Vapour in Equilibrium with a Liquid or Solid

When the vapour of a substance is in thermodynamic equilibrium with its liquid or solid phase at a given temperature, the pressure exerted by the vapour is called the saturated vapour pressure.

The dependence of saturated vapour pressure on temperature is described by the Clapeyron equation.

For two phases in equilibrium, the Clapeyron equation is

\[ \boxed{ \frac{dP}{dT} = \frac{L}{T(V_2-V_1)} } \tag{1} \]

where

• \(L\) is the molar latent heat of transition.
• \(V_1\) is the molar volume of the condensed phase.
• \(V_2\) is the molar volume of the vapour phase.
• \(T\) is the absolute temperature.

For vaporization or sublimation at sufficiently low pressure, the molar volume of the vapour is much larger than that of the liquid or solid:

\[ V_2\gg V_1. \]

Therefore,

\[ V_2-V_1\approx V_2. \]

Equation (1) becomes

\[ \frac{dP}{dT} \approx \frac{L}{TV_2}. \tag{2} \]

Assuming that the vapour behaves as an ideal gas, the equation of state for one mole is

\[ PV_2=RT. \]

Hence,

\[ V_2=\frac{RT}{P}. \tag{3} \]

Substituting equation (3) into equation (2),

\[ \frac{dP}{dT} = \frac{L}{T\left(\frac{RT}{P}\right)}. \]

Simplifying,

\[ \boxed{ \frac{dP}{dT} = \frac{LP}{RT^2} } \tag{4} \]

Rearranging,

\[ \frac{dP}{P} = \frac{L}{R}\frac{dT}{T^2}. \]

If the latent heat \(L\) is approximately constant over the temperature interval, integration gives

\[ \int\frac{dP}{P} = \frac{L}{R}\int T^{-2}\,dT. \]

Therefore,

\[ \ln P = -\frac{L}{RT}+C, \]

where \(C\) is an integration constant.

Writing \(C=\ln P_0\), where \(P_0\) is a constant having the dimensions of pressure,

\[ \ln P = -\frac{L}{RT}+\ln P_0. \]

Hence,

\[ \ln\left(\frac{P}{P_0}\right) = -\frac{L}{RT}. \]

Taking the exponential of both sides,

\[ \boxed{ P=P_0\exp\left(-\frac{L}{RT}\right) } \tag{5} \]

This equation gives the approximate dependence of saturated vapour pressure on temperature when the latent heat is constant and the vapour behaves ideally.

Vapour Pressure at Two Temperatures

For two equilibrium states at temperatures \(T_1\) and \(T_2\), with corresponding vapour pressures \(P_1\) and \(P_2\),

\[ \ln P_1=-\frac{L}{RT_1}+C \]

and

\[ \ln P_2=-\frac{L}{RT_2}+C. \]

Subtracting the first equation from the second,

\[ \ln P_2-\ln P_1 = -\frac{L}{RT_2}+\frac{L}{RT_1}. \]

Therefore,

\[ \boxed{ \ln\left(\frac{P_2}{P_1}\right) = \frac{L}{R} \left( \frac{1}{T_1}-\frac{1}{T_2} \right) } \tag{6} \]

This integrated Clausius–Clapeyron equation is widely used to estimate vapour pressure and latent heat over a limited temperature range.

Physical significance: For a positive latent heat of vaporization or sublimation, the saturated vapour pressure increases with increasing temperature. A plot of \(\ln P\) against \(1/T\) is approximately linear, with slope \(-L/R\), provided \(L\) remains constant.