4.4.3. Reversible Reactions#
4.4.3.1. Motivation#
Reactions that can proceed both in the forward and reverse directions are important in chemistry, as we saw in the equilibrium section of Thermodynamics. In these notes we will discuss how to write rate laws for simple “reversible” reactions.
4.4.3.2. Learning Goals#
After working through these notes, you should be able to:
Define a reversible reaction in the context of kinetics
Write out the derivative of concentration with respect to time for species in reversible reactions
Derive and use the first order integrated rate law for a simple reversible process
4.4.3.3. Coding Concepts#
The following coding concepts are used in this notebook:
4.4.3.4. Rate Laws for Reversible Reactions#
Most chemical processes should be considered to go both in the forward and reverse directions. In the context of Kinetics, we describe this as a reversible reaction (distinct from a reversible process in Thermodynamics). A reaction that is considered to go both in the forward and reverse direction is denoted with two arrows (as we have seen in the Equilibrium portion of the Thermodynamics section). As an example
we say the
To achieve equilibrium, we must allow the reaction to achieve the steady-state. That is, the concentrations of
Note that this does not mean that the forward and reverse reactions are not happening. It just means that they are happening in such a way that and
We now consider the special case in which the forward and reverse processes are both first order. In such a case we can write
If we want to equate these rate laws to the time derivatives of concentration of
Notice that the forward depletion is denoted by a negative sign infront of
Notice that, at equilibrium, we have that
This is clearly an important relationship that states that the equilibrium constant is equal to a ratio of the forward and reverse rate constants. We will use this in our derivation below for the integrated rate law.
In order to derive an integrated rate law from the differential expression, we must write
in the last step we have set
The final expression can be considered an integrated rate law for reversible first order processes. A slight rearrangement yields
demonstrating that a plot of
4.4.3.4.1. Example: Time Dependence of for Reversible First Order Processes#
Consider a generic equilibrium
with
We will need to do some slight rearrangements of the equation we have above to achieve this. First, we want to plot
Next, we recognize that by dictating the rate constants, the problem has also dictated the relative amounts of the species at equilibrium:
Now we use this relationship in the equation for
Now we can plot this function with just
Show code cell source
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
# setup plot parameters
fontsize=16
fig = plt.figure(figsize=(8,8), dpi= 80, facecolor='w', edgecolor='k')
ax = plt.subplot(111)
ax.grid(which='major', axis='both', color='#808080', linestyle='--')
ax.set_xlabel("$t$ (s)",size=fontsize)
ax.set_ylabel("$[A]/[A]_0$",size=fontsize)
plt.tick_params(axis='both',labelsize=fontsize)
def rev_first_order(t,k1,k2):
K = k1/k2
return K*np.exp(-(k1+k2)*t)/(K+1) + 1/(K+1)
k1 = 2.25e-2
k2 = 1.50e-2
t = np.arange(0,200,0.01)
ax.plot(t,rev_first_order(t,k1,k2),lw=3, label="A")
ax.plot(t,1-rev_first_order(t,k1,k2),lw=3, label="B")
plt.legend(fontsize=16);
