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Reaction equilibrium conditions at interior and boundary amounts

Physical scope

Local C1 constrained Gibbs-potential charts for a declared internal reaction network at fixed positive reservoir temperature and pressure; interior conclusions use the admitted process-record model, while boundary conclusions require the explicit boundary-inclusive entropy and auxiliary extension stated here.

Statement

Fix $C \geq 1$ species and $R \geq 1$ declared internal reaction extents. Let $ν \in R^{C \times R}$ be the dimensionless stoichiometric matrix, let $n^{0} \in [0, \infty)^{C}$ have units mol, and let $ξ \in R^{R}$ have units mol. The amount vector is $n (ξ) = n^{0} + ν ξ$ , and the nonnegative-amount feasible extent set is

F_{ν} : = {ξ \in R^{R} : n^{0} + ν ξ \in [0, \infty)^{C}} .

Assume the exact local reaction-feasible extent set is $F_{ν} \cap U$ ; no additional local constraint removes any direction in its tangent cone. Fix $T_{0} > 0$ and $p_{0} > 0$ in a declared natural-variable chart. On an open neighborhood $U$ of a candidate endpoint $ξ_{*}$ , let $Φ : U \to R$ be a finite $C^{1}$ constrained Gibbs-potential extension, in joules, satisfying

D Φ (ξ) [h] = \sum_{j = 1}^{C} μ_{j} (ξ) (ν h)_{j} = (ν^{T} μ (ξ)) \cdot h,

where every $μ_{j}$ is in $J m o l^{- 1}$ . At positive amounts, require $Φ (ξ) = U_{c} (x (ξ)) + p_{0} V_{c} (x (ξ)) - T_{0} S_{c} (x (ξ))$ for the system state $x (ξ)$ , with $U_{c}, V_{c}, S_{c}$ from Equilibrium state domains and Concave macroscopic fundamental entropy; its derivative is that of the local Gibbs chart supplied by Local thermodynamic Legendre potentials and Differentials of the thermodynamic potentials.

If all amounts at $ξ_{*}$ are positive, assume $ξ_{*}$ is the system endpoint of a closed system process record and its admitted isolated-compound extension exactly as in Fixed-temperature and fixed-pressure potential minimum with internal reactions, is a global entropy-maximizing endpoint on that extension's exact feasible set, and the record's system projection contains every state $x (ξ)$ for $ξ \in F_{ν} \cap U$ . If at least one amount is zero, it is outside Equilibrium state domains; in that boundary case, assume instead that the model supplies a boundary-inclusive state domain with finite $C^{1}$ maps ${\overset{ˉ}{U}}_{c}, {\overset{ˉ}{V}}_{c}, {\overset{ˉ}{S}}_{c} : U \to R$ in joules, $m^{3}$ , and $J K^{- 1}$ , respectively, that $Φ (ξ) = {\overset{ˉ}{U}}_{c} (ξ) + p_{0} {\overset{ˉ}{V}}_{c} (ξ) - T_{0} {\overset{ˉ}{S}}_{c} (ξ)$ , and that its exact isolated-compound model has fixed total energy, an ideal reservoir with $S_{R} (E_{R}) = S_{R, 0} + E_{R} / T_{0}$ , a constant-entropy pressure store satisfying $E_{B} - p_{0} {\overset{ˉ}{V}}_{c} (ξ) = C_{B}$ , and a feasible-set projection equal to $F_{ν} \cap U$ . Assume $ξ_{*}$ is an attained global entropy maximum on that boundary-inclusive set. This additional boundary model premise is not supplied by the positive-mole state chart or by Fixed-temperature and fixed-pressure potential minimum with internal reactions.

Under the relevant interior or boundary assumptions above, $ξ_{*}$ locally minimizes $Φ$ on $F_{ν} \cap U$ . Let $J_{*} : = {j : n_{j}^{0} + (ν ξ_{*})_{j} = 0}$ be the active species set. Then the first-order necessary condition is

(ν^{T} μ (ξ_{*})) \cdot h \geq 0 for every h \in T_{F_{ν}} (ξ_{*}),

where

T_{F_{ν}} (ξ_{*}) : = {h \in R^{R} : (ν h)_{j} \geq 0 for every j \in J_{*}} .

If $ξ_{*}$ is interior to $F_{ν}$ , this reduces to $ν^{T} μ (ξ_{*}) = 0$ . If, in addition, $Φ$ is convex and differentiable on a convex neighborhood containing the relevant feasible set, the same tangent inequality is sufficient for global minimization there.

Facts & Assumptions

Given: The finite species list, stoichiometric matrix, nonnegative-amount feasible set, fixed $T_{0} > 0, p_{0} > 0$ , potential $Φ$ and its C1 extension and gradient identity. In the interior case, $ξ_{*}$ is a positive-amount endpoint of the admitted process record used in the fixed-potential theorem. In the boundary case, use a separately declared boundary-inclusive state and auxiliary model as described in [A2]; zero-amount points are not in the positive-mole state chart.

[A1]

Boundary-chart hypothesis: $Φ$ is finite and C1 on an open subset $U \subseteq R^{R}$ containing $ξ_{*}$ , and its derivative along the affine extent map is the displayed stoichiometric chemical-potential expression throughout $U$ . At positive amounts, the expression follows from the local Gibbs differential of [F2]; at zero amounts it is an added model extension.

