1. The reversible reaction A = B has the following rate equation: k2 dA k2B – kA (1) dt where A = A(t) and B = B(t) are, respectively, the molar concentrations of A and B. a) Use (1) and mass conservation to find a differential equation for A(t). b) Use direct integration of the equation found in (a) to find A(t) in the case that equal amounts, xo, of A and B are mixed in a closed container. c) Verify your solution for A(t) using the expression for the general solution of a FOLDE. d) What are the equilibrium concentrations A(t → ∞) and B(t → ∞)?
1. The reversible reaction A = B has the following rate equation: k2 dA k2B – kA (1) dt where A = A(t) and B = B(t) are, respectively, the molar concentrations of A and B. a) Use (1) and mass conservation to find a differential equation for A(t). b) Use direct integration of the equation found in (a) to find A(t) in the case that equal amounts, xo, of A and B are mixed in a closed container. c) Verify your solution for A(t) using the expression for the general solution of a FOLDE. d) What are the equilibrium concentrations A(t → ∞) and B(t → ∞)?
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
Section: Chapter Questions
Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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![**Transcription for Educational Website**
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**Reversible Reaction and Rate Equation Analysis**
1. Consider the reversible reaction \( \text{A} \overset{k_1}{\underset{k_2}{\rightleftharpoons}} \text{B} \) which follows the rate equation:
\[
\frac{dA}{dt} = k_2B - k_1A \tag{1}
\]
where \( A = A(t) \) and \( B = B(t) \) denote the molar concentrations of A and B, respectively.
**Tasks:**
a) Utilize equation (1) along with the principle of mass conservation to derive a differential equation for \( A(t) \).
b) Apply direct integration to the derived equation from part (a) to determine \( A(t) \), assuming equal initial amounts, \( x_0 \), of A and B are placed in a closed container.
c) Verify the solution for \( A(t) \) by employing the formulation for the general solution of a First-Order Linear Differential Equation (FOLDE).
d) Determine the equilibrium concentrations \( A(t \to \infty) \) and \( B(t \to \infty) \).
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This transcription explains the chemical kinetics involved in reversible reactions and guides you through finding the solutions to differential equations associated with them.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f6272ad-1a25-4a67-b389-4d6bf92e8dd2%2Fe1c55a1f-e2a9-4241-b8fd-bed873df37f8%2F43776y_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website**
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**Reversible Reaction and Rate Equation Analysis**
1. Consider the reversible reaction \( \text{A} \overset{k_1}{\underset{k_2}{\rightleftharpoons}} \text{B} \) which follows the rate equation:
\[
\frac{dA}{dt} = k_2B - k_1A \tag{1}
\]
where \( A = A(t) \) and \( B = B(t) \) denote the molar concentrations of A and B, respectively.
**Tasks:**
a) Utilize equation (1) along with the principle of mass conservation to derive a differential equation for \( A(t) \).
b) Apply direct integration to the derived equation from part (a) to determine \( A(t) \), assuming equal initial amounts, \( x_0 \), of A and B are placed in a closed container.
c) Verify the solution for \( A(t) \) by employing the formulation for the general solution of a First-Order Linear Differential Equation (FOLDE).
d) Determine the equilibrium concentrations \( A(t \to \infty) \) and \( B(t \to \infty) \).
---
This transcription explains the chemical kinetics involved in reversible reactions and guides you through finding the solutions to differential equations associated with them.
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