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.