[A2]

Boundary-inclusive physical model: for the boundary route only, the model declares a state domain containing the zero-amount state at $ξ_{*}$ and finite C1 maps ${\overset{ˉ}{U}}_{c}, {\overset{ˉ}{V}}_{c}, {\overset{ˉ}{S}}_{c} : U \to R$ with units J, $m^{3}$ , and $J K^{- 1}$ , and $Φ = {\overset{ˉ}{U}}_{c} + p_{0} {\overset{ˉ}{V}}_{c} - T_{0} {\overset{ˉ}{S}}_{c}$ . Its isolated compound has an exact feasible set of tuples $(ξ, y_{R}, y_{B})$ whose projection onto extents is $F_{ν} \cap U$ , fixed total energy ${\overset{ˉ}{U}}_{c} (ξ) + E_{R} (y_{R}) + E_{B} (y_{B})$ , ideal-reservoir entropy $S_{R} (E_{R}) = S_{R, 0} + E_{R} / T_{0}$ , constant store entropy, and pressure-store relation $E_{B} (y_{B}) - p_{0} {\overset{ˉ}{V}}_{c} (ξ) = C_{B}$ throughout that feasible set. The endpoint corresponding to $ξ_{*}$ is an attained global maximum of the sum of system, reservoir, and store entropies on this exact set. These are explicit additional model premises, not consequences of the positive-mole state domain.

[F1]

Fixed-temperature and fixed-pressure potential minimum with internal reactions: For an admitted closed process record with the specified ideal reservoir and pressure store, its system potential decreases on every record; an attained global entropy maximum on that exact isolated-compound feasible set gives a global minimum on the projected system state set. For this interior application, that projection contains the system states corresponding to every extent in $F_{ν} \cap U$ . The supplier does not cover zero amounts outside $D_{c}$ .

[F2]

Differentials of the thermodynamic potentials: On a valid single-phase energy-representation and Gibbs natural-variable chart, $d G = - S d T + V d p + \sum_{j} μ_{j} d n_{j}$ . The identity is used only at positive amounts; at the boundary, [A1] is the separate derivative premise.

[F3]

The feasible tangent cone for nonnegative stoichiometric amounts: For the nonnegative stoichiometric feasible polyhedron, the tangent cone consists exactly of $h$ with $(ν h)_{j} \geq 0$ for every active species. At a differentiable local minimum, the directional derivative is nonnegative on that cone; for a differentiable convex objective on a convex domain, this condition is sufficient for a global minimum.

Proof

1.1givenF1

In the positive-amount case, [F1] gives a local minimum of $Φ$ relative to the exact reaction-feasible states near $ξ_{*}$ .

1.2givenA2algebra

In the boundary case, fixed total energy and $E_{B} - p_{0} {\overset{ˉ}{V}}_{c} = C_{B}$ in [A2] give $E_{R} = C_{1} - {\overset{ˉ}{U}}_{c} (ξ) - p_{0} {\overset{ˉ}{V}}_{c} (ξ)$ for a constant $C_{1}$ . Therefore the sum of system and reservoir entropies, plus the constant store entropy, is $S_{t o t} = C_{0} + {\overset{ˉ}{S}}_{c} (ξ) - ({\overset{ˉ}{U}}_{c} (ξ) + p_{0} {\overset{ˉ}{V}}_{c} (ξ)) / T_{0} = C_{0} - Φ (ξ) / T_{0}$ on the exact boundary-inclusive feasible set. Since $T_{0} > 0$ , an attained global entropy maximum at $ξ_{*}$ is a global, hence local, minimum of $Φ$ there.

2.1step 1.1step 1.2

Thus in either case $ξ_{*}$ is a local minimum relative to $F_{ν} \cap U$ .

3.1givenA1F2F3step 2.1

Apply [F3] at $ξ_{*}$ . It identifies the feasible tangent directions as exactly those $h$ satisfying $(ν h)_{j} \geq 0$ for all $j \in J_{*}$ . The differentiable local-minimum inequality gives $D Φ (ξ_{*}) [h] \geq 0$ on this cone; substituting [A1]'s gradient identity gives $(ν^{T} μ (ξ_{*})) \cdot h \geq 0$ .

4.1givenstep 3.1algebra

If $ξ_{*}$ is interior to $F_{ν}$ , an open neighborhood of it lies in $F_{ν}$ , so every direction is feasible for sufficiently small positive extent and $T_{F_{ν}} (ξ_{*}) = R^{R}$ . For each coordinate $k$ , let $e_{k}$ be the $k$ th standard basis vector and use the two directions $h = \pm (1 m o l) e_{k}$ . Step 3.1 gives $\pm (1 m o l) (ν^{T} μ (ξ_{*}))_{k} \geq 0$ , so each component vanishes and $ν^{T} μ (ξ_{*}) = 0$ .

5.1givenF3step 3.1∎

If $Φ$ is convex and differentiable on a convex neighborhood containing the relevant feasible set, the sufficiency clause of [F3] makes the tangent inequality sufficient for global minimization there. Without convexity this sufficiency conclusion is not asserted.

Remarks

For the interior route, the potential minimum follows from the admitted process record and its reservoir/store model. A boundary point with a zero species amount is not in the positive-mole chart. The boundary route therefore needs the explicitly extended entropy and endpoint model in [A2]; a finite C1 extension of $Φ$ alone would not establish physical stable equilibrium.

Reaction equilibrium conditions at interior and boundary amounts

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources