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Thermodynamics filled in class notes_Part_107

# Thermodynamics filled in class notes_Part_107 - 221 6.4...

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6.4. ADIABATIC, ISOCHORIC KINETICS 221 Note finally that it can be shown that D , of dimension N × ( N L ), and ν , of dimension N × J , share the same column space, which is of dimension ( N L ); consequently, both matrices map vectors into the same space. 6.3.2 Species dominant: J< ( N L ) Next consider the case in which J< ( N L ). This often arises in models of simple chemistry, for example one- or two-step kinetics. The fundamental reaction dynamics are most concisely governed by the J equations which form 1 V d ζ dt = r . (6.153) However, r is a function of the concentrations; one must therefore recover ρ as a function of reaction progress ζ . In vector form, Eq. (6.143) is written as d ρ dt = 1 V ν · d ζ dt . (6.154) Take as an initial condition that the reaction progress is zero at t = 0 and that there are an appropriate set of initial conditions on the species concentrations ρ : ζ = 0 , t = 0 , (6.155) ρ = ρ o , t = 0 . (6.156) Then, since ν is a constant, Eq. (6.154) is easily integrated. After applying the initial conditions, Eq. (6.156), one gets ρ = ρ o + 1 V ν · ζ .

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