For a chemical reaction in a closed vessel

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

For a chemical reaction in a closed vessel, the rate at which product accumulates is modeled by R(t)=ke^{-0.3t} \text{mol\,h}^{-1}, t\ge0, where

k>0

is a constant that depends on initial reactant concentration. Let

P(t)

be the total number of moles of product formed during the first

t

hours. (a) Obtain an explicit formula for

P(t)

in terms of

k

. (b) Given a target amount

N

(moles) with

0< N< \dfrac{k}{0.3}

, solve for the time

t

at which

P(t)=N

. Express your answer as a function

t(N)

involving

k

and

N

. (c) Compute

\dfrac{dt}{dN}

and simplify the expression. (d) Interpret the sign and magnitude of

\dfrac{dt}{dN}

in practical terms for the reaction. Discuss how sensitive the completion time is to changes in the target amount of product.