5.3: Phase Transitions
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If we have a single constituent, for thermodynamic equilibrium, we should have equality of T, p and µ for different subparts of the system. If we have different phases of the system, such as gas, liquid, or solid, in equilibrium, we have
T1=T2=T3=...=Tp1=p2=p3=...=pµ1=µ2=µ3=...
where the subscripts refer to various phases.
We will consider the equilibrium of the two phases in more detail. In this case, we have
µ1(p,T)=µ2(p,T)
If equilibrium is also obtained for a state defined by (p+dp, T+dT), then we have
µ1(p+dp,T+dT)=µ2(p+dp,T+dT)
These two equations yield
Δμ1=Δμ2,Δμ=μ(p+dp,T+dT)−μ(p,T)
This equation will tell us how p should change when T is altered (or vice versa) so as to preserve equilibrium. Expanding to first order in the variations, we find
−s1dT+v1dp=−s2dT+v2dp
where we have used
∂µ∂T=−SN≡s,∂µ∂p=−VN≡v
Equation ??? reduces to
dpdT=s1−s2v1−v2=LT(v1−v2)
where L=T(s1−s2) is the latent heat of the transition. This equation is known as the Clausius-Clapeyron equation. It can be used to study the variation of saturated vapor pressure with temperature (or, conversely, the variation of boiling point with pressure). As an example, consider the variation of boiling point with pressure, when a liquid boils to form gaseous vapor. In this case, we can take v1=vg≫v2=vl. Further, if we assume, for the sake of the argument, that the gaseous phase obeys the ideal gas law, vg=kTp, then the Clausius-Clapeyron Equation ??? becomes
dpdT≈pLkT2
Integrating this from one value of T to another,
log(pp0)=Lk(1T0−1T)
Thus for p>p0, T must be larger than T0; this explains the increase of boiling point with pressure.
If ∂µ∂T and ∂µ∂p are continuous at the transition, s1=s2 and v1=v2. In this case, we have to expand µ to second order in the variations. Such a transition is called a second order phase transition. In general, if the first (n−1) derivatives of µ are continuous, and the n-th derivatives are discontinuous at the transition, the transition is said to be of the n-th order. Clausius-Clapeyron equation, as we have written it, applies to the first order phase transitions. These have a latent heat of transition.