d) The back reaction in Equation 1, in which \( \mathrm{NO}_{2} \) reacts to give NO and \( \mathrm{O}_{2} \), was investigated in two separate kinetic studies at different temperatures. Below are data from these studies, both of which begin with pure \( \mathrm{NO}_{2} \) (initial concentration of NO and \( \mathrm{O}_{2} \) is zero). Determine the rate law for the back reaction, and calculate the rate constants \( k \) at \( 300^{\circ} \mathrm{C} \) and \( 383^{\circ} \mathrm{C} \).
Table 2. Concentration of \( \mathrm{NO}_{2} \) as a function of time at \( 300^{\circ} \mathrm{C} \).
\begin{tabular}{ll}
Time (s) & {\( \left[\mathrm{NO}_{2}\right](\mathrm{M}) \)} \\
0 & 0.01000 \\
50 & 0.00787 \\
100 & 0.00649 \\
200 & 0.00481 \\
300 & 0.00380
\end{tabular}
Table 3. Concentration of \( \mathrm{NO}_{2} \) as a function of time at \( 383^{\circ} \mathrm{C} \).
\begin{tabular}{ll}
Time \( (\mathrm{s}) \) & {\( \left[\mathrm{NO}_{2}\right](\mathrm{M}) \)} \\
0 & 0.100 \\
5 & 0.017 \\
10 & 0.0090 \\
15 & 0.0062 \\
20 & 0.0047
\end{tabular}