First order phase transition/constancy of Gibbs

[CAF123]

Homework Statement

a) Starting from the statement that total entropy (S_system+S_surr) 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 T_C when heated. Its volume increases by ΔV and it absorbs latent heat L as its temperature is changed from just below T_C to just above T_C. Explain why at T_C, 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 T_C is expected to increase or decrease with pressure.

Homework Equations

Clausius-Clapyeron Equation

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 T_C the value of T on the boundary line between the two phases at that particular pressure? If the case, then at T_C 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 T_C to decrease with pressure. Did I do this right?

Many thanks.

 

[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 T_c to just above T_c, so it is therefore changing temperature, albeit infinitesimally if that is what 'just above (below)' means.