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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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