Synthesis Problem: 6. Consider a chemical reaction where A- as: (a) Use na, n, nc to...

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Synthesis Problem: 6. Consider a chemical reaction where A- as: (a) Use na, nв, nc to represent the populations of A, B, and C. Write down the coupled rate equations describing the populations of A, B, and C as a function of time. (b) Find the eigenvalues of the coefficient matrix in terms of ki, k2, and k3. (You should get λ = -kı, -(k2 + k3), and 0) اور شام (c) The eigenvector corresponding to the eigenvalue λ= -k₁ is: n₁(t) = e = k₁1 k₂ k, k₁ ng(t)= + K₁ K₁ K₁-K₁ -k₂. Find the eigenvectors for the other two eigenvalues. The top row in both cases will be a zero. (d) For an initial condition such that only A is present at t=0 (i.e. nA(0) = 1, nв(0) = 0, nc(0) = 0), the populations (after a fair bit of algebra) are given by: nc(t)= 1-e k₁ where k4= k2 + k3. BC. This could represent a reaction such -kal_k₁-k₂ e-k₁². k₂-k₁ e k₂ k₁ - K₁ -k,t =(e^₁² - e²^²) [k₁-(k₂ + k₂)] k₂-k₁ Use Matlab to plot the populations of A, B, and C as a function of time from t=0 to t-10 ps if ki = 5 ps¹, k₂= 0.3 ps¹ and k3 = 0.1 ps¹. What is the ratio of nв to nc at very long times (i.e. as t→∞0)? (e) Plot and compare the results in (d) with results for ki = 5 ps¹, k2= 0.1 ps¹ and k3= 0.2 ps-¹. Interpret your plots in terms of the reaction. The rate constants we obtain from our measurements for the reaction shown above are k₁ = 6.25 ps¹ (1/k₁ = 0.16 ps = 160 fs) and k4= 0.16 ps¹ (1/k4= 6.25 ps). Synthesis Problem: 6. Consider a chemical reaction where A- as: (a) Use na, nв, nc to represent the populations of A, B, and C. Write down the coupled rate equations describing the populations of A, B, and C as a function of time. (b) Find the eigenvalues of the coefficient matrix in terms of ki, k2, and k3. (You should get λ = -kı, -(k2 + k3), and 0) اور شام (c) The eigenvector corresponding to the eigenvalue λ= -k₁ is: n₁(t) = e = k₁1 k₂ k, k₁ ng(t)= + K₁ K₁ K₁-K₁ -k₂. Find the eigenvectors for the other two eigenvalues. The top row in both cases will be a zero. (d) For an initial condition such that only A is present at t=0 (i.e. nA(0) = 1, nв(0) = 0, nc(0) = 0), the populations (after a fair bit of algebra) are given by: nc(t)= 1-e k₁ where k4= k2 + k3. BC. This could represent a reaction such -kal_k₁-k₂ e-k₁². k₂-k₁ e k₂ k₁ - K₁ -k,t =(e^₁² - e²^²) [k₁-(k₂ + k₂)] k₂-k₁ Use Matlab to plot the populations of A, B, and C as a function of time from t=0 to t-10 ps if ki = 5 ps¹, k₂= 0.3 ps¹ and k3 = 0.1 ps¹. What is the ratio of nв to nc at very long times (i.e. as t→∞0)? (e) Plot and compare the results in (d) with results for ki = 5 ps¹, k2= 0.1 ps¹ and k3= 0.2 ps-¹. Interpret your plots in terms of the reaction. The rate constants we obtain from our measurements for the reaction shown above are k₁ = 6.25 ps¹ (1/k₁ = 0.16 ps = 160 fs) and k4= 0.16 ps¹ (1/k4= 6.25 ps).

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