Equilibrium between phases

A substance can be in several different states. Take water for example, we can find it as ice, liquid water and water vapour. Each of these states has different physical properties. We know from the discussions in the previous chapters that at equilibrium the system will adopt the physical state that maximises its entropy, if it is isolated from the environment. By that we mean that the removal of any internal constraint would increase the entropy of the system or, which is the same, the introduction of any internal constraint would reduce it. Similarly, we have seen that if we maintain the system at constant volume, constant temperature and constant number of particles, for example by putting it in contact with a heat bath and allow energy to be exchanged, then the Helmholtz free energy of the system will always decrease upon removal of internal constraints. A corollary of this statement is that a system kept at constant $V,T,N$ will choose the physical state that minimises its Helmholtz free energy. If the system were kept at constant $P,T,N$ then it would be its Gibbs free energy to be minimum. These statements lead us naturally to the discussion of equilibrium between different phases. For example, we know that liquid water at ambient pressure freezes at a temperature of zero Celsius. This is a consequence of the Gibbs free energy of the system becoming lower if all liquid transforms into solid. The total free energy of the system is $G = G_l + G_s$, with $G_l$ and $G_s$ the Gibbs free energies of solid and liquid. The equilibrium condition is therefore expressed by $dG = 0$:

$$dG_l + dG_s = (V_l + V_s) dP - (S_l + S_s) dT + \mu_s dN_s + \mu_l dN_l = 0$$ (6.1)

If the system is kept at constant $T$ and constant $P$, and if the total number of particles is kept constant so that we have $d N_s = - dN_l = dN$, we have:

$$dG = (\mu_s - \mu_l) dN = 0,$$ (6.2)

which implies

$$\mu_s = \mu_l,$$ (6.3)

which we can also write:

$$g_s = g_l,$$ (6.4)

where $g_s = G_s / N$ and $g_l = G_l / N$. The chemical potentials $\mu_s,\mu_l$, or Gibbs free energies per particle $g_s,g_l$, depend on pressure and temperature. In Fig.  we show an example of how $g_s$ and $g_l$ vary with temperature at some fixed value of pressure. The temperature $T_m$ for which $g_{ls} = g_l - g_s = 0$ is the melting temperature at that value of pressure. The derivative of the chemical potential w.r.t. temperature is the negative of the entropy (per particle):

$$\left(\frac{\partial g}{\partial T} \right)_P = -s,$$ (6.5)

Figure: The Gibbs free energy per particle (chemical potential) as function of temperature of a system in the solid state (solid black line) and liquid state (red dashed line), at some fixed pressure. At low temperature the chemical potential of the solid is lower than that of the liquid and the system is found in the solid phase. The opposite is true at high temperature. The temperature for which $g_l = g_s$ is the melting temperature at this pressure.

and since the entropy must be positive, the slope of the chemical potential must be negative, as shown in Fig. . Moreover, the requirement for the system to be solid below the melting temperature and liquid above it means that the entropy of the liquid must be larger than that of the solid at and near the melting point. The difference of the two slopes means that, as the system transforms from one phase to the other, the entropy changes in a discontinuous way and the transformation is called first order phase transition, where first indicates that there is a discontinuity in the first derivative. The entropy change on melting is $s_{ls} = s_l - s_s$, computed at the melting temperature. At this temperature $g_{ls} = 0$, which implies $h_{ls} -T_m s_{ls} = 0$, where $h_{ls} = h_l - h_s$ is the enthalpy change on melting, with $h_l$ and $h_s$ the enthalpies of the two phases. The enthalpy change on melting is therefore $h_{ls} = T_m s_{ls}$. This quantity is usually called $L$ and known as the latent heat of fusion, which is the amount of heat that has to be provided to the system to change its phase from solid to liquid at the melting temperature. Conversely, in a crystallisation process the latent heat has to be extracted from the system to allow it to crystallise.

Figure: The chemical potential as function of pressure of a system in the solid state (solid black line) and liquid state (red dashed line), at some fixed temperature. The pressure for which $g_l = g_s$ is the melting pressure at this temperature.

Instead of plotting $g_l$ and $g_s$ as function of temperature at some fixed pressure, we could just as well plot them as function of pressure at some fixed temperature, as in Fig. . For this particular temperature we can then obtain the melting pressure from the point where $g_{ls} = 0$. The derivative of the chemical potential w.r.t to pressure at constant temperature is equal to the volume per particle $v = V/N$:

$$\left(\frac{\partial g}{\partial P} \right)_T = v,$$ (6.6)

which therefore must be positive, as displayed in Fig. . However, the volume change on melting is not required to be of a particular sign, because there is no need for the system to be either necessarily solid or necessarily liquid below the melting pressure. Most system have a larger volume in the liquid phase, such as the one showed in Fig. , so that the volume change on melting is positive, but there are notable exceptions, e.g. water. This has interesting consequences for how the melting temperature depends on pressure, as we shall see in the next section.

The condition of equilibrium between two phases defines a one-dimensional locus of points (a line) in $P,T$ space. For example, for a melting transition this locus of points is known as the melting line, or melting curve. If we also consider the vapour phase we can introduce two additional equilibrium conditions, the sublimation line, defined by $g_s = g_v$, with $g_v$ the chemical potential of the vapour phase, and the vapour line, defined by $g_l = g_v$. It is also possible to find equilibrium between all three phases at once $g_s = g_l = g_v$, which defines a zero-dimensional triple point. This unique point in $(P,T)$ space for water is used to define the Kelvin temperature scale (see Fig. ), as we mentioned in Chapter 1.

Figure: The phase diagram of water.