Question

Please help me to derive that `((delS)/(delT))_P = C_P/T` and `((delS)/(delT))_V = C_V/T`?

Answer 1

Okay, so I will assume you know or can figure out the Maxwell relations. Those are a common starting point. Furthermore, you have by definition that `C_P = ((delH)/(delT))_P` and `C_V = ((delU)/(delT))_V`, so you do not have to derive those.

---

Recall that `G = H - TS`. Then, the derivative is:

`dG = dH - d(TS)`

`= dH - SdT - TdS`
(use product rule)

Now, for the first derivation, you can divide through by `delT` at a constant `P` to generate two derivatives that can be figured out, and the third one is what you're looking for:

`((delG)/(delT))_P = ((delH)/(delT))_P - cancel(S((delT)/(delT))_P)^(1) - T((delS)/(delT))_P`

`" "" "bb((1))`

where the third term goes to `1` because `T` does not change with respect to `T`; it is one-to-one with itself.

By definition, `((delH)/(delT))_P = C_P`. `" "" "bb((2))`

Next, recall that for the natural variables `T` and `P` correspond to the Gibbs' free energy, so that the Maxwell relation is:

`dG = -SdT + VdP` `" "" "" "bb((3))`

We do not yet know what `((delG)/(delT))_P` is, but using the Maxwell relation, we get:

`((delG)/(delT))_P = -S` `" "" "" "bb((4))`

Therefore. plugging `bb((4))` and `bb((2))` into `bb((1))`:

`cancel(-S) = C_P - cancel(S) - T((delS)/(delT))_P`

It follows that if `C_P = T((delS)/(delT))_P`, it must be that `color(blue)(((delS)/(delT))_P = C_P/T)`.

---

A similar process follows for deriving `C_V/T = ((delS)/(delT))_V`. Using:

`A = U - TS`,

take the derivative to get:

`dA = dU - SdT - TdS`

Almost like before, divide by `delT` at a constant `V` instead of `P` to generate two derivatives that can be figured out, and the third one is what you're looking for:

`((delA)/(delT))_V = ((delU)/(delT))_V - Scancel(((delT)/(delT))_V)^(1) - T((delS)/(delT))_V`

We know by definition that `((delU)/(delT))_V = C_V`, `((delT)/(delT))_V = 1`, and that `A` is a function of the natural variables `T` and `V`.

So from the Maxwell relation:

`dA = -SdT - PdV`

we have that:

`((delA)/(delT))_V = -S`,

so plugging back into the main equation, we get:

`cancel(-S) = C_V - cancel(S) - T((delS)/(delT))_V`

Since `C_V = T((delS)/(delT))_V`, it follows that `color(blue)(((delS)/(delT))_V = C_V/T)`.