Much better, but I still need an answer.

Considering the equilibrium reaction starting with either 1.00 mol of either A or B,

A

B

I have thus far I've worked out that...

**ΔG**_{state} = xΔG_{A} + (1-x)ΔG_{B}x: Percentage of A at a certain state.

But since

**ΔG = ΔH - TΔS** then,

**ΔG**_{state} = x(ΔH_{A}-TΔS_{A}) + (1-x)(ΔH_{B}-TΔS_{B})Expanding and rearranging, we get...

**ΔG**_{state} = xΔH_{A} - xTΔS_{A} -TΔS_{B} - xΔH_{B} + xTΔS_{B} + ΔH_{B}I know to that...

1. ΔG

_{equilibrium} is when ΔG

_{state} is a minimum.

2. δΔG

_{state}/δx should still have an x-term in it, and would be equal to zero at equilibrium

So, using implicit differentiation we get...

**δΔG**_{state}/δx = ΔH_{A} - Tδ(xΔS_{A})/δx -TδΔS_{B}/δx - ΔH_{B} + Tδ(xΔS_{B})/δx

δΔG_{state}/δx = ΔH_{A} - TΔS_{A} - TxδΔS_{A}/δx -TδΔS_{B}/δx - ΔH_{B} + TΔS_{B} + TxδΔS_{B}/δxTaking

**ΔH = ΔH**_{A} - ΔH_{B} **δΔG**_{state}/δx = ΔH + T(ΔS_{B} - ΔS_{A} - δΔS_{B}/δx) + Tx(δΔS_{B}/δx - δΔS_{A}/δx) From here I'm stuck. I'm not aware of how ΔS varies w.r.t. x. I know I probably have to re-express it before I can differentiate to show that the curve has a minimum, or am I missing something that is totally obvious? I'm also not sure if there's a relation between ΔS

_{A} and ΔS

_{B}. I've found the equation ΔS

_{state} = ΔS

^{o} - Rln[C], but have no idea how to use it and clearly this is for a one-element system.