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Nonzero pressure work around a closed model cycle

Physical scope

Constructed smooth concave positive-temperature chart for a closed fixed-composition system in a fixed Galilean chart, coupled to an ideal planar constant-area piston in frictionless mechanical equilibrium; the work and heat values are model calculations, not reported observations.

Example

Fix a material constitution and amount $n$ and stipulate, on $U > 0$ and $0 < V < 1 / b$ , the entropy chart

$S (U, V) = C \ln (U / U_{*}) + a V - \frac{a b}{2} V^{2} + S_{0},$

where $C > 0$ is in $J K^{- 1}$ , $U_{*} > 0$ is in joules, $a > 0$ is in $J K^{- 1} m^{- 3}$ , $b > 0$ is in $m^{- 3}$ , and $S_{0}$ is in $J K^{- 1}$ . Let $s$ be a closed system of this constitution and amount. Stipulate that the process model admits a record $P \in P_{c l} (s)$ as in Thermodynamic process records and transfer conventions, using one fixed Galilean inertial chart and satisfying $Δ K_{c m} = Δ U_{e x t} = 0$ . Its equilibrium state path is piecewise-C1, starting at $(T_{1}, V_{1})$ and following, in order, a constant-temperature leg $T = T_{1}$ from $V_{1}$ to $V_{2}$ , a constant-volume leg $V = V_{2}$ to $T_{2}$ , a constant-temperature leg $T = T_{2}$ from $V_{2}$ to $V_{1}$ , and a constant-volume leg $V = V_{1}$ back to $T_{1}$ , where $T_{1} > T_{2} > 0$ and $0 < V_{1} < V_{2} < 1 / b$ . The moving boundary is a planar piston of constant area. Throughout the volume-changing legs it remains in frictionless mechanical equilibrium with a load tracking the equilibrium pressure, as required by Mechanical pressure-volume work coupling; controlled heat exchange is permitted, and no other work coordinate acts. In this model, the finite work transfer $W_{o n} [P]$ is defined as the line integral of the incremental piston work along this piecewise-C1 path. This is a stipulated idealized model record, not a reported observation. Its pressure-work integral is not an endpoint increment.

The displayed entropy is $S^{S I}$ in physical SI coordinates. For this constructed local model, choose the positive-amount open neighborhood stated here (or any positive neighborhood of the fixed amount, extending this entropy independently of amount) and let $O_{c}$ be the numerical preimage of that product patch under $Φ_{c} (\hat{U}, \hat{V}, \hat{n}) = (\hat{U} J, \hat{V} m^{3}, \hat{n} m o l)$ . Declare $O_{c}$ as the model's local state-domain patch and set $S_{c} = S^{S I} \circ Φ_{c}$ . The derivatives below are those of $S^{S I} = S_{c} \circ Φ_{c}^{- 1}$ on $Φ_{c} (O_{c})$ , exactly as in Entropic temperature, pressure, and molar chemical potentials. This supplies the numerical-to-SI chart identification for the formal witness, with no assertion of a measured material or a global constitutive extension.

Facts & Assumptions

Given: The stipulated entropy chart, constants and SI units in the Example, and the model assumption that its process family admits the closed core record $P \in P_{c l} (s)$ described there, in one fixed Galilean chart and with $Δ K_{c m} = Δ U_{e x t} = 0$ . Its state path is piecewise-C1; the moving boundary is a planar constant-area piston in frictionless mechanical equilibrium with a load tracking the model pressure. Controlled heat exchange is permitted and no other work coordinate acts. No empirical realization is claimed.

[F1]

Equilibrium state domains and SI quantities and unit conventions: The chart uses positive physical energy and volume in joules and cubic metres; entropy is in $J K^{- 1}$ .

[F2]

The natural logarithm as the inverse of the exponential function and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\ln$ is defined on positive arguments and has derivative $1 / x$ .

[F3]

Entropic temperature, pressure, and molar chemical potentials: $T = 1 / S_{U}$ and $p = T S_{V}$ on a positive-temperature chart.

[F4]

Mechanical pressure-volume work coupling: On the admitted quasistatic frictionless piston path, the incremental work on the system is $δ W_{o n} = - p d V$ ; this constructed protocol defines its finite $W_{o n} [P]$ as the path integral of those increments along the four legs.

[F5]

First-law energy balance for closed process records and Thermodynamic process records and transfer conventions: For the admitted closed cycle, $Δ U = Q + W_{o n}$ with heat positive into and work positive on the system.

[L1]

Reversal, concatenation, closed paths, and oriented piecewise-C1 reparametrizations and Scalar line integrals with respect to arc length and vector-field line integrals: For a piecewise-C1 path $γ$ in the $(T, V)$ chart, $\int_{γ} (0, p) \cdot d (T, V)$ is the Riemann line integral $\int p d V$ .

[L2]

Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative and A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion: Continuous integrands on each compact segment are Riemann integrable and their polynomial antiderivatives give the segment integrals.

[L3]

The gradient theorem: the line integral of a gradient is the endpoint increment: The integral of a gradient along a closed piecewise-C1 path is zero.

[L4]

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.

Verification

1.1givenF1F2F3L4algebra

By [F2] and [L4], the entropy derivatives are $S_{U} = C / U$ and $S_{V} = a (1 - b V)$ . Thus [F3] gives $T = U / C > 0$ and $p = a T (1 - b V) > 0$ on the stated domain. The Hessian is diagonal with entries $- C / U^{2}$ and $- a b$ , so the chart is C2 and concave.

2.1givenL1L2step 1.1algebra

On each isotherm the vector field in [L1] is continuous, so [L2] gives $\int_{V_{1}}^{V_{2}} p (T_{1}, V) d V = a T_{1} [(V_{2} - V_{1}) - \frac{b}{2} (V_{2}^{2} - V_{1}^{2})]$ ; the reverse isotherm contributes the negative of the same bracket multiplied by $a T_{2}$ , and the isochoric legs contribute zero.

3.1givenF4step 2.1algebra

Summing the four legs yields $\oint p d V = a (T_{1} - T_{2}) (V_{2} - V_{1}) [1 - \frac{b}{2} (V_{1} + V_{2})] > 0$ , since $T_{1} > T_{2}$ and $V_{1}, V_{2} < 1 / b$ . Thus $W_{o n} = - \oint p d V < 0$ by [F4].

4.1F5L3step 3.1algebra∎

The cycle returns to the same $U$ , so [F5] gives $Q = - W_{o n} = \oint p d V > 0$ . If there were a $C^{1}$ scalar potential $ϕ$ on an open neighborhood of the loop with $d ϕ = p d V$ (equivalently, $\nabla_{(T, V)} ϕ = (0, p)$ ), [L3] would force its closed integral to be zero, contradicting step 3.1. Thus no such endpoint potential represents pressure work on a neighborhood containing this loop; the work depends on the path.

Nonzero pressure work around a closed model cycle

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources