# Duhem–Margules Equation

## Duhem–Margules Equation

Duhem–Margules equation 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.Let consider a binary liquid mixture of two component in equilibrium with their vapor at constant temperature and pressure. Then from Gibbs–Duhem equation is

n

_{A}dμ

_{A}+ n

_{B}dμ

_{B}= 0 -----(equation-1)

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 equation-1 by n

_{A}+ n

_{B}we get-

n

_{A}/(n

_{A}+ n

_{A}) dμ

_{A}+ n

_{B}/(n

_{A}+ n

_{B}) dμ

_{B}= 0

or, X

_{A}dμ

_{A}+ X

_{A}dμ

_{B}= 0 -----(equation-2)

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-

dμ

_{A}= (dμ

_{A}/dX

_{A})

_{T,P}. dX

_{A}-----(equation-3)

and

dμ

_{B}= (dμ

_{B}/dX

_{B})

_{T,P}. dX

_{B}-----(equation-4)

Putting these values in (equation-2), then-

X

_{A}(dμ

_{A}/dX

_{A})

_{T,P}. dX

_{A}+ X

_{B}(dμ

_{B}/dX

_{B})

_{T,P}. dX

_{B}= 0 -----(equation-5)

we know that sum of mole fraction of all component in the mixture is unity i.e., X

_{A}+ X

_{B}= 1

Hence, dX

_{A}+ dX

_{B}=0

So, equation -5 becomes

X

_{A}(dμ

_{A}/dX

_{A})

_{T,P}= X

_{B}(dμ

_{B}/dX

_{B})

_{T,P}-----(equation-6)

We know that the chemical potential of any component in mixture is

μ = μ

_{o}+ RTlnP

where P is partial pressure of component. By differentiating this equation with respect to the mole fraction of a component we get-

dμ/dX = RT(dlnP/dX)

so for two component A and B-

dμ

_{A}/dX

_{A}= RT(dlnP

_{A}/dX

_{A})

dμ

_{B}/dX

_{B}= RT(dlnP

_{B}/dX

_{B})

Now substituting these values in equation-6 we get-

X

_{A}(dlnP

_{A}/dX

_{A}) = X

_{B}(dlnP

_{B}/dX

_{B}) -----(equation-7)

or, (dlnP

_{A}/dlnX

_{A})

_{T,P}= (dlnP

_{B}/dlnX

_{B})

_{T,P}-----(equation-8)

Equation-7 and 8 are two forms of Duhem–Margules equation. and these equations are used in the derivation of Konowaloff's rule which is useful in the binary liquid mixtures by distillation.

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