Statistical Physics

8.4 The Clausius–Clapeyron Law 123such a metastable state. Interaction with the heat bath makes the systemfluctuate around the corresponding minimum with a probability given by thecanonical distribution. A rather large perturbation is needed from the heatbath to make the system move to a configuration around the other (true)minimum. Therefore, the system can remain around the higher minimum forsome time.8.4 The Clausius–Clapeyron LawThe slope of the coexistence line in the phase diagram, dP/dT ,canberelatedto measurable quantities. Along the coexistence line, the Gibbs free energiesof the two phases are the same. Therefore, we have two equations that applyat adjacent points on the coexistence line:G I (T,P,N)=G II (T,P,N) ,G I (T +∆T,P +∆P, N) =G II (T +∆T,P +∆P, N) , (8.8)where I and II are labels used to distinguish the two phases. We expand bothsides of the second equation with respect to ∆T and ∆P to the lowest orderand, using the relation in the first line, obtain∆TSince( ∂GI∂T)P,N+∆P( ∂GI∂P)T,N( ) ∂GI∂T( ) ∂GI∂P=∆TP,NT,N(∂GII∂T)P,N+∆P(∂GII∂P)T,N.(8.9)= −S I , (8.10)= V I , (8.11)and so on, we obtain∆T ∆S =∆P ∆V , (8.12)where ∆S ≡ S II − S I and ∆V ≡ V II − V I .SinceT ∆S = Q L is the latent heatof the transition,dPdT = lim ∆P∆T →0 ∆T = Q LT ∆V . (8.13)Thus the volume change and the latent heat determine the slope of the coexistenceline.In the case of a gas–liquid transition, where phase I is the liquid andphase II is the gas, ∆V > 0and∆S>0. Therefore, the slope is positive. Aswe increase the temperature, the saturated vapor pressure increases. In thecase of water, the vapor pressure reaches atmospheric pressure at 100 ◦ C, andwater begins to boil at that temperature at sea level.