# Contour Plots for Reaction Rates

Contour Plots for Reaction Rates

Consider a reversible reaction with the reaction rate and the equilibrium constant given by:

A⇋B

r

K

eq

r=r-r=exp--C(1-X)-exp--CX

f

b

E

af

R

1

T

1

T

0

A0

E

ab

R

1

T

1

T

0

A0

K=exp--

eq

E-E

af

ab

R

1

T

1

T

0

where is the conversion fraction, is the inlet concentration (taken to be 10 moles/liter), is the universal gas constant (1.987 cal/mol K), and is the temperature (in kelvin).

X

C

A0

R

T

Use the sliders to vary the activation energies for the forward () and reverse () reactions. Both of these activation energies are expressed in cal/mol.

E

af

E

ab

This Demonstration plots the contour lines for the reaction rate either for exothermic reactions or for endothermic reactions . The equilibrium conversion is plotted versus the temperature (see the black curve, for which we have ). It can be easily shown that . Thus, for endothermic reactions, and , so and will increase monotonically with if you move along a horizontal line (i.e. at a constant conversion fraction). On the other hand, for exothermic reactions we have , thus we have along a horizontal line (i.e. at a constant value of ): (1) at low temperature and ; and (2) at higher temperature and . In conclusion, for an exothermic reaction, the reaction rate initially increases with increasing , reaches a maximum value when , then starts to decrease until it reaches the equilibrium conversion curve where . The loci of the points where are indicated by the gray dots and curve.

r

ΔH=E-E<0

af

ab

ΔH=E-E>0

af

ab

X=

eq

K

eq

1+K

eq

r=0

=(Er-Er)

∂r

∂T

X

1

RT

2

af

f

ab

b

E>E

af

ab

r>r

f

b

>0

∂r

∂T

X

r

T

E<E

af

ab

X

Er>Er

af

f

ab

b

>0

∂r

∂T

X

Er<Er

af

f

ab

b

<0

∂r

∂T

X

r

T

r

max

=0

∂r

∂T

X

r=0

r=r

max