# How to Determine the Rate Law knowing time and concentrations

Consider the hypothetical reaction described by the following equation:

$$\ce{3 A + 2 B -> C}$$

Reagent B was added in excess and the following time concentration data was obtained.

\begin{array}{cr} A\ [\mathrm{10^{-3}\ mol\cdot L^{-1}}] & \mathrm{time [s]} \\ \hline 5.0 & 10\\ 2.9 & 50\\ 1.9 & 80\\ 1.1 & 120\\ 0.7 & 150\\ \end{array}

(i) Derive the relevant integrated rate law and hence find the rate constant.

I am thinking that to solve this question you just sub the values into the integrate rate laws and see if it produces a linear relationship.

Is this correct? Also could you please suggest a faster method of solving this type of question.

Thank you.

You could plot the data to determine the order of the reaction. I would use Microsoft Excel to do this. Reproduce the data and then add columns that calculate the $\ln [A]$ and $\frac{1}{[A]}$. Then plot $[A]$, $\ln [A]$ ,and $\frac{1}{[A]}$ vs. time (I would use separate plots). The plot that gives the "best" linear fit (which you can assess using the $R^2$ value), corresponds to the appropriate integrated rate law.

• If $[A]$ vs. time is linear, then the reaction is zero order.
• If $\ln [A]$ vs. time is linear, then the reaction is first order.
• If $\frac{1}{[A]}$ vs. time is linear, then the reaction is second order.

Hope this helps.

You might want to give Python, the IPython Notebook and matplotlib a try:

import numpy as np
import matplotlib.pyplot as plt

t = np.array([10, 50, 80, 120, 150])
a = np.array([5.0, 2.9, 1.9, 1.1, 0.7])

a_int = np.log(a)

plt.figure()
plt.subplot(121)
plt.plot(t, a, 'ro-')
plt.xlabel('t [s]')
plt.ylabel('A')

plt.subplot(122)
plt.plot(t, a_int, 'gs-')
plt.xlabel('t [s]')
plt.ylabel('ln(A)')

plt.show()


Note that the plots are not fine-tuned and that I omitted a further fitting ;-)