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:

\left({\frac {d\,\ln \,P_{A}}{d\,\ln \,x_{A}}}\right)_{T,P}=\left({\frac {d\,\ln \,P_{B}}{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.

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

n_Adμ_A + n_Bdμ_B = 0 ...(I)

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

Dividing equ. (i) by n_A + n_B , ththen

(n_A / n_A + n_B ) dμ_A + (n_B / n_A + n_B) dμ_B= 0

Or

x_Adμ_A + x_Bdμ_B = 0. ...(ii)

Now the chemical potential of any componentv in mixture is depend upon temperature, pressure and composition of mixture. Hence if temperature and pressure taking constant then chemical potential

dμ_A = (dμ_A / dx_A )_{T, P} dx_A ...(iii)

dμ_B = (dμ_B / dx_B )_{T, P} dx_B. ...(iv)

Putting these values in equ. (ii), then

x_A(dμ_Α / dx_A)_{T, P} dx_A + x_B(dμ_B / dx_B)_{T, P} dx_B = 0 ...(v)

Because the sum of mole fraction of all component in the mixture is unity i.e.,

x₁ + x₂ = 1

Hence

dx₁ + dx₂ = 0 or dx₁ = -dx₂

puting these value in equ. (v), then

x_A(dμ_Α / dx_A)_{T, P} = x_B(dμ_B / dx_B)_{T, P} …(vi)

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

μ = μ_o + RT In P , where P is partial pressure of component. Now differentiating this equ.

dμ / dx = RT (d In P / dx)

Above equ.can be written for component A and B is

dμ_A / dx_A = RT (d In P_A / dx_A) …(vii)

dμ_B / dx_B = RT (d In P_B / dx_B) …(viii)

Substituting these value in equ.(vi), then

x_A (d In P_A / dx_A) = x_B (d In P_B / dx_B)

or

(d In P_A / dx_A) = (d In P_B / dx_B)

this is the final equation of 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.

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