Mathematics

Problem

In a chemical reactor, the instantaneous reaction rate

R

(in mol/L·s) depends on the reactant concentration

c

R(c)=k c^{\alpha},

where

k

and

\alpha

are positive constants. The concentration depends on temperature

T

through an empirical relation

c(T)=c_{0}e^{-\beta/T},

with constants

c_{0}>0

and

\beta>0

. The reactor’s temperature in turn depends on the volume of coolant circulating through an external jacket, with

T(V)=T_{0}-\gamma V,

where

\gamma>0

and

V

is coolant volume in liters. Assume that at the present operating point the coolant volume is being increased at a rate of

1.5\text{ L/min}

. (a) Without performing detailed calculations, describe qualitatively how

R

is expected to change immediately as the coolant volume increases. Your explanation should reference the signs of the relevant derivatives at the operating point. (b) Explain why the magnitudes of

\dfrac{dT}{dV}

\dfrac{dc}{dT}

, and

\dfrac{dR}{dc}

together determine whether small adjustments to

V

significantly influence the overall reaction rate.

Intuitive explanation

The reaction rate ultimately depends on coolant volume through a chain of three nested functions: volume controls temperature, temperature controls concentration, and concentration controls reaction rate. You need to use the chain rule to connect these layers—finding $\frac{dR}{dV}$ by multiplying the derivatives at each stage. This tells you how fast the reaction is slowing down as more coolant flows through.

Intuitive explanation

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