Problem 93 - Chemistry: The Central Science in SI Units, Global Edition

Question

Consider the following reaction between mercury(II) chloride and oxalate ion: $$ 2 \mathrm{HgCl}_{2}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}^{2-}(a q) \longrightarrow 2 \mathrm{Cl}^{-}(a q)+2 \mathrm{CO}_{2}(g)+\mathrm{Hg}_{2} \mathrm{Cl}_{2}(s) $$ The initial rate of this reaction was determined for several concentrations of \(\mathrm{HgCl}_{2}\) and \(\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}\), and the following rate data were obtained for the rate of disappearance of \(\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}\) : $$ \begin{array}{llll} \hline \text { Experiment } & {\left[\mathrm{HgCl}_{2}\right](M)} & {\left[\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\right](M)} & \text { Rate }(M / \mathrm{s}) \\ \hline 1 & 0.164 & 0.15 & 3.2 \times 10^{-5} \\ 2 & 0.164 & 0.45 & 2.9 \times 10^{-4} \\ 3 & 0.082 & 0.45 & 1.4 \times 10^{-4} \\ 4 & 0.246 & 0.15 & 4.8 \times 10^{-5} \\ \hline \end{array} $$ (a) What is the rate law for this reaction? (b) What is the value of the rate constant with proper units? (c) What is the reaction rate when the initial concentration of \(\mathrm{HgCl}_{2}\) is \(0.100 \mathrm{M}\) and that of \(\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\) is \(0.25 \mathrm{M}\) if the temperature is the same as that used to obtain the data shown?

Step-by-Step Solution

Answer

(a) The rate law for this reaction is Rate = \(k[HgCl_2]^1[C_2O_4^{2-}]^2\). (b) The rate constant, k, is ≈ \(2.4 \times 10^{2}\) M⁻²s⁻¹. (c) The reaction rate when [HgCl2] = 0.100 M and [\(C_2O_4^{2-}\)] = 0.25 M is \(1.5 \times 10^{-4}\) M/s.

Step 1: Determine the order with respect to each reactant

Let's start by comparing experiment 1 (E1) and experiment 2 (E2). The concentrations of HgCl2 remain constant at 0.164 M, while the concentration of \(C_2O_4^{2-}\) increases by a factor of 3 (0.15 M to 0.45 M). The rate increases from \(3.2 \times 10^{-5}\) M/s to \(2.9 \times 10^{-4}\) M/s, which is a factor of approximately 9. Similarly, we can compare experiment 1 (E1) and experiment 4 (E4). This time, the concentration of \(C_2O_4^{2-}\) remains constant at 0.15 M, while the concentration of HgCl2 increases by a factor of 1.5 (0.164 M to 0.246 M). The rate increases from \(3.2 \times 10^{-5}\) M/s to \(4.8 \times 10^{-5}\) M/s, which is a factor of 1.5. Now we can determine the order of the reaction for each reactant. Since the rate increases by a factor of 9 when the concentration of \(C_2O_4^{2-}\) is tripled, the order with respect to \(C_2O_4^{2-}\) must be 2. Likewise, since the rate increases by a factor of 1.5 when the concentration of HgCl2 increases by a factor of 1.5, the order with respect to HgCl2 must be 1.

Step 2: Write the rate law

Now that we know the order of the reaction for each reactant, we can write the rate law: Rate = \(k[HgCl_2]^1[C_2O_4^{2-}]^2\)

Step 3: Calculate the rate constant (k)

We can use one of the experiments to determine the rate constant. Let's use experiment 1 (E1) for this purpose. From E1, we have: Rate = \(3.2 \times 10^{-5}\) M/s, [HgCl2] = 0.164 M, [\(C_2O_4^{2-}\)] = 0.15 M Plug in these values into the rate law: \(3.2 \times 10^{-5}\) M/s = \(k(0.164 \thinspace M)^1(0.15 \thinspace M)^2\) Solve for k: k ≈ \(2.4 \times 10^{2}\) M⁻²s⁻¹

Step 4: Calculate the reaction rate for given initial concentrations

To find the reaction rate when [HgCl2] = 0.100 M and [\(C_2O_4^{2-}\)] = 0.25 M, plug in these concentrations and the value of k into the rate law: Rate = \((2.4 \times 10^{2}\) M⁻²s⁻¹)(0.100 M)(0.25 M)²\) Rate = \(1.5 \times 10^{-4}\) M/s So, the answers are: (a) Rate law: Rate = \(k[HgCl_2]^1[C_2O_4^{2-}]^2\) (b) Rate constant, k ≈ \(2.4 \times 10^{2}\) M⁻²s⁻¹ (c) Reaction rate: \(1.5 \times 10^{-4}\) M/s

Key Concepts

Rate Law

The rate law is a mathematical expression that links the rate of a chemical reaction to the concentrations of its reactants. For the reaction of mercury(II) chloride and oxalate ion, we determine the rate law by examining experimental data. The rate law for a reaction is determined by finding how the rate depends on the concentration of each reactant. In our example, the determined rate law is:

• Rate = \(k[\text{HgCl}_2]^1[\text{C}_2\text{O}_4^{2-}]^2\)

This expression tells us how changes in the concentrations of the reactants will affect the reaction rate at a given moment. Here, \(k\) is the rate constant, \([\text{HgCl}_2]\) and \([\text{C}_2\text{O}_4^{2-}]\) are the concentrations of the reactants. Understanding the rate law is crucial for understanding reaction kinetics.

Order of Reaction

The order of a reaction with respect to a particular reactant tells us how the rate is affected by changing its concentration. In the exercise, the order with respect to each reactant was obtained through analysis of the experimental data. By comparing how the reaction rate changes with different concentrations, we determine:

• Order with respect to \(\text{HgCl}_2\) is 1.
• Order with respect to \(\text{C}_2\text{O}_4^{2-}\) is 2.

This means that doubling the concentration of \(\text{HgCl}_2\) will double the rate, while tripling the concentration of \(\text{C}_2\text{O}_4^{2-}\) will increase the rate by a factor of nine. Reaction order provides insight into the mechanism of the reaction and helps predict how changes in conditions can affect the reaction speed.

Rate Constant

The rate constant, represented by \(k\), is a crucial factor in the rate law equation. It is a proportionality constant that relates the rate of reaction to the concentrations of the reactants raised to their respective orders. The value of \(k\) is specific to a particular reaction at a given temperature. For our reaction, it was determined using the provided experimental data:

• \(k \approx 2.4 \times 10^{2}\ \text{M}^{-2}\text{s}^{-1}\)

This value was calculated by substituting known concentration values and the reaction rate into the rate law. The units of the rate constant reflect the overall order of the reaction. In this case, the reaction is third-order overall (with orders of one and two from each reactant), explaining the units of \(\text{M}^{-2}\text{s}^{-1}\). The rate constant can change with temperature and is key to determining the speed of the reaction.

Chemical Reaction Rates

Chemical reaction rates describe how quickly a reaction occurs, impacted by several factors including temperature, concentration of reactants, and the presence of catalysts. In the exercise, the rates were determined experimentally for different concentrations and used to derive the rate law and rate constant.

• The rate was calculated as \(1.5 \times 10^{-4}\ \text{M/s}\) for specific reactant concentrations.

Understanding reaction rates is important for controlling reactions in industrial and laboratory settings. It helps in designing processes to maximize efficiency and productivity while minimizing costs. Reaction rates are also vital in understanding reaction mechanisms, shedding light on how different steps in the reaction contribute to the overall speed.