Question

Derive the integrated form of Clausius-Clapeyron equation.

Answer 1

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Answer 2

The equilibrium between water and water vapor depends upon the temperature of the system. If the temperature increases the saturation pressure of the water vapor increases. The rate of increase in vapor pressure per unit increase in temperature is given by the Clausius-Clapeyron equation. Let p be the saturation vapor pressure and T the temperature. The Clausius-Clapeyron equation for the equilibrium between liquid and vapor is then

dp/dT = L/(T(Vv-Vl))

where L is the latent heat of evaporation, and Vv and Vl are the specific volumes at temperature T of the vapor and liquid phases, respectively.

More generally the Clausius-Clapeyron equation pertains to the relationship between the pressure and temperature for conditions of equilibrium between two phases. The two phases could be vapor and solid for sublimation or solid and liquid for melting.

The material below is an examination of the application of the Clausius-Clapeyron equation to meteorology where the most relevant systems are water and water vapor and salt water and water vapor. But the equation also applies to ice and water vapor and ice and water. Finally there is a derivation of the Clausius-Clapeyron equation from thermodynamic principles.

Note that Vv is much greater than Vl so that to a good approximation

dp/dT = L/(TVv)

Furthermore the ideal gas equation applies to the vapor; i.e.,

pVv = RT
and hence
Vv = RT/p

where R is the universal gas constant.

Thus

dp/dT = L/(RT²/p) 
or, equivalently
(1/p)(dp/dT) = L/(RT²)

In differential form this is

dp/p = (L/R)(dT/T²) 
or, equivalently
d(ln(p)) = (L/R)d(-1/T)

If L is independent of temperature then the solution to the differential equation is

ln(p) = c0 - (L/RT) 
or, equivalently
p = c1exp(-L/RT)

where c1 is a constant.

The shape of this function is given below:

The empirical curves have the opposite curvature but clearly the equation dp/dT = L/(RT²/p) = Lp/T² indicates that as temperature increases the slope (dp/dT) decreases. Of course, the empirical curves take into account that the latent heat L can change with temperature, but the change in L with T is relative small. There is a fundamental discrepancy between the Clausius-Clapeyron Equation and the empirical relationships and the published derivations of the equation blithely show an empirical curve in which dp/dT increases with T and an equation in which dp/dT is inversely proportional to T. However this discrepancy will be left unresolved here and the implications of the Clausius-Clapeyron equation will will be further explored.

For some purposes the density of the vapor is of more interest than the pressure. By the ideal gas equation, the molecular density D is given by

D(T) = 1/Vv = p/RT 
which for the above
pressure-temperature
relationship is
D(T) = c1exp(-L/RT)/RT