What is a Binary solution?
Binary solutions mean solutions have only two liquids, and both liquids are to be volatile.
Gibbs-Duhem-Margules give a relation between binary solution components A and B equilibrium with vapour at constant temperature and pressure.
nA and nB = number of components A and B
According to Gibbs- Duhem equation,
\[\displaystyle {{n}_{A}}d{{\mu }_{A}}+{{n}_{B}}d{{\mu }_{B}}=0------(1)\]
\[\displaystyle \begin{array}{l}{{\mu }_{A}}=\text{Chemical potential of A}\\{{\mu }_{B}}=\text{Chemical potential of B}\end{array}\]
Equation (1) is divided by nA + nB and,
\[\displaystyle \frac{{{{n}_{A}}}}{{{{n}_{A}}+{{n}_{B}}}}d{{\mu }_{A}}+\frac{{{{n}_{B}}}}{{{{n}_{A}}+{{n}_{B}}}}d{{\mu }_{B}}=0\]
or
\[\displaystyle {{x}_{A}}d{{\mu }_{A}}+{{x}_{B}}d{{\mu }_{B}}=0-------(2)\]
\[\displaystyle \begin{array}{l}{{x}_{A}}=\text{mole fraction of component A}\\{{x}_{B}}=\text{mole fraction of component B}\end{array}\]
The chemical potential of any constituent of a liquid mixture is denoted as thermodynamically,
\[\displaystyle {{\mu }_{l}}={{\mu }_{i}}+RT\ln {{f}_{i}}------(3)\]
\[\displaystyle \begin{array}{l}{{f}_{i}}=\text{Fugacity of given constituent in liquid or vapour phase in equilibrium}\\{{\mu }_{i}}=\text{Constant for constant temperature }\end{array}\]
Above equation (3) differentiating at the constant temperature we get,
\[\displaystyle \text{d}{{\mu }_{i}}=RTd\ln {{f}_{i}}------(4)\]
Inserting equation (4) in equation (2) and we have,
\[\displaystyle {{x}_{A}}RTd\ln {{f}_{A}}+{{x}_{B}}RTd\ln {{f}_{B}}=0------(5)\]
Above equation (5) dividing throughout dxA and we get,
\[\displaystyle \frac{{{{x}_{A}}RTd\ln {{f}_{A}}}}{{d{{x}_{A}}}}+\frac{{{{x}_{B}}RTd\ln {{f}_{B}}}}{{d{{x}_{B}}}}=0------(6)\]
\[\displaystyle Now,{{x}_{A}}+{{x}_{B}}=1,\]
\[\displaystyle d{{x}_{A}}=-d{{x}_{B}}\]
Equation (6) taken the form,
\[\displaystyle \frac{{d\ln {{f}_{A}}}}{{\frac{{d{{x}_{A}}}}{{{{x}_{A}}}}}}-\frac{{d\ln {{f}_{B}}}}{{\frac{{d{{x}_{B}}}}{{{{x}_{B}}}}}}=0\]
or
\[\displaystyle \frac{{d\ln {{f}_{A}}}}{{d\ln {{x}_{A}}}}-\frac{{d\ln {{f}_{B}}}}{{d\ln {{x}_{B}}}}=0\]
or
\[\displaystyle \frac{{d\ln {{f}_{A}}}}{{d\ln {{x}_{A}}}}=\frac{{d\ln {{f}_{B}}}}{{d\ln {{x}_{B}}}}\]
The above equation (7) is known as the Gibbs-Duhem-Margules equation.
