Q1. Consider the following batch chemical system which we wish to model similarly to A=B-P The time evolution of the system is given according to the following model equations: =-k₁[A] + k-1[B], d[B] dt = k₁[A]-(K-1 + K₂)[B], d[P] dt d[A] dt = K₂[B]. The rate coefficients are k₁= 10-4-¹, k_1 = 10-²5-¹, and k₂ We are interested in the situation where initially (timet。), [A] = 10−7 mol/L, [B] = 0, and [P] = 0. = 10-3 S-1 (a) Write an analytical solution for [P] in terms of [B]. (b) Show how eigen-decomposition of the first two differential equations IVPs can be used to determine [A] and [B] as a function of time. dv(t) (c) Write the entire set of equations in matrix vector form =Mv(t) where M is a 3*3 square matrix and v is a column vector. What is the initial condition vector v(t)? dt
Q1. Consider the following batch chemical system which we wish to model similarly to A=B-P The time evolution of the system is given according to the following model equations: =-k₁[A] + k-1[B], d[B] dt = k₁[A]-(K-1 + K₂)[B], d[P] dt d[A] dt = K₂[B]. The rate coefficients are k₁= 10-4-¹, k_1 = 10-²5-¹, and k₂ We are interested in the situation where initially (timet。), [A] = 10−7 mol/L, [B] = 0, and [P] = 0. = 10-3 S-1 (a) Write an analytical solution for [P] in terms of [B]. (b) Show how eigen-decomposition of the first two differential equations IVPs can be used to determine [A] and [B] as a function of time. dv(t) (c) Write the entire set of equations in matrix vector form =Mv(t) where M is a 3*3 square matrix and v is a column vector. What is the initial condition vector v(t)? dt
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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![Q1. Consider the following batch chemical system which we
wish to model similarly to
A=B-P
The time evolution of the system is given according to the
following model equations:
=-k₁[A] + K-1[B],
d[B]
dt
= k₁[A]-(k-1 + K₂)[B],
d[P]
dt
d[A]
dt
= K₂[B].
The rate coefficients are k₁ = 10-4s-¹, k-₁ = 10-² S-¹, and k₂
We are interested in the situation where initially
(timeto), [A] = 10-7 mol/L, [B] = 0, and [P] = 0.
= 10-3 S-1
(a) Write an analytical solution for [P] in terms of [B].
(b) Show how eigen-decomposition of the first two differential
equations IVPS can be used to determine [A] and [B] as a
function of time.
(c) Write the entire set of equations in matrix vector form
dv(t)
=Mv(t)
dt
where M is a 3*3 square matrix and v is a column
vector. What is the initial condition vector v(t)?
(d) Although it is convenient to transform the matrix M to a
diagonal form that is not always possible. If M is singular, does
that imply it is not diagonalizable? Is M being singular a
necessary condition for this system of differential equations to
have a non-trivial steady state solution (other than v(t))?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F405e7a69-2f84-44eb-b08d-4db515620f63%2F95bc6483-d3cc-4e68-9a6b-b213e4d174ac%2Faq4zlxl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q1. Consider the following batch chemical system which we
wish to model similarly to
A=B-P
The time evolution of the system is given according to the
following model equations:
=-k₁[A] + K-1[B],
d[B]
dt
= k₁[A]-(k-1 + K₂)[B],
d[P]
dt
d[A]
dt
= K₂[B].
The rate coefficients are k₁ = 10-4s-¹, k-₁ = 10-² S-¹, and k₂
We are interested in the situation where initially
(timeto), [A] = 10-7 mol/L, [B] = 0, and [P] = 0.
= 10-3 S-1
(a) Write an analytical solution for [P] in terms of [B].
(b) Show how eigen-decomposition of the first two differential
equations IVPS can be used to determine [A] and [B] as a
function of time.
(c) Write the entire set of equations in matrix vector form
dv(t)
=Mv(t)
dt
where M is a 3*3 square matrix and v is a column
vector. What is the initial condition vector v(t)?
(d) Although it is convenient to transform the matrix M to a
diagonal form that is not always possible. If M is singular, does
that imply it is not diagonalizable? Is M being singular a
necessary condition for this system of differential equations to
have a non-trivial steady state solution (other than v(t))?
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