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Helmholtz and Gibbs potential minimum principles for ideal reservoirs

Physical scope

Conditional endpoint deductions for closed, fixed-composition macroscopic systems coupled to the explicitly defined ideal thermal and pressure reservoirs; other work coordinates, open matter flow, and nonideal finite reservoirs are excluded.

Statement

Fix a closed macroscopic system with composition fixed and two admitted equilibrium endpoints. In the fixed-volume model, couple it only to an ideal thermal reservoir at constant $T_{0} > 0$ . In the fixed-pressure model, also couple it to an ideal pressure store at constant $p_{0} > 0$ . Define the constrained potentials

$F_{T_{0}} (x) : = U (x) - T_{0} S (x), G_{T_{0}, p_{0}} (x) : = U (x) + p_{0} V (x) - T_{0} S (x) .$

Assume the ideal-reservoir bookkeeping specified in [A1]–[A3] of Facts & Assumptions and the adopted isolated-compound second law Macroscopic second law for isolated compounds. Then for every admitted process record of the isolated composite consisting of the system and its stated reservoir, with the pressure store included in the fixed-pressure case, $F_{T_{0}}$ does not increase at fixed volume, and $G_{T_{0}, p_{0}}$ does not increase at fixed pressure. If an endpoint is an attained global entropy maximum on the same feasible set, the corresponding constrained potential attains a global minimum there. If the system endpoint is on its $T = T_{0}$ or $(T, p) = (T_{0}, p_{0})$ natural-variable chart, these constrained functions agree with the Helmholtz or Gibbs potentials of Local thermodynamic Legendre potentials.

The statement is conditional on the ideal bath/load model and is not a claim that every physical process has such a reservoir or reaches the minimum.

Facts & Assumptions

Given: A fixed composition, an explicit feasible endpoint set for a closed system plus its named reservoirs, equilibrium endpoints, fixed excluded kinetic and external-field energy changes, and the following ideal reservoir model. Energies are in joules, entropy in $J K^{- 1}$ , $T_{0}$ in kelvins, $V$ in $m^{3}$ , and $p_{0}$ in pascals.

[A1]

Ideal thermal reservoir: its energy $E_{R}$ ranges in an open interval and its entropy is $S_{R} (E_{R}) = S_{R, 0} + E_{R} / T_{0}$ there. Heat $Q$ positive into the system changes its energy by $Δ E_{R} = - Q$ , so $Δ S_{R} = - Q / T_{0}$ .

[A2]

Ideal pressure store: at fixed external pressure $p_{0}$ , its stored energy changes by $p_{0} Δ V$ when the system volume changes by $Δ V$ ; the only system work is $W_{o n} = - p_{0} Δ V$ . The store's entropy is constant. At fixed volume this store is absent. No other work coordinate or matter transfer is admitted.

[A3]

The system and bath, with the pressure store when present, form a complete isolated compound with fixed total energy. In the fixed-volume case $Δ U + Δ E_{R} = 0$ ; in the fixed-pressure case $Δ U + Δ E_{R} + p_{0} Δ V = 0$ . These system energy balances are adopted ideal-environment constraints. Each operation includes an admitted core closed record of the complete compound as in Thermodynamic process records and transfer conventions, with positive fixed total mass, defined endpoint center-of-mass velocities in one fixed Galilean chart, and zero endpoint changes of compound center-of-mass kinetic and separately tracked external-field energies. The model supplies the full constituent and auxiliary endpoint sets with their energy and entropy maps, and an endpoint association $ρ_{P} : {-, +} \to F$ into one exact feasible set, representing both aggregate equilibrium endpoints and their system, bath and store coordinates. Every named auxiliary is retained; the endpoint split is admitted by the additive rule required by Macroscopic second law for isolated compounds. Any additional retained auxiliary has constant total energy and entropy contributions throughout $F$ . On every tuple of $F$ , total energy is one fixed constant and total entropy is the sum of the system, bath and constant store or other entropy-neutral contributions. In the fixed-pressure case require $E_{p} - p_{0} V = C_{p}$ on every tuple of that same $F$ for one fixed constant $C_{p}$ ; at fixed volume $V$ is fixed and the pressure store is absent. Thus the constraints used for global minimization hold throughout the feasible set, including disconnected components. No core-record assumption is imposed on the subsystem alone: its displayed energy balances are already adopted above.

[F2]

Macroscopic second law for isolated compounds: on the admitted isolated core compound record with its full endpoint association, admitted additive split and zero excluded-energy changes specified in [A3], $Δ S_{t o t} \geq 0$ .

[F3]

Stable equilibrium requires an attained global entropy maximum: Stable equilibrium is an attained global maximum of the sum of constituent entropies on the explicitly declared feasible set; uniqueness is not assumed.

[F4]

Local thermodynamic Legendre potentials: The local Helmholtz and Gibbs potentials are $F = U - T S$ and $G = U - T S + p V$ on their valid natural-variable charts.

[F5]

SI quantities and unit conventions: $U$ , $Q$ , and work transfers are in joules, $T$ in kelvins, $S$ in $J K^{- 1}$ , $p$ in pascals, and $V$ in $m^{3}$ , with $P a m^{3} = J$ .

Proof

1.1givenA1A2A3F2F5algebra

In the fixed-volume case, the adopted balance [A3] and bath relation [A1] give $Δ U = Q$ . The bath relation [A1] gives $Δ S_{t o t} = Δ S - Q / T_{0}$ . By [F2], $Δ S_{t o t} \geq 0$ , and therefore $Δ F_{T_{0}} = Q - T_{0} Δ S = - T_{0} Δ S_{t o t} \leq 0$ .

1.2givenA1A2A3F2F5algebra

In the fixed-pressure case, [A1] and the adopted balance [A3] give $Δ U = Q - p_{0} Δ V$ , while [A1] gives $Δ S_{t o t} = Δ S - Q / T_{0}$ . Hence $Δ G_{T_{0}, p_{0}} = Δ U + p_{0} Δ V - T_{0} Δ S = Q - T_{0} Δ S = - T_{0} Δ S_{t o t} \leq 0$ .

1.3A1A3F3F4algebra

For fixed volume, [A1] and [A3] express total entropy on the feasible set as $S_{t o t} = c o n s t a n t - F_{T_{0}} / T_{0}$ . Since $T_{0} > 0$ , maximizing total entropy is equivalent to minimizing $F_{T_{0}}$ ; on the natural-variable chart the function is the Helmholtz potential by [F4].

2.1A1A2A3F3F4algebra∎

For fixed pressure, [A1]–[A3] express total entropy as $S_{t o t} = c o n s t a n t - G_{T_{0}, p_{0}} / T_{0}$ . Again $T_{0} > 0$ , so every attained global entropy maximizer on that feasible set is a global minimizer of $G_{T_{0}, p_{0}}$ ; on the natural-variable chart it is the Gibbs potential by [F4]. Neither equivalence implies uniqueness or attainment beyond [F3].

Remarks

The assumptions define ideal reservoirs, not measured reservoirs. A finite bath generally has varying temperature and does not satisfy [A1] exactly. The result does not cover an open flowing control volume or other generalized work coordinates.

Helmholtz and Gibbs potential minimum principles for ideal reservoirs

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources