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A regular two-phase three-species reaction model

Physical scope

A constructed smooth local two-phase Gibbs model with one formal internal reaction, used to verify the local equation-map nonsingular-minor hypothesis only; no stable or realized coexistence claim, empirical system, or chemical data are made.

Example

Take $C = 3$ species and $P = 2$ phases. Write $x_{3} : = 1 - x_{1} - x_{2}$ , and center the open composition charts at $x_{1} = x_{2} = x_{3} = 1 / 3$ . Fix $T_{0} > 0$ , $p_{0} > 0$ , $s_{0}, c, v_{0}, κ, q > 0$ , and $η = 1 J m o l^{- 1}$ . The parameters have units $s_{0}, c \in J m o l^{- 1} K^{- 1}$ , $v_{0} \in m^{3} m o l^{- 1}$ , $κ \in m^{3} m o l^{- 1} P a^{- 1}$ , and $q \in m^{3} m o l^{- 1}$ . For $γ \in {α, β}$ , set $a_{α} = 2 η$ , $a_{β} = η$ , $q_{α} = 0$ , and $q_{β} = q$ , and define molar Gibbs potentials

g^{γ} (T, p, x) = - s_{0} (T - T_{0}) - \frac{c}{2 T_{0}} (T - T_{0})^{2} + v_{0} (p - p_{0}) - \frac{κ}{2} (p - p_{0})^{2} + \frac{a_{γ}}{2} (x_{1}^{2} + x_{2}^{2}) - \frac{a_{γ}}{3} (x_{1} + x_{2}) + \frac{a_{γ}}{9} + q_{γ} (p - p_{0}) x_{3} .

Set $G^{γ} = m_{γ} g^{γ}$ on each fixed-tag branch and adopt the natural Gibbs assignments $S = - G_{T}$ , $V = G_{p}$ , and $μ_{j} = G_{n_{j}}$ , with $U = G + T S - p V$ . These smooth charts are scale-invariant in amount. Near the centered point the volumes are positive, and the $(U, V)$ chart Jacobian at fixed composition and amount is nonsingular: $U_{T} = m_{γ} c$ , $V_{p} = - m_{γ} κ$ , and $V_{T} = 0$ there, so its determinant is $- m_{γ}^{2} c κ \neq 0$ . The reaction vector is $ν = (- 1, - 1 / 2, 1)^{T}$ , of rank $R = 1$ . At $(T_{0}, p_{0}, x_{1}, x_{2}) = (T_{0}, p_{0}, 1 / 3, 1 / 3)$ , both phases have $μ^{γ} = (0, 0, 0)$ , so the phase-equality equations and reaction affinity vanish.

For each branch, fix a reference amount $m_{*} > 0$ in moles. Choose an open neighborhood $O$ of $(T_{0}, p_{0}, 1 / 3, 1 / 3, m_{*})$ in $(0, \infty)^{2} \times {(x_{1}, x_{2}) : x_{1}, x_{2} > 0, x_{1} + x_{2} < 1} \times (0, \infty)$ , using $n = (m x_{1}, m x_{2}, m (1 - x_{1} - x_{2}))$ . Shrink this neighborhood to the inverse-function patch verified below; all component amounts remain positive. The constructed branch's local physical state map is $χ : (T, p, x_{1}, x_{2}, m) \mapsto (U, V, m x_{1}, m x_{2}, m (1 - x_{1} - x_{2}))$ , on the positive amount and interior composition domain just specified. On the inverse patch verified below, define its numerical domain $O_{c} = Φ_{c}^{- 1} (χ (O))$ and declare this patch to be the local model's $D_{c}$ patch, where $Φ_{c} (\hat{U}, \hat{V}, {\hat{n}}_{1}, {\hat{n}}_{2}, {\hat{n}}_{3}) = (\hat{U} J, \hat{V} m^{3}, {\hat{n}}_{1} m o l, {\hat{n}}_{2} m o l, {\hat{n}}_{3} m o l)$ is the unit map of Entropic temperature, pressure, and molar chemical potentials. Define the numerical entropy by $S_{c} = S^{S I} \circ Φ_{c}$ , with $S^{S I}$ supplied by the Gibbs assignments through this inverse. Thus the displayed derivatives are derivatives in physical SI coordinates on $Φ_{c} (O_{c})$ ; the model patch is not an empirical claim about an independently specified material domain.

Facts & Assumptions

Given: The two Gibbs branches, their conjugate assignments and parameter units in the Example, the open positive-temperature and positive-pressure simplex charts, and the reaction vector $ν$ . These define a constructed local model for checking the rank condition; no measured or realized material is claimed.

[A1]

Adopted local model: The C2 function $G (T, p, n)$ has conjugates $S = - G_{T}$ , $V = G_{p}$ , and $μ_{j} = G_{n_{j}}$ , with $U = G + T S - p V$ . These relations define the model's natural Gibbs chart.

[F1]

Regular rank form of the finite-component phase dimension count: For $C = 3$ , $P = 2$ , $R = 1$ , a full-rank combined equation map has local equation-zero-set dimension $C - P + 2 - R = 2$ .

