# First order phase transition/constancy of Gibbs

1. May 13, 2014

### CAF123

1. The problem statement, all variables and given/known data
a) Starting from the statement that total entropy (Ssystem+Ssurr) can only increase, show that G = U - TS +pV will attain its minimum value for a system in equilibrium with a fixed pressure and temperature reservoir.

b)At atmospheric pressure, a particular substance is found to undergo a discontinuous change between two states at temperature TC when heated. Its volume increases by ΔV and it absorbs latent heat L as its temperature is changed from just below TC to just above TC. Explain why at TC, the value of G is the same for the two states with different volumes.

c)Explain why L must be positive and comment on whether TC is expected to increase or decrease with pressure.

2. Relevant equations
Clausius-Clapyeron Equation

3. The attempt at a solution
a)Taking the differential of the given equation, I get dG = dU - TdS -SdT +pdV + Vdp. Eliminate two terms because the system is in thermal equilibrium at constant pressure/temperature. This gives dG = dU - TdS + pdV = 0 using the first law. Hence G is mimimum when the boundary conditions of the system permit a constant pressure/temperature environment. I did not really use the fact that the total S ≥ 0 though, so is there another derivation?

b)So is this process occurring at constant pressure and is TC the value of T on the boundary line between the two phases at that particular pressure? If the case, then at TC the two phases instantaneously have the same pressure/temperature. dG = Vdp - SdT = 0, so G is constant over the boundary line.

c) V increases upon heating at constant pressure. So I would imagine this would correspond to an increase in entropy of the system. So l = T(S2-S1) > 0. dP/dT is usually +ve for most substances. So dT/dP is decreasing, so in most cases expect TC to decrease with pressure. Did I do this right?

Many thanks.

2. May 13, 2014

### CAF123

Also, I thought a latent heat was associated with a change of phase transition at constant temperature. It says that the substance absorbs latent heat from just below Tc to just above Tc, so it is therefore changing temperature, albeit infinitesimally if that is what 'just above (below)' means.