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(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.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...
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(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:(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.
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.
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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