Question

The following set of data was obtained by the method of initial rates for the reaction: $$ \begin{aligned} 2 \mathrm{HgCl}_{2}(\mathrm{aq})+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(\mathrm{aq}) \rightarrow \\ 2 \mathrm{Cl}^{-}(\mathrm{aq})+2 \mathrm{CO}_{2}(\mathrm{~g})+\mathrm{Hg}_{2} \mathrm{Cl}_{2}(\mathrm{~s}) \end{aligned} $$ What is the rate law for the reaction? $$ \begin{array}{lll} \hline\left[\mathrm{HgCl}_{2}\right], \mathrm{M} & {\left[\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\right], \mathrm{M}} & \text { Rate, } \mathrm{M} / \mathrm{s} \\ \hline 0.10 & 0.10 & 1.3 \times 10^{-7} \\ 0.10 & 0.20 & 5.2 \times 10^{-7} \\ 0.20 & 0.20 & 1.0 \times 10^{-6} \\ \hline \end{array} $$ a. Rate \(=\mathrm{k}\left[\mathrm{HgCl}_{2}\right]\left[\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\right]^{2}\) b. Rate \(=\mathrm{k}\left[\mathrm{HgCl}_{2}\right]^{2}\left[\mathrm{C}_{2} \mathrm{O}_{4}{ }^{2-}\right]\) c. Rate \(=\mathrm{k}\left[\mathrm{HgCl}_{2}\right]\left[\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\right]^{2-}\) d. Rate \(=\mathrm{k}\left[\mathrm{HgCl}_{2}\right]\left[\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\right]^{-1}\)

Short answer

The rate law is: a. Rate \( = k[\text{HgCl}_2][\text{C}_2\text{O}_4^{2-}]^2 \).

Step-by-step solution

Analyze Experimental Data

We have three experiments with varying concentrations of \([\text{HgCl}_2]\) and \([\text{C}_2\text{O}_4^{2-}]\). We will use these to determine the reaction order with respect to each reactant.

Determine Order with Respect to \([\text{C}_2\text{O}_4^{2-}]\)

Compare Trial 1 and Trial 2 where \([\text{HgCl}_2]\) is constant at 0.10 M. The rate increases from \(1.3 \times 10^{-7}\) M/s to \(5.2 \times 10^{-7}\) M/s as the concentration of \([\text{C}_2\text{O}_4^{2-}]\) doubles (from 0.10 to 0.20 M). This is a fourfold increase in rate, indicating that the reaction is second order with respect to \([\text{C}_2\text{O}_4^{2-}]\).

Determine Order with Respect to \([\text{HgCl}_2]\)

Compare Trial 2 and Trial 3 where \([\text{C}_2\text{O}_4^{2-}]\) is constant at 0.20 M. The rate increases from \(5.2 \times 10^{-7}\) M/s to \(1.0 \times 10^{-6}\) M/s as the concentration of \([\text{HgCl}_2]\) doubles (from 0.10 to 0.20 M). The rate also doubles, indicating that the reaction is first order with respect to \([\text{HgCl}_2]\).

Write Rate Law Expression

The rate law expression is determined by the orders of the reactants established. Since the rate is first order in \([\text{HgCl}_2]\) and second order in \([\text{C}_2\text{O}_4^{2-}]\), the rate law is: \[ \text{Rate} = k[\text{HgCl}_2][\text{C}_2\text{O}_4^{2-}]^2. \]

Match the Rate Law with the Options

Compare the derived rate law \(\text{Rate} = k[\text{HgCl}_2][\text{C}_2\text{O}_4^{2-}]^2\) with the provided options: a. \( k[\text{HgCl}_2][\text{C}_2\text{O}_4^{2-}]^2 \). This matches our derived expression, confirming that option (a) is correct.

Key concepts

Reaction Order

In chemical kinetics, reaction order is crucial. It defines how the concentration of a reactant affects the rate of reaction. For each reactant in a rate law, there is an exponent—known as the order with respect to that reactant. The overall reaction order is the sum of these exponents. This indicates how various concentration changes influence the reaction rate. Reactants can exhibit: - **Zero order:** The rate doesn't depend on the concentration of the reactant. Changes in concentration don't affect the rate. - **First order:** The rate is directly proportional to the concentration of the reactant. Doubling the concentration doubles the rate. - **Second order:** The rate is proportional to the square of the reactant's concentration. Doubling the concentration increases the rate by a factor of four.

Initial Rates Method

The initial rates method involves conducting a series of experiments where the concentrations of reactants are varied, and the rate of reaction is measured initially. It helps determine the reaction order and rate law. Here's how it typically works: - Keep the concentration of one reactant constant while varying the other and measure the change in initial rate. - Calculate the reaction order by comparing how changes in concentrations affect the rate. In the example provided, comparing trials where one reactant concentration was kept constant revealed how the other reactant's concentration influenced the reaction rate, allowing determination of the specific order with respect to each reactant.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and factors affecting them. Understanding kinetics helps chemists control reactions to maximize yields and efficiency. Key elements include: - **Reaction Rate:** How fast reactants convert to products. Affected by temperature, concentration, surface area, and catalysts. - **Rate Law:** Mathematical expression representing the relationship between reactant concentrations and reaction rate. - **Activation Energy:** Minimum energy required for a reaction to proceed. It's a barrier that reactants must overcome. Kinetics is essential in industrial processes, pharmaceuticals, and environmental science, influencing everything from drug development to pollution control.

Reaction Mechanism

A reaction mechanism provides a step-by-step description of how reactants transform into products. It outlines each elementary step, helping us understand the overall transformation. Reaction mechanisms often include: - **Elementary Steps:** Simple reactions that occur within the overall reaction. Each step has its own rate law. - **Intermediates:** Species formed in one step and consumed in another. Not present in the overall balanced equation. - **Rate-Determining Step:** The slowest step that dictates the reaction rate. Understanding this helps in controlling reaction speeds. Knowing the mechanism allows chemists to propose and test hypotheses about how a reaction proceeds, which aids in designing more efficient chemical processes.