At \(40^{\circ} \mathrm{C}, \mathrm{H}_{2} \mathrm{O}_{2}(a q)\) will decompose according to the following reaction: $$ 2 \mathrm{H}_{2} \mathrm{O}_{2}(a q) \longrightarrow 2 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{O}_{2}(\mathrm{~g}) $$ The following data were collected for the concentration of $\mathrm{H}_{2} \mathrm{O}_{2}$ at various times. $$ \begin{array}{|cc|} \hline \begin{array}{c} \text { Time } \\ (\mathbf{s}) \end{array} & \begin{array}{c} {\left[\mathrm{H}_{2} \mathrm{O}_{2}\right]} \\ (\mathrm{mol} / \mathrm{L}) \end{array} \\ \hline 0 & 1.000 \\ \hline 2.16 \times 10^{4} & 0.500 \\ \hline 4.32 \times 10^{4} & 0.250 \\ \hline \end{array} $$ a. Calculate the average rate of decomposition of $\mathrm{H}_{2} \mathrm{O}_{2}\( between 0 and \)2.16 \times 10^{4} \mathrm{~s}$. Use this rate to calculate the average rate of production of \(\mathrm{O}_{2}(g)\) over the same time period. b. What are these rates for the time period \(2.16 \times 10^{4} \mathrm{~s}\) to \(4.32 \times 10^{4} \mathrm{~s} ?\)

Step by step solution

01

Calculate the rate of decomposition of H2O2 in the first time interval

According to the data given: Initial concentration of H2O2: \([H_2O_2]_i = 1.000 \, mol/L\) Initial time: \(t_i = 0\,s\) Final concentration of H2O2: \([H2O2]_f = 0.500 \, mol/L\) Final time: \(t_f = 2.16 \times 10^4 \, s\) Now we can apply the formula for the average rate of reaction: Average rate = \(\frac{[H_2O_2]_f - [H_2O_2]_i}{t_f - t_i}\)

02

Calculate the average rate of production of O2 in the first time interval

To find the average rate of production of O2, analyze the stoichiometry of the reaction: \(2H_2O_2(aq) \longrightarrow 2H_2O(l) + O_2(g)\) The ratio of H2O2 to O2 according to the reaction coefficients is 2:1. Thus, the rate of production of O2 gas will be half the rate of H2O2 decomposition.

03

Calculate the rates for the second time interval

Use the same approach to find the rates for the second time interval (2.16 x 10^4 s to 4.32 x 10^4 s). Initial concentration of H2O2: \([H_2O_2]_i = 0.500\, mol/L\) Initial time: \(t_i = 2.16 \times 10^4\,s\) Final concentration of H2O2: \([H2O2]_f = 0.250 \, mol/L\) Final time: \(t_f = 4.32 \times 10^4 \, s\) Apply the formula for the average rate of decomposition and use stoichiometry to calculate the rate of production of O2 in the second time interval, following the same steps as in the first time interval.

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