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The first-order thermal decomposition of chlorocyclohexane is as follows: $\mathrm{C}_6 \mathrm{H}_{11} \mathrm{Cl}(g) \rightarrow$ $\mathrm{C}_6 \mathrm{H}_{10}(g)+\mathrm{HCl}(g)$ . For a constant volume system the following total pressures were measured as a function of time:

a. Derive the following relationship for a first-order reaction: $P\left(t_2\right)-P\left(t_1\right)=\left(P\left(t_{\infty}\right)-P\left(t_0\right)\right) e^{-k t_1}\left(1-e^{-k\left(t_2-t_1\right)}\right)$ . In this relation, $P\left(t_1\right)$ and $P\left(t_2\right)$ are the pressures at two specific times; $\mathrm{P}\left(t_0\right)$ is the initial pressure when the reaction is initiated, $P\left(t_{\infty}\right)$ is the pressure at the completion of the reaction, and $k$ is the rate constant for the reaction. To derive this relationship do the following: i. Given the first-order dependence of the reaction, write the expression for the pressure of chlorocyclohexane at a specific time $t_1$ . ii. Write the expression for the pressure at another time $t_2$ , which is equal to $t_1+\Delta$ where delta is a fixed quantity of time. iii. Write expressions for $P\left(t_{\infty}\right)-P\left(t_1\right)$ and $P\left(t_{\infty}\right)-P\left(t_2\right)$ . iv. Subtract the two expressions from part (iii). b. Using the natural log of the relationship from part (a) and the data provided in the table given earlier in this problem, determine the rate constant for the decomposition of chlorocyclohexane. (Hint: Transform the data in the table by defining $t_2-t_1$ to be a constant value, for example, $9 \mathrm{~s}$ .)

The disaccharide lactose can be decomposed into its constituent sugars galactose and glucose. This decomposition can be accomplished through acid-based hydrolysis or by the enzyme lactase. Lactose intolerance in humans is due to the lack of lactase production by cells in the small intestine. However, the stomach is an acidic environment; therefore, one might expect lactose hydrolysis to still be an efficient process. The following data were obtained on the rate of lactose decomposition as a function of acid and lactose concentration. Using this information, determine the rate law expression for the acid-based hydrolysis of lactose.

Initial Rate $\left(\mathbf{M}^{-\mathbf{1}} \mathbf{s}^{-\mathbf{1}}\right)$	{ $[ \text { lactose }]_{\mathbf{0}}(\mathbf{M})$ }	{ $\left[\mathbf{H}^{+}\right](\mathbf{M})$ }
$0.00116$	$0.01$	$0.001$
$0.00232$	$0.02$	$0.001$
$0.00464$	$0.01$	$0.004$

A proposed two-step mechanism for the destruction of ozone in the upper atmosphere is What is the overall balanced equation for the ozone destruction reaction?

Question

A certain reaction is first order, and $540 \mathrm{~s}$ after initiation of the reaction, $32.5 \%$ of the reactant remains. a. What is the rate constant for this reaction? b. At what time after initiation of the reaction will $10 \%$ of the reactant remain?

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In this task, we need to determine the rate constant for the first order reaction where $32.5\ \%$ of reactant remains after $540\ \text s$ of reaction and the time where $10\ \%$ of reactant remains.