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Maxwell relations on valid thermodynamic potential charts

Physical scope

Equality of mixed response derivatives on positive-temperature C2 single-phase charts where the named thermodynamic potentials are C2 functions of their local natural coordinates; no relation crosses a singular chart or phase boundary.

Statement

Let $U, H, F, G, Ω$ and their natural-coordinate charts be as in Local thermodynamic Legendre potentials, with the exact differentials supplied by Differentials of the thermodynamic potentials. Assume each potential is C2 on its named open chart. Every pair of conjugate coefficients in its exact differential has equal cross derivatives. Explicitly, with all unlisted natural coordinates held fixed, for each species $i, j$ :

From $U (s, V, n)$ : $T_{V} ∣_{s, n} = - p_{s} ∣_{V, n}$ ; $T_{n_{j}} ∣_{s, V, n} = μ_{j, s} ∣_{V, n}$ ; $- p_{n_{j}} ∣_{s, V, n} = μ_{j, V} ∣_{s, n}$ ; and $μ_{i, n_{j}} ∣_{s, V} = μ_{j, n_{i}} ∣_{s, V}$ .
From $H (s, p, n)$ : $T_{p} ∣_{s, n} = V_{s} ∣_{p, n}$ ; $T_{n_{j}} ∣_{s, p, n} = μ_{j, s} ∣_{p, n}$ ; $V_{n_{j}} ∣_{s, p, n} = μ_{j, p} ∣_{s, n}$ ; and $μ_{i, n_{j}} ∣_{s, p} = μ_{j, n_{i}} ∣_{s, p}$ .
From $F (T, V, n)$ : $s_{V} ∣_{T, n} = p_{T} ∣_{V, n}$ ; $s_{n_{j}} ∣_{T, V, n} = - μ_{j, T} ∣_{V, n}$ ; $- p_{n_{j}} ∣_{T, V, n} = μ_{j, V} ∣_{T, n}$ ; and $μ_{i, n_{j}} ∣_{T, V} = μ_{j, n_{i}} ∣_{T, V}$ .
From $G (T, p, n)$ : $- s_{p} ∣_{T, n} = V_{T} ∣_{p, n}$ ; $- s_{n_{j}} ∣_{T, p, n} = μ_{j, T} ∣_{p, n}$ ; $V_{n_{j}} ∣_{T, p, n} = μ_{j, p} ∣_{T, n}$ ; and $μ_{i, n_{j}} ∣_{T, p} = μ_{j, n_{i}} ∣_{T, p}$ .
From $Ω (T, V, μ)$ : $s_{V} ∣_{T, μ} = p_{T} ∣_{V, μ}$ ; $s_{μ_{j}} ∣_{T, V, μ_{k \neq j}} = n_{j, T} ∣_{V, μ}$ ; $p_{μ_{j}} ∣_{T, V, μ_{k \neq j}} = n_{j, V} ∣_{T, μ}$ ; and $n_{i, μ_{j}} ∣_{T, V, μ_{k \neq j}} = n_{j, μ_{i}} ∣_{T, V, μ_{k \neq i}}$ .

Facts & Assumptions

Given: The local C2 potential charts and their exact differentials. Temperature is positive; derivatives are taken only inside the relevant open natural-coordinate chart.

[F1]

Differentials of the thermodynamic potentials: The coefficient lists are $(T, - p, μ)$ for $U$ , $(T, V, μ)$ for $H$ , $(- s, - p, μ)$ for $F$ , $(- s, V, μ)$ for $G$ , and $(- s, - p, - n)$ for $Ω$ .

[F2]

Local thermodynamic Legendre potentials: The listed natural variables are local inverse coordinates only on the stated valid transform charts.

[L1]

Clairaut--Schwarz theorem for continuous second partial derivatives: For a C2 scalar function on an open Euclidean set, every pair of mixed second coordinate partials is equal.

Proof

1.1givenF1F2L1

For $U (s, V, n)$ the coefficient of $d s$ is $T$ , of $d V$ is $- p$ , and of $d n_{j}$ is $μ_{j}$ by [F1]. Apply [L1] to each unordered pair of its natural coordinates; the resulting equalities are exactly the four relations in the first bullet of the Statement.

1.2givenF1F2L1

For $H (s, p, n)$ the coefficients from [F1] are $T, V, μ_{j}$ . Equality of the mixed derivatives in [L1] for the pairs $(s, p)$ , $(s, n_{j})$ , $(p, n_{j})$ , and $(n_{i}, n_{j})$ gives the second bullet.

1.3givenF1F2L1algebra

For $F (T, V, n)$ the coefficients are $- s, - p, μ_{j}$ . Applying [L1] to $(T, V)$ , $(T, n_{j})$ , $(V, n_{j})$ , and $(n_{i}, n_{j})$ yields the third bullet, including its minus signs.

1.4givenF1F2L1algebra

For $G (T, p, n)$ the coefficients are $- s, V, μ_{j}$ . Applying [L1] to the same four types of coordinate pairs yields the fourth bullet.

2.1givenF1F2L1algebra∎

For $Ω (T, V, μ)$ the coefficients are $- s, - p, - n_{j}$ . Applying [L1] to $(T, V)$ , $(T, μ_{j})$ , $(V, μ_{j})$ , and $(μ_{i}, μ_{j})$ yields the fifth bullet. Each equality is confined to the same valid chart by [F2].

Maxwell relations on valid thermodynamic potential charts

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources