Phase coexistence and general properties of phase transitions

In order to understand a little bit more about phase transitions and the coexistence of different phases, let us consider a $PVT$$PVT$ system in the liquid-gaseous transition, and let us focus on the region of liquid-gas coexistence; physically, what happens (and what can also be understood from the figure) is that the system exhibits both a liquid and a gaseous phase, and it is possible to change the volume of the system without changing its pressure at fixed temperature. This means that the work we do on the system is used only to change the proportion between the phases (breaking or forming molecular bonds, on a microscopic level).

Let us now consider a slightly more general situation: suppose we have an isolated single-component thermodynamic system described by the internal energy $U$$U$ and extensive variables $x_{i}$$x_i$ (generalized displacements); if the system is subject to reversible processes then from the expression of $dU$$dU$ we get:

$$dS={\frac {1}{T}}dU-\sum _{i}{\frac {X_{i}}{T}}dx_{i}$$

$$dS={\frac {1}{T}}dU-\sum _{i}{\frac {X_{i}}{T}}dx_{i}$$

where $X_{i}$$X_i$ are the generalized forces relative to the generalized displacements $x_{i}$$x_i$. Let us now suppose that two phases $\alpha$$\alpha$ and $\beta$$\beta$ coexist and that they can exchange $U$$U$ and $x$$x$ (namely, they are systems in contact with each other). At equilibrium the total entropy $S=S_{\alpha }+S_{\beta }$$S=S_{\alpha }+S_{\beta }$ must be maximized, and since the whole system is in equilibrium we also have $dU_{\alpha }+dU_{\beta }=0$$dU_{\alpha }+dU_{\beta }=0$ and $dx_{i,\alpha }+dx_{i,\beta }=0$$dx_{i,\alpha }+dx_{i,\beta }=0$. Substituting in ${\textstyle dS}$${\textstyle dS}$ we get:

$$dS=\left({\frac {1}{T_{\alpha }}}-{\frac {1}{T_{\beta }}}\right)dU_{\alpha }-\sum _{i}\left({\frac {X_{i,\alpha }}{T_{\alpha }}}-{\frac {X_{i,\beta }}{T_{\beta }}}\right)dx_{i,\alpha }=0$$

$$dS=\left({\frac {1}{T_{\alpha }}}-{\frac {1}{T_{\beta }}}\right)dU_{\alpha }-\sum _{i}\left({\frac {X_{i,\alpha }}{T_{\alpha }}}-{\frac {X_{i,\beta }}{T_{\beta }}}\right)dx_{i,\alpha }=0$$

Therefore, since $dU_{\alpha }$$dU_{\alpha }$ and $dx_{i,\alpha }$$dx_{i,\alpha }$ are arbitrary positive quantities, we must have:

$$T_{\alpha }=T_{\beta }\quad \qquad X_{i,\alpha }=X_{i,\beta }$$

$$T_{\alpha }=T_{\beta }\quad \qquad X_{i,\alpha }=X_{i,\beta }$$

Considering our $PVT$$PVT$ system again, since the generalized displacements that are needed in order to describe it are the volume $V$$V$ and the number of particles $N$$N$, when two phases coexist we have:

$$T_{\alpha }=T_{\beta }\quad \quad P_{\alpha }=P_{\beta }\quad \quad \mu _{\alpha }=\mu _{\beta }$$

$$T_{\alpha }=T_{\beta }\quad \quad P_{\alpha }=P_{\beta }\quad \quad \mu _{\alpha }=\mu _{\beta }$$

