# CfE: Clapeyron Equation

### Clapeyron Equation

Taking the criteria for phase equilibrium as out starting point:

$g_l = g_v$

We can write:

$dg_l = dg_v$

Using one of our fundamental property relations, for phase "k"

$dg_k = v_kdP - s_kdT$

so

$v_ldP - s_vdT=v_ldP - s_ldT$

$\frac{dP}{dT} = \frac{s_l-s_v}{v_l-v_v}$

Recalling that we could also write

$g_k = h_k - Ts_k$

we have

$h_l - Ts_l = h_v-Ts_v$

$s_l - s_v = \frac{h_l-h_v}{T}$

combining we get

##### Definition:

The Clapeyron equation relates the phase equilibrium pressure (often called $P_{sat}$) to the temperature for any combination of phases:

$\frac{dP}{dT} = \frac{h_l-h_v}{T(v_l-v_v)}$

##### Outcome:

Use the Clapeyron equation and/or the Clausius-Clapeyron equation to relate T and P for pure species phase equilibrium