 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/17/a/solution

U = U (S, V)

T = U_{S} ∣_{V}

and

- P = U_{V} ∣_{S}

. Equality of mixed second derivatives gives

(\frac{\partial T}{\partial V})_{S} = - (\frac{\partial P}{\partial S})_{V} .

For the Helmholtz free energy, subtract the product differential

d (TS)

d F = - S d T - P d V

. Its natural variables are

(T, V)

, so

F_{T} = - S

and

F_{V} = - P

. Exactness therefore gives

(\frac{\partial S}{\partial V})_{T} = (\frac{\partial P}{\partial T})_{V} .

d G = - S d T + V d P

. Thus

G_{T} = - S

G_{P} = V

, and

(\frac{\partial S}{\partial P})_{T} = - (\frac{\partial V}{\partial T})_{P} .

For the enthalpy,

d H = T d S + V d P

H_{S} = T

H_{P} = V

and

(\frac{\partial T}{\partial P})_{S} = (\frac{\partial V}{\partial S})_{P} .

These are the four Maxwell relations, each following from exactness in the appropriate pair of natural variables.

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