Since, as we have shown in Thermodynamic potentials, the Gibbs free energy is $G(T,P)=\mu N$$G(T,P)=\mu N$ (for a single-component $PVT$$PVT$ system) then $g(T,P)=\mu$$g(T,P)=\mu$ and so when two phases coexist we also have:

$$g_{\alpha }(T,P)=g_{\beta }(T,P)$$

$$g_{\alpha }(T,P)=g_{\beta }(T,P)$$

This equality must hold along the whole coexistence line in $(T,P)$$(T,P)$ space and so if we know a point in this space where the two phases coexist we can, at least locally, "reconstruct" the coexistence line. In fact we have $dg_{\alpha }=dg_{\beta }$$dg_{\alpha }=dg_{\beta }$, namely:

$${\begin{aligned}dg_{\alpha }=-s_{\alpha }dT+v_{\alpha }dP=dg_{\beta }=-s_{\beta }dT+v_{\beta }dP\quad \Rightarrow \\\Rightarrow \quad {\frac {dP}{dT}}_{|{\text{coex.}}}={\frac {s_{\alpha }-s_{\beta }}{v_{\alpha }-v_{\beta }}}={\frac {L_{\alpha ,\beta }}{T(v_{\alpha }-v_{\beta })}}\end{aligned}}$$

$${\begin{aligned}dg_{\alpha }=-s_{\alpha }dT+v_{\alpha }dP=dg_{\beta }=-s_{\beta }dT+v_{\beta }dP\quad \Rightarrow \\\Rightarrow \quad {\frac {dP}{dT}}_{|{\text{coex.}}}={\frac {s_{\alpha }-s_{\beta }}{v_{\alpha }-v_{\beta }}}={\frac {L_{\alpha ,\beta }}{T(v_{\alpha }-v_{\beta })}}\end{aligned}}$$

where $s=S/N$$s=S/N$ and $v=V/N$$v=V/N$, and by definition $L_{\alpha ,\beta }$$L_{\alpha ,\beta }$ is the molar latent heat needed to bring the system from phase $\beta$$\beta$ to phase $\alpha$$\alpha$. This is known as Clausius-Clapeyron equation.

From the expression of $dg_{\alpha }=dg_{\beta }$$dg_{\alpha }=dg_{\beta }$ we can also understand some very general properties of phase transitions. In fact, from the expressions of ${\textstyle dg_{\alpha }}$${\textstyle dg_{\alpha }}$ and ${\textstyle dg_{\beta }}$${\textstyle dg_{\beta }}$ at phase coexistence we have:

$${\frac {\partial g_{\beta }}{\partial T}}_{|P}-{\frac {\partial g_{\alpha }}{\partial T}}_{|P}=\Delta s>0\quad \quad {\frac {\partial g_{\beta }}{\partial P}}_{|T}-{\frac {\partial g_{\alpha }}{\partial P}}_{|T}=\Delta v>0$$

$${\frac {\partial g_{\beta }}{\partial T}}_{|P}-{\frac {\partial g_{\alpha }}{\partial T}}_{|P}=\Delta s>0\quad \quad {\frac {\partial g_{\beta }}{\partial P}}_{|T}-{\frac {\partial g_{\alpha }}{\partial P}}_{|T}=\Delta v>0$$

This means that when the system undergoes a phase transition its volume and its entropy have a jump discontinuity; as we will later see in more detail, since in this transition the first derivatives of a thermodynamic potential have a jump discontinuity, we call it a first order transition. A similar behaviour can be encountered in magnetic systems, where the magnetization $M$$M$ has a jump at $H=0$$H=0$ for temperatures lower that the critical one; in this case since $M=-\partial F/\partial H$$M=-\partial F/\partial H$ we see that the first derivative of the free energy $F$$F$ with respect to $H$$H$ has a jump discontinuity.

Furthermore, if we consider our system at the critical point we see from the $(V,P)$$(V,P)$ projection of the phase diagram that the isothermal compressibility:

$$K_{T}=-{\frac {1}{V}}{\frac {\partial V}{\partial P}}_{|T}$$

$$K_{T}=-{\frac {1}{V}}{\frac {\partial V}{\partial P}}_{|T}$$

diverges when $V=V_{c}$$V=V_{c}$. Similarly, the magnetic susceptibility $\chi _{T}$$\chi _{T}$ of a magnetic system at its critical temperature diverges when $H=0$$H=0$ and $M=0$$M=0$. As we will see further on in Long range correlations, the divergence of response functions is a typical behaviour of thermodynamic systems in the neighbourhood of critical points and has important consequences.