By Sunil Bhardwaj

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Consider an open system composed of i constituents. Let $${ n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i }$$ be the number of moles of constituents 1, 2, 3, ...., i respectively. Let G be the free energy of the system. Its value is determined by the state of the system (i.e. Temperature and Pressure) and the amount of various constituents in the system. It means free energy G is a function of temperature, pressure and amount of various constituents. $$G = f\left( T, P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } \right) \qquad .... (1)$$ The change in free energy dG due to small change in temperature, pressure and amount of constituents is given by, $$dG = { \left( \frac { \partial G }{ \partial T } \right) }_{ P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dp + { \left( \frac { \partial G }{ \partial P } \right) }_{ T, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dt + { \left( \frac { \partial G }{ \partial { n }_{ 1 } } \right) }_{ T, P, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }d{ n }_{ 1 } + { \left( \frac { \partial G }{ \partial { n }_{ 2 } } \right) }_{ T, P, { n }_{ 1 }, { n }_{ 3 }, ...., { n }_{ i } }d{ n }_{ 2 } + { \left( \frac { \partial G }{ \partial { n }_{ 3 } } \right) }_{ T, P, { n }_{ 1 }, { n }_{ 2 }, ...., { n }_{ i } }d{ n }_{ 3 } + .... + { \left( \frac { \partial G }{ \partial { n }_{ i } } \right) }_{ T, P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., }d{ n }_{ i }$$ $$\therefore \boxed { dG = { \left( \frac { \partial G }{ \partial T } \right) }_{ P, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dp + { \left( \frac { \partial G }{ \partial P } \right) }_{ T, { n }_{ 1 }, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }dt + \overline { { G }_{ 1 } } d{ n }_{ 1 } + \overline { { G }_{ 2 } } d{ n }_{ 2 } + \overline { G_{ 3 } } d{ n }_{ 3 } + .... + \overline { G_{ i } } d{ n }_{ i } }$$ Where $$\overline { { G }_{ 1 } }$$ is the partial molal free energy and $$\overline { { G }_{ 1 } } = { \left( \frac { \partial G }{ \partial { n }_{ 1 } } \right) }_{ T, P, { n }_{ 2 }, { n }_{ 3 }, ...., { n }_{ i } }$$ And at constant Temperature and Pressure first tow terms becomes zero, $$\boxed { dG = \overline { G_{ 1 } } d{ n }_{ 1 } + \overline { G_{ 2 } } d{ n }_{ 2 } + \overline { G_{ 3 } } d{ n }_{ 3 } + .... + \overline { G_{ i } } d{ n }_{ i } }$$ The partial molal free energy is, however, identical with chemical potential $$\mu$$ described by Gibbs. $$\therefore dG = \mu _{ 1 }d{ n }_{ 1 } + \mu _{ 2 }d{ n }_{ 2 } + \mu _{ 3 }d{ n }_{ 3 } + .... + \mu _{ i }d{ n }_{ i }\qquad .... (2)$$ If the system is having definite composition, the integration of equation (2) gives, $${ \left( G \right) }_{ T, P, N } = \mu _{ 1 }{ n }_{ 1 } + \mu _{ 2 }{ n }_{ 2 } + \mu _{ 3 }{ n }_{ 3 } + .... + \mu _{ i }{ n }_{ i }$$ where N indicates definite composition of the system.

On complete differentiation of the above equation. $$dG = \left( \mu _{ 1 }d{ n }_{ 1 } + { n }_{ 1 }d\mu _{ 1 } \right) + \left( \mu _{ 2 }d{ n }_{ 2 } + { n }_{ 2 }d\mu _{ 2 } \right) + \left( \mu _{ 3 }d{ n }_{ 3 } + { n }_{ 3 }d\mu _{ 3 } \right) + .... + \left( \mu _{ i }d{ n }_{ i } + { n }_{ i }d\mu _{ i } \right)$$ or $$dG = \left[ \mu _{ 1 }d{ n }_{ 1 } + \mu _{ 2 }d{ n }_{ 2 } + \mu _{ 3 }d{ n }_{ 3 } + .... + \mu _{ i }d{ n }_{ i } \right] + \left[ { n }_{ 1 }d\mu _{ 1 } + { n }_{ 2 }d\mu _{ 2 } + { n }_{ 3 }d\mu _{ 3 } + .... + { n }_{ i }d\mu _{ i } \right]$$ from equation (2) $$\therefore dG = dG + \left[ { n }_{ 1 }d\mu _{ 1 } + { n }_{ 2 }d\mu _{ 2 } + { n }_{ 3 }d\mu _{ 3 } + .... + { n }_{ i }d\mu _{ i } \right]$$ or $$\boxed { \left[ { n }_{ 1 }d\mu _{ 1 } + { n }_{ 2 }d\mu _{ 2 } + { n }_{ 3 }d\mu _{ 3 } + .... + { n }_{ i }d\mu _{ i } \right] = 0 }$$ also $$\boxed { \sum { nd\mu } = 0 }$$ This equation is known as Gibbs - Duhem Equation.

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