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Heat-capacity relation on a stable response chart

Physical scope

Whole-system heat capacities and response coefficients on a C2, positive-temperature, fixed-composition simple-compressible chart with a valid pressure-volume response inverse and positive isothermal compressibility; singular and critical charts are excluded.

Statement

On an open C2 Helmholtz chart $F (T, V, n)$ at fixed composition $n$ , suppose $T > 0$ , $V > 0$ , the response coordinates $(T, p, n)$ have the local inverse required by Thermodynamic response functions on valid local charts, and $κ_{T} > 0$ . Then

$C_{p} - C_{V} = \frac{T V α_{p}^{2}}{κ_{T}} .$

Here $C_{V}, C_{p}$ are the whole-system heat capacities, $α_{p} = V^{- 1} (\partial V / \partial T)_{p, n}$ , and $κ_{T} = - V^{- 1} (\partial V / \partial p)_{T, n}$ , all as in Thermodynamic response functions on valid local charts. The formula is a conditional model identity, not an empirical law or a claim at singular response points.

Facts & Assumptions

Given: A fixed-composition open C2 chart with $T, V > 0$ , the valid local pressure response inverse, and finite derivatives in the Statement.

[F1]

Thermodynamic response functions on valid local charts: $C_{V} = (\partial U / \partial T)_{V, n}$ , $C_{p} = (\partial H / \partial T)_{p, n}$ , $κ_{T} = - V^{- 1} (\partial V / \partial p)_{T, n}$ , and $α_{p} = V^{- 1} (\partial V / \partial T)_{p, n}$ ; the definitions assert no response identities.

[F2]

Differentials of the thermodynamic potentials: $d F = - s d T - p d V + \sum_{j} μ_{j} d n_{j}$ and $d H = T d s + V d p + \sum_{j} μ_{j} d n_{j}$ on valid charts.

[F3]

Maxwell relations on valid thermodynamic potential charts: $s_{V} ∣_{T, n} = p_{T} ∣_{V, n}$ on the Helmholtz chart.

[F4]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : Derivatives of compositions obey the total-derivative chain rule.

[F5]

Chain sum product and composition rules for Banach derivatives: The derivative of a product of differentiable scalar maps is the sum of the two product terms.

Proof

1.1givenF2F5algebra

At fixed $V, n$ , $U = F + T s$ and $F_{T} = - s$ from [F2], so the product rule [F5] gives $C_{V} = U_{T} ∣_{V, n} = T s_{T} ∣_{V, n}$ . At fixed $p, n$ , $d H = T d s + V d p$ from [F2], hence $C_{p} = H_{T} ∣_{p, n} = T s_{T} ∣_{p, n}$ .

1.2givenF1F4algebra

On the valid isothermal response inverse, write $V = V (T, p, n)$ , so $p (T, V (T, p, n), n) = p$ , where the rightmost $p$ is the independent pressure coordinate. Differentiating in that pressure coordinate at fixed $T, n$ gives $1 = p_{V} ∣_{T, n} V_{p} ∣_{T, n}$ . By [F1], $V_{p} ∣_{T, n} = - V κ_{T}$ , hence $p_{V} ∣_{T, n} = - 1 / (V κ_{T})$ . Differentiating the same identity in $T$ at fixed $p, n$ gives $0 = p_{T} ∣_{V, n} + p_{V} ∣_{T, n} V_{T} ∣_{p, n}$ . Again by [F1], $V_{T} ∣_{p, n} = V α_{p}$ , so $p_{T} ∣_{V, n} = α_{p} / κ_{T}$ . The stated inverse chart and $κ_{T} > 0$ make these derivatives and reciprocals valid.

2.1F1F2F3F4step 1.1algebra

Along a constant- $p$ , fixed- $n$ curve, [F4] gives $s_{T} ∣_{p, n} = s_{T} ∣_{V, n} + s_{V} ∣_{T, n} V_{T} ∣_{p, n}$ . Subtract the two identities in step 1.1 and use [F3] to obtain $C_{p} - C_{V} = T p_{T} ∣_{V, n} V_{T} ∣_{p, n}$ .

3.1F1step 2.1step 1.2algebra∎

Substitute $p_{T} ∣_{V, n} = α_{p} / κ_{T}$ and $V_{T} ∣_{p, n} = V α_{p}$ into step 2.1 to obtain $C_{p} - C_{V} = T V α_{p}^{2} / κ_{T}$ . Since $κ_{T} > 0$ , no zero denominator is used. The units are $J K^{- 1}$ on both sides because $P a m^{3} = J$ .

Heat-capacity relation on a stable response chart

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources