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Water formation at a boundary of the feasible extent interval

Example

For the reaction $2 H_{2} + O_{2} \to 2 H_{2} O$ , let extent $ξ$ and species amounts use moles. The dimensionless stoichiometric column is $ν = (- 2, - 1, 2)^{T}$ in the species order $(H_{2}, O_{2}, H_{2} O)$ , and take $n^{0} = (2, 1, 0)^{T}$ . Then the feasible extent interval is $[0, 1]$ , and at $ξ = 0$ the tangent cone permits only forward directions $h \geq 0$ . The matrix of hydrogen and oxygen atom counts annihilates $ν$ . For the explicitly mathematical objective $Φ (n) = n_{H_{2} O}$ , the induced extent function is $f (ξ) = 2 ξ$ and has a boundary minimum at zero. This is a stoichiometric calculation, not a claim about chemical equilibrium or a measured system.

Facts & Assumptions

Given: The reaction vector, initial amounts, extent convention, and linear objective in the Example.

[L1]

The feasible tangent cone for nonnegative stoichiometric amounts: the feasible tangent cone is determined by nonnegative changes in species whose amounts are zero at the point; the extent derivative is $ν^{T} μ$ for a differentiable potential.

Verification

technique · direct calculation

1.1givenalgebra

The species-amount vector is $n (ξ) = n^{0} + ν ξ = (2 - 2 ξ, 1 - ξ, 2 ξ)^{T}$ . Nonnegativity of its third component requires $ξ \geq 0$ , and nonnegativity of its first two components requires $ξ \leq 1$ ; hence the feasible set is exactly $[0, 1]$ .

1.2givenalgebra

The elemental count matrix is $D = (\begin{array}{ccc} 2 & 0 & 2 \\ 0 & 2 & 1 \end{array})$ , with first row counting hydrogen atoms and second row oxygen atoms per molecule. Direct multiplication gives $D ν = (- 4 + 4, - 2 + 2)^{T} = (0, 0)^{T}$ , so every change $n (ξ) - n^{0} = ν ξ$ preserves both atom totals.

2.1L1step 1.1algebra

At $ξ_{*} = 0$ , only the water component is zero. Its directional change is $(ν h)_{3} = 2 h$ , so [L1] gives $T_{F} (0) = {h : 2 h \geq 0} = [0, \infty)$ . A negative direction is infeasible because it would make the water amount $2 ξ$ negative.

3.1L1step 1.1step 2.1algebra∎

For $Φ (n) = n_{3}$ , the induced function is $f (ξ) = Φ (n (ξ)) = 2 ξ$ . Its derivative at $0$ is $D f (0) h = 2 h \geq 0$ for every feasible tangent $h \geq 0$ , and $f (ξ) \geq f (0)$ for every $ξ \in [0, 1]$ . This verifies the boundary first-order condition and minimum directly for this chosen mathematical objective.

Depends on

The feasible tangent cone for nonnegative stoichiometric amounts

Dependency tree · 2 levels

Sources

David Bindel, Optimization Theory, §6.1.4, Numerical Methods for Data Science lecture notes