[F2]

Entropic temperature, pressure, and molar chemical potentials: The derivative-defined quantities obey $S_{U} = 1 / T$ , $S_{V} = p / T$ , and $S_{n_{j}} = - μ_{j} / T$ in a valid physical state chart.

[F3]

The model and calculation verify only the equation-map rank and dimension; no stable or realized coexistence endpoint is assumed.

[L1]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : The derivative of a differentiable composition is the composition of its derivatives.

[L2]

The Euclidean inverse function theorem: A C1 map between open Euclidean neighborhoods with invertible derivative at a point has a local C1 inverse.

[L3]

$C^{k}$ Euclidean maps are closed under componentwise algebra and composition preserves $C^{1}$ under finite algebra and composition. Continuity and derivatives of positive-base real powers gives $(t^{- 1})^{'} = - t^{- 2}$ on $t > 0$ , hence the reciprocal is $C^{1}$ there.

Verification

1.1givenA1F2L1L2L3algebra

From the displayed $g^{γ}$ and [A1], at the center $S_{T} = m_{γ} c / T_{0}$ , $U_{T} = m_{γ} c$ , $V_{T} = 0$ , and $V_{p} = - m_{γ} κ$ , so the fixed-composition/amount $(T, p)$ Jacobian for $(U, V)$ is $- m_{γ}^{2} c κ$ . The map $(x_{1}, x_{2}, m_{γ}) \mapsto (n_{1}, n_{2}, n_{3})$ has determinant $m_{γ}^{2} (x_{1} + x_{2} + x_{3}) = m_{γ}^{2}$ . Hence the full state-map Jacobian is $- m_{γ}^{4} c κ \neq 0$ , and [L2] gives the specified open full-state inverse patch, which can be shrunk to retain positive temperature, volume and amounts. On this patch, [L1] and the conjugate assignments give $d G = - S d T + V d p + \sum_{j} μ_{j} d n_{j}$ and $d U = T d S - p d V + \sum_{j} μ_{j} d n_{j}$ , so $d S = T^{- 1} d U + (p / T) d V - \sum_{j} (μ_{j} / T) d n_{j}$ . The inverse coordinates and chemical potentials are $C^{1}$ ; by [L3], all these coefficients are $C^{1}$ . Thus the entropy representative is $C^{2}$ and has $S_{U} = 1 / T > 0$ , $S_{V} = p / T$ , and $S_{n_{j}} = - μ_{j} / T$ , verifying the derivative-defined quantities of [F2].

2.1givenA1L1step 1.1algebra

For $m = n_{1} + n_{2} + n_{3}$ and $x_{j} = n_{j} / m$ , [L1] at fixed $T, p$ and the other two species amounts gives

μ_{1} = g + (1 - x_{1}) g_{x_{1}} - x_{2} g_{x_{2}}, μ_{2} = g - x_{1} g_{x_{1}} + (1 - x_{2}) g_{x_{2}}, μ_{3} = g - x_{1} g_{x_{1}} - x_{2} g_{x_{2}} .

Substitution of the two stated $g^{γ}$ at the centered point gives $μ^{α} = μ^{β} = (0, 0, 0)$ . Thus $F_{i} = μ_{i}^{β} - μ_{i}^{α} = 0$ and $A = ν^{T} μ^{α} = 0$ .

3.1givenstep 2.1algebra

From the formulas in step 2.1, at the centered point

D_{(x_{1}, x_{2})} μ^{γ} = (\begin{array}{cc} 2 a_{γ} / 3 & - a_{γ} / 3 \\ - a_{γ} / 3 & 2 a_{γ} / 3 \\ - a_{γ} / 3 & - a_{γ} / 3 \end{array}), \partial_{p} μ^{γ} = (v_{0}, v_{0}, v_{0} + q_{γ})^{T} .

Using $a_{β} = η$ , $a_{α} = 2 η$ , $q_{β} = q$ , and $q_{α} = 0$ , the derivative minor of $(F_{1}, F_{2}, F_{3}, A)$ in columns $(x_{1}^{β}, x_{2}^{β}, p, x_{1}^{α})$ is

(\begin{array}{cccc} 2 η / 3 & - η / 3 & 0 & - 4 η / 3 \\ - η / 3 & 2 η / 3 & 0 & 2 η / 3 \\ - η / 3 & - η / 3 & q & 2 η / 3 \\ 0 & 0 & - v_{0} / 2 & - 5 a_{α} / 6 \end{array}) .

The cofactor of the last-row pressure entry is zero because, after deleting that row and column, the fourth column is $- 2$ times the first. Expansion along the last row therefore gives determinant $(- 5 a_{α} / 6) q (η^{2} / 3) \neq 0$ . The combined equation map therefore has the required rank four.

4.1F1F3step 3.1algebra∎

The ambient coordinate dimension is $d = 2 + P (C - 1) = 6$ , and the number of equations is $q_{e q} + R = C (P - 1) + R = 3 + 1 = 4$ , where $q_{e q} : = C (P - 1)$ is a dimensionless equation count distinct from the volume-per-mole parameter $q$ . By [F1], the local equation zero set has dimension $6 - 4 = 2$ , agreeing with $C - P + 2 - R = 3 - 2 + 2 - 1 = 2$ . This is only the full-rank equation-set calculation.

A regular two-phase three-species reaction model

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

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