Chapter 4

The activity of the antibiotic penicillin slowly decomposes when stored in a buffer at \(\mathrm{pH} 7.0,298 \mathrm{~K}\). The time dependence of the penicillin antibiotic activity is given in the table below. $$ \begin{array}{lc} \text { Time (weeks) } & \text { Penicillin Activity (arbitrary units) } \\ \hline 0 & 10,100 \\ 1.00 & 8,180 \\ 2.00 & 6,900 \\ 3.00 & 5,380 \\ 5.00 & 3,870 \\ 8.00 & 2,000 \\ 10.00 & 1.330 \\ 12.00 & 898 \\ 15.00 & 403 \\ 20.00 & 167 \\ \hline \end{array} $$What is the rate law for this reaction, that is, what is the order of the reaction with respect to the penicillin concentration? Calculate the rate constant from the data if possible. (Data adapted from Ref. I.)

The kinetics of the reaction $$ 2 \mathrm{Fe}^{3+}+\mathrm{Sn}^{2+} \rightarrow 2 \mathrm{Fe}^{2+}+\mathrm{Sn}^{4+} $$ has been studied extensively in acidic aqueous solutions. When \(\mathrm{Fe}^{2+}\) is added initially at relatively high concentrations, the rate law is $$ R=k\left[\mathrm{Fe}^{3+}\right]^{2}\left[\mathrm{Sn}^{2+}\right] /\left[\mathrm{Fe}^{2+}\right] $$ Postulate a mechanism that is consistent with this rate law. Show that it is consistent by deriving the rate law from the proposed mechanism.

The radioactive decay rates of naturally occurring radioactive elements can t used to determine the age of very old materials. For example, \({ }_{6}^{14} \mathrm{C}\) is radioactiv and emits a low-energy electron with a half-life of about 5730 years. Throug a balance of natural processes, the ratio of \({ }^{14} \mathrm{C} /{ }^{12} \mathrm{C}\) is constant in living orgar isms. However, in dead organisms or material, this ratio decreases as the decays. Since the radioactive decay is known to be a first order reaction, the ag of the material can be estimated by measuring the decrease in the \({ }^{14} \mathrm{C}^{12} \mathrm{C}\) ratio Suppose a piece of ancient wool is found in which the ratio has been found decrease by \(20 \%\). What is the age of the wool?

A hydrogen bonded dimer is formed between 2 -pyridone according to the reaction O=c1cccc[nH]1 O=c1cccc[nH]1 The relaxation time for this reaction, which occurs in nanoseconds, has been determined in chloroform at \(298 \mathrm{~K}\) at various concentrations of 2 -pyridone. The data obtained are [G. G. Hammes and A. C. Park, J. Am. Chem. Soc. 91 , 956 (1969)): \begin{tabular}{lc} 2-Pyridone (M) & \(10^{9} \tau(\mathrm{s})\) \\ \hline \(0.500\) & \(2.3\) \\ \(0.352\) & \(2.7\) \\ \(0.251\) & \(3.3\) \\ \(0.151\) & \(4.0\) \\ \(0.101\) & \(5.3\) \\ \hline \end{tabular} From these data calculate the equilibrium and rate constants characterizing this reaction. Hint: If the expression for the relaxation time is squared, the concentration dependence can be expressed as a simple function of the total concentration of 2-pyridone.