Thermodynamics filled in class notes_Part_134

Thermodynamics filled in class notes_Part_134 - 7.2...

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Unformatted text preview: 7.2. ADIABATIC, ISOCHORIC KINETICS 275 Now, let us take the limit of high activation energy by defining ǫ to be ǫ ≡ 1 Θ . (7.338) Let us let the assume the remaining parameters, q and k are both O (1) constants. When Θ is large, ǫ will be small. With this definition, Eq. (7.337) becomes exp parenleftbigg − Θ 1 + qλ parenrightbigg ∼ e − 1 /ǫ exp ( qλ 1 + O ( ǫ 2 ) ) . (7.339) With these assumptions and approximations, Eq. (7.331) can be written as d dτ ( ǫλ 1 + ... ) = e − 1 /ǫ exp ( qλ 1 + O ( ǫ 2 ) ) × parenleftbigg (1 − ǫλ 1 − ... ) − ( ǫλ 1 + ... ) exp parenleftbigg − q ( k − 1)(1 + qǫλ 1 + ... ) parenrightbiggparenrightbigg . (7.340) Now let us rescale time via τ ∗ = 1 ǫ e − 1 /ǫ τ. (7.341) With this transformation, the chain rule shows how derivatives transform: d dτ = dτ ∗ dτ d dτ ∗ = 1 ǫe 1 /ǫ d dτ ∗ . (7.342) With this transformation, Eq. (7.340) becomes 1 ǫe 1 /ǫ d dτ ∗ ( ǫλ 1 + ... ) = 1 e 1 /ǫ exp ( qλ 1...
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Thermodynamics filled in class notes_Part_134 - 7.2...

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