Question

The reaction: $2\mathrm{HI}\to {\mathrm{H}}_2+{\mathrm{I}}_2$, is second order and the rate constant at $800\mathrm{K}$ is $9.70\times {10}^{-2}{\mathrm{M}}^{-1}{\mathrm{s}}^{-1}$. How long will it take for $8.00\times {10}^{-2}$ mol/litre of HI to decrease to one-fourth of its initial concentration? a. $587\mathrm{s}$ b. $387\mathrm{s}$ c. $148\mathrm{s}$ d. $687\mathrm{s}$

Short answer

It takes 387 s for the concentration to decrease to one-fourth of its initial concentration.

Step-by-step solution

Write the Integrated Rate Law for Second Order Reactions

For a second-order reaction, the integrated rate law is given by $$\frac{1}{[A]}-\frac{1}{[A{]}_0}=kt$$, where - [A] is the concentration of the reactant at time $t$.- $[A{]}_0$ is the initial concentration.- $k$ is the rate constant, and- $t$ is the time required for the reaction to change from $[A{]}_0$ to $[A]$.

Identify Given Values and Desired Outcome

We know the following:- Initial concentration $[HI{]}_0=8.00\times {10}^{-2}\text{mol/L}$.- Final concentration $[HI]=\frac{1}{4}\times 8.00\times {10}^{-2}\text{mol/L}=2.00\times {10}^{-2}\text{mol/L}$.- Rate constant $k=9.70\times {10}^{-2}{\text{M}}^{-1}{\text{s}}^{-1}$.We need to find $t$.

Substitute Values into the Rate Law

Substituting the given values into the equation $$\frac{1}{[A]}-\frac{1}{[A{]}_0}=kt$$ gives us:$$\frac{1}{2.00\times {10}^{-2}}-\frac{1}{8.00\times {10}^{-2}}=(9.70\times {10}^{-2})\times t$$.

Solve for Time $t$

First, calculate $\frac{1}{2.00\times {10}^{-2}}=50$ and $\frac{1}{8.00\times {10}^{-2}}=12.5$.Substitute these into the equation:$$50-12.5=9.70\times {10}^{-2}\times t$$$$37.5=9.70\times {10}^{-2}\times t$$.Then, solve for $t$:$$t=\frac{37.5}{9.70\times {10}^{-2}}\approx 386.6$$.

Round to Closest Option

Round 386.6 seconds to the closest number in the options, which is 387. Hence, the answer is effectively $387\text{ s}$.

Key concepts

Integrated Rate Law

The integrated rate law is a crucial concept for understanding how concentrations change over time in chemical reactions. In the context of second-order reactions, the integrated rate law has a specific form:

• $\frac{1}{[A]}-\frac{1}{[A{]}_0}=kt$ is the equation used, where
  • $[A]$ represents the concentration of the reactant at any time $t$.
  • $[A{]}_0$ is the initial concentration.
  • $k$ denotes the rate constant.
  • $t$ is the time elapsed during the reaction.
  This integrated rate law helps to establish a relationship between the concentration of reactants and time, providing insights into reaction dynamics. Particularly for a second-order reaction, it shows how the reaction rate is directly proportional to the square of the reactant concentration. By using this law, one can determine how long it will take for a reactant to reach a certain concentration from its initial state.

Rate Constant

The rate constant, symbolized as $k$, is an integral part of reaction kinetics and carries significant meaning for the speed of a reaction. For second-order reactions, the rate constant has the following characteristics:

• It is usually expressed in terms of ${\text{M}}^{-1}{\text{s}}^{-1}$, reflecting its dependency on concentration and time.
• It provides an indication of how rapidly a reaction proceeds under given conditions.
• The value of $k$ is determined experimentally and can vary with temperature, which is why it's crucial to state the temperature (in this case, 800 K) when expressing $k$.

In the provided problem, the rate constant $k=9.70\times {10}^{-2}{\text{ M}}^{-1}{\text{s}}^{-1}$ helps to quantify the reaction rate and allows us to calculate how long it takes for the concentration to decrease to a specified level. This underscores the utility of $k$ in predicting how changes in concentration will affect reaction time.

Reaction Kinetics

Reaction kinetics is the study of the rates of chemical processes and encompasses concepts such as reaction order, rate laws, and mechanisms. In second-order reactions, understanding the kinetics involves:

• Analyzing how the rate of reaction is influenced by changes in reactant concentrations, as depicted by the rate law $r=k[A{]}^2$ for a simple reaction.
• Employing integrated rate laws, like the one used in the solved problem, to derive more complex relationships between concentration and time.
• Utilizing the rate constant and derived equations to make predictions about reaction rates and to estimate half-lives or times required for specific concentration changes.

Through studying reaction kinetics, chemists can decipher not only the speed of reactions but also gain insights into reaction mechanisms and potential pathways that reactions might follow. This enables targeted manipulation of reaction conditions to achieve desired outcomes in various chemical industries and laboratory settings.