}}{\mathrm {d} x_{B}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d1eef2679c59b61a206580f2664372792712767) | | (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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/608a01049fd4539df2de00ec7c28d0abcc774b0a)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1bd0be668886f0e7270ab9b5fbc65aaea175ff6)
This final equation is the Duhem–Margules equation.
Sources
-
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.