### Derive expression for Clapeyron Clausius Equation.

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Let us consider a closed system in which a pure liquid water and its vapors are in equilibrium with each other. $$Water (liquid) \Longleftrightarrow Water (vapor)$$ For this system Clapeyron equation can be written as, $$\boxed { \frac { dP }{ dT } = \frac { \Delta H }{ T\left( { V }_{ v } - { V }_{ l } \right) } }$$ or $$\boxed { \frac { dP }{ dT } = \frac { \Delta { L }_{ v } }{ T\left( { V }_{ v } - { V }_{ l } \right) } }$$ Where $${ L }_{ v }$$ is latent heat of vaporization. $${ V }_{ v }$$ is Volume of one mole of water in vapor phase. $$V_{ l }$$ is Volume of one mole of water in liquid phase.

If temerature is not near the critical temerature, $${ V }_{ v }>>{ V }_{ l }$$ and hence $${ V }_{ v } - { V }_{ l } \simeq { V }_{ v }$$ $$\therefore \frac { dP }{ dT } = \frac { \Delta { L }_{ v } }{ T{ V }_{ v } } \qquad ....(1)$$ Assuming that the vapor obeys ideal gas law, $$P{ V }_{ v } = nRT$$ for one mole $$P{ V }_{ v } = RT i.e. { V }_{ v } = \frac { RT }{ P } \qquad ....(2)$$ Lets substitute it in equation (1) $$\frac { dP }{ dT } = \frac { \Delta { L }_{ v } }{ T } \frac { P }{ RT }$$ $$\therefore \boxed { \frac { dP }{ P } = \frac { \Delta { L }_{ v } }{ R } \frac { dT }{ { T }^{ 2 } } } \qquad ....(3)$$ This equation is known as Clapeyron Clausius Equation .

The equation (3) can be integrated between the limits $${ P }_{ 1 }$$ to $${ P }_{ 2 }$$ and $$T_{ 1 }$$ to $${ T }_{ 2 }$$ assuming $$\Delta { L }_{ v }$$ remains constant for small range of temerature. $$\int _{ { P }_{ 1 } }^{ { P }_{ 2 } }{ \frac { dP }{ P } } = \frac { \Delta { L }_{ v } }{ R } \int _{ T_{ 1 } }^{ T_{ 2 } }{ \frac { dT }{ { T }^{ 2 } } }$$ i.e. $$ln\frac { { P }_{ 2 } }{ { P }_{ 1 } } = \frac { -\Delta { L }_{ v } }{ R } \left( \frac { 1 }{ { T }_{ 2 } } -\frac { 1 }{ T_{ 1 } } \right)$$ i.e. $$2.303log\frac { { P }_{ 2 } }{ { P }_{ 1 } } = \frac { \Delta { L }_{ v } }{ R } \left( \frac { 1 }{ T_{ 1 } } -\frac { 1 }{ { T }_{ 2 } } \right)$$ $$\boxed { log\frac { { P }_{ 2 } }{ { P }_{ 1 } } = \frac { \Delta { L }_{ v } }{ 2.303R } \left( \frac { { T }_{ 2 } - { T }_{ 1 } }{ T_{ 1 }{ T }_{ 2 } } \right) }$$ This equation is integrated form of Clapeyron Clausius Equation .

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