In a chemical reaction taking place in a well–stirred tank

MATHEMATICS · Multi-part · CBSE · Mathematics · Differential Equations · Differential Equations

Problem

In a chemical reaction taking place in a well–stirred tank, the concentration

C(t)

(in moles per liter) of a reactant at time

t

minutes satisfies the differential equation

\frac{dC}{dt}= -\frac{6t+4}{t^{2}+4t+3},\qquad t>0.

Assume the concentration at the initial time is

C(0)=12\ \text{mol/L}

. (a) Rewrite the right–hand side of the differential equation in the form

\displaystyle \frac{A}{t+1}+\frac{B}{t+3}

and state the polynomial identity obtained after clearing denominators. (b) Determine the values of

A

and

B

. (c) Solve the differential equation to obtain an explicit expression for

C(t)

that satisfies the initial condition. (d) Evaluate

\displaystyle \lim_{t\to \infty} C(t)

and interpret this limit in the context of the reaction.