(b) Assume the previous part, show that z satisfy the equation dz/dt = a +bz+cz², where a, b, c are constants depending on xo, Yo, Zo- (c) Let us assume that xo = 3, yo = 4, zo = 0. What is the limiting concentration lim→∞o z(t)? You do not need to solve the equation for that. (d) Suppose now that k+ = 1,k_ = 5. Explain why the limiting concentration of z(t) would be lower in that case. (e) Verify your answer to the previous part by solving for z(t) and calculating the limiting concentration.
(b) Assume the previous part, show that z satisfy the equation dz/dt = a +bz+cz², where a, b, c are constants depending on xo, Yo, Zo- (c) Let us assume that xo = 3, yo = 4, zo = 0. What is the limiting concentration lim→∞o z(t)? You do not need to solve the equation for that. (d) Suppose now that k+ = 1,k_ = 5. Explain why the limiting concentration of z(t) would be lower in that case. (e) Verify your answer to the previous part by solving for z(t) and calculating the limiting concentration.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(b) Assume the previous part, show that z satisfy the equation
dz/dt = a + bz+cz²,
where a, b, c are constants depending on xo, yo, zo.
(c) Let us assume that xo = 3, yo = 4, zo = 0. What is the limiting concentration lim/→∞ z(t)?
You do not need to solve the equation for that.
(d) Suppose now that k+ = 1,k_ = 5. Explain why the limiting concentration of z(t) would
be lower in that case.
(e) Verify your answer to the previous part by solving for z(t) and calculating the limiting
concentration.
Transcribed Image Text:4. Consider a chemical equilibrium X + Y ⇒ Z, for three materials whose concentrations
by time are given by x(t), y(t) and z(t), respectively. We may model z(t) by the equation
dz/dt =k+xy - k_z,
for some constants k+,k_ (the rate constants of the forward and backward interactions). We
will take k+ = k = 1, so dz/dt = xy - z.
(a) Explain with a physical argument why we must have x − xo = y — yo = zo - z where
xo, yo, zo are the initial concentrations of X, Y, Z respectively.
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