The Clausius-Clapeyron relation

A direct consequence of the first order phase transitions discussed in the previous section is that the slopes of the lines separating two different phases in $(P,T)$ space are related to the discontinuities in the first derivatives of the Gibbs free energy. Consider for example the melting line defined by $g_{ls}(P_m,T_m)=0$. For variations of $dP$ and $dT$ on the melting line we have:

$$g_{ls}(P_m+dP,T_m+dT) = g_{ls}(P_m,T_m) + \left (\frac{\partial g... ...al P} \right )_T dP + \left (\frac{\partial g_{ls}}{\partial T} \right )_P dT =$$

$$v_{ls} dP - s_{ls} dT = 0,$$ (6.7)

from which we get the Clausius-Clapeyron relation:

$$\left (\frac{dT}{dP}\right )_m = \frac{v_{ls}}{s_{ls}},$$ (6.8)

where $(\cdot)_m$ means variation on the melting line. As we mentioned in the previous section, $s_{ls}$ is always positive, but $v_{ls}$ can be either positive or negative. For most materials $v_{ls}$ is positive, but for some materials, most notably water, $v_{ls}$ is negative and so the slope of the melting curve is negative. This means that in a gravitational field 'standard' material freeze from the inside out, e.g. iron in the Earth's core, which is why the Earth has a solid inner core surrounded by a liquid outer core. By contrast, the Jupiter moon Europa has a water ocean beneath an icy crust, which freezes from the outside. This of course is also the environment of the artic ocean on Earth, which is (for now) solid on the outside, with liquid water underneath.