Biordi (15) employed conductivity measurements to monitor the progress of the methanolysis of benzoyl chloride at 25^{\circ}C in methanol solution:
Under the conditions of the experiment, it is known that the reaction may be considered to be irreversible. From the data that follow, determine the order of the reaction with respect to benzoyl chloride and the apparent rate constant.
| 70 | 0.9 |
| 79 | 0.969 |
| 86 | 1.07 |
| 93 | 1.12 |
| 100 | 1.21 |
| 105 | 1.26 |
| 114 | 1.33 |
| 120 | 1.4 |
| . | . |
| . | . |
| . | . |
| 10,800 | 3.5 |
Because a vast excess of methanol (the solvent) was used, the concentration of methanolwas essentially invariant, and it is appropriate to assume a rate expression of the form r=k^{'}C_{A}^{\beta ^{A}}. Since the conductivity of the solution is a property that is an additive function of contributions that are linear in concentration, the generalized physical property approach may be used:
\frac{k-k_{0}}{k_{\infty }-k_{0}}=\frac{\xi ^{\ast }}{\xi _{\infty }^{\ast }}=\frac{\xi ^{\ast }}{C_{A0}} (A)
where k_{\infty } is taken to be the conductivity reading at 10,800 s. The integrated form of the rate law for a first-order reaction is given by equation (3.1.4):
-kt=\ln \left ( \frac{C_{A0}-\xi ^{\ast }}{C_{A0}} \right )=\ln \left ( 1-\frac{\xi ^{\ast }}{C_{A0}} \right ) (B)
Combination of equations (A) and (B) gives
-kt=\ln \left ( 1-\frac{k-k_{0}}{k_{\infty }-k_{0}} \right )=\ln \left ( \frac{k_{\infty }-k}{k_{\infty }-k_{0}} \right )
or
\ln (k_{\infty }-k)=\ln (k_{\infty }-k_{0})-kt (C)
If a plot of the left side of equation (C) versus time is linear, the reaction is first-order and the slope of this plot is equal to (-k). A plot of the data in this form indicates that the reaction is indeed first-order with respect to benzoyl chloride. The slope of the line corresponds to a pseudo first-order rate constant of (4.24 \pm 0.041)\times 10^{-3}s^{-1}.