In a chemical reaction vessel

MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations

Problem

In a chemical reaction vessel, the concentration

C

(measured in mol/L) of a reactant changes according to

\frac{dC}{dt}=k\,(C-4)\,(10-C),\qquad k>0,

where

t

is measured in minutes and

k

is a positive constant that depends on temperature. (a) Determine all equilibrium concentrations. (b) Classify each equilibrium concentration as stable or unstable, providing clear reasoning based on the sign of

\dfrac{dC}{dt}

in intervals between the equilibrium values. (c) Suppose

C(0)=1

. Predict the concentration

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

and justify your conclusion. (d) If the vessel temperature is increased so that the value of

k

doubles, explain whether your stability classifications in part (b) change. Provide a brief justification.

Related theory

Introduction to Differential Equations
Foundations
Qualitative Analysis
Differential Equations

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