In a chemical reactor

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

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.

Related theory

Browse the problem library