WOLFRAM|DEMONSTRATIONS PROJECT

Contour Plots for Reaction Rates

​
reaction type
exothermic
endothermic
activation energies
exothermic reaction
forward reaction
20000
reverse reaction
50000
endothermic reaction
forward reaction
60000
reverse reaction
20000
Consider a reversible reaction
A⇋B
with the reaction rate
r
and the equilibrium constant
K
eq
given by:
r=
r
f
-
r
b
=exp-
E
af
R
1
T
-
1
T
0
C
A0
(1-X)-exp-
E
ab
R
1
T
-
1
T
0
C
A0
X
,
K
eq
=exp-
E
af
-
E
ab
R
1
T
-
1
T
0
,
where
X
is the conversion fraction,
C
A0
is the inlet concentration (taken to be 10 moles/liter),
R
is the universal gas constant (1.987 cal/mol K), and
T
is the temperature (in kelvin).
Use the sliders to vary the activation energies for the forward (
E
af
) and reverse (
E
ab
) reactions. Both of these activation energies are expressed in cal/mol.
This Demonstration plots the contour lines for the reaction rate
r
either for exothermic reactions
ΔH=
E
af
-
E
ab
<0
or for endothermic reactions
ΔH=
E
af
-
E
ab
>0
. The equilibrium conversion
X
eq
=
K
eq
1+
K
eq
is plotted versus the temperature (see the black curve, for which we have
r=0
). It can be easily shown that
∂r
∂T
X
=
1
R
2
T
(
E
af
r
f
-
E
ab
r
b
)
. Thus, for endothermic reactions,
E
af
>
E
ab
and
r
f
>
r
b
, so
∂r
∂T
X
>0
and
r
will increase monotonically with
T
if you move along a horizontal line (i.e. at a constant conversion fraction). On the other hand, for exothermic reactions we have
E
af
<
E
ab
, thus we have along a horizontal line (i.e. at a constant value of
X
): (1) at low temperature
E
af
r
f
>
E
ab
r
b
and
∂r
∂T
X
>0
; and (2) at higher temperature
E
af
r
f
<
E
ab
r
b
and
∂r
∂T
X
<0
. In conclusion, for an exothermic reaction, the reaction rate
r
initially increases with increasing
T
, reaches a maximum value
r
max
when
∂r
∂T
X
=0
, then starts to decrease until it reaches the equilibrium conversion curve where
r=0
. The loci of the points where
r=
r
max
are indicated by the gray dots and curve.