Question

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

Short answer

The rate law for this reaction is: \( Rate=k[HgCl_{2}][C_{2}O_{4}^{2-}]^{2} \). The rate constant, k, is approximately \(7.16 \times 10^{-3} M^{-2}s^{-1}\). When the concentration of HgCl2 is 0.100 M and the C2O42- concentration is 0.25 M, the reaction rate is approximately \(8.95 \times 10^{-5} M/s\).

Step-by-step solution

Compare the Initial Rates

To determine the rate law for the reaction, we need to find the order of the reaction with respect to HgCl2 and C2O42-. We can do this by comparing the initial rates between two experiments with the same concentration of one reactant and varying concentrations of the other. Examine experiments 1 and 4, where the concentration of C2O42- is held constant. If the reaction rate varies linearly with the change in HgCl2 concentration, this indicates that the reaction order is first-order in HgCl2. If the reaction rate changes by a factor of N when the HgCl2 concentration changes by a factor of N, the reaction is second-order. Now, compare experiments 1 and 2. The concentration of HgCl2 is held constant in these experiments. Analyze the relationship between the change in C2O42- concentration and the change in reaction rate. Again, if the relationship is linear, the reaction is first-order in C2O42-, and if the relationship is a multiple of the concentration increase, the reaction is second-order.

Determine the Reaction Orders

From experiments 1 and 4: \[ \frac{Rate_{4}}{Rate_{1}} = \frac{4.8 \times 10^{-5}}{3.2 \times 10^{-5}} = 1.5 \] \[ \frac{[HgCl_{2}]_{4}}{[HgCl_{2}]_{1}} = \frac{0.246}{0.164} = 1.5 \] Since the ratio of rates is equal to the ratio of HgCl2 concentrations, the reaction is first-order in HgCl2. From experiments 1 and 2: \[ \frac{Rate_{2}}{Rate_{1}} = \frac{2.9 \times 10^{-4}}{3.2 \times 10^{-5}} = 9.0625 \] \[ \frac{[C_{2}O_{4}^{2-}]_{2}}{[C_{2}O_{4}^{2-}]_{1}} = \frac{0.45}{0.15} = 3 \] Since the ratio of rates is equal to the square of the ratio of C2O42- concentrations, the reaction is second-order in C2O42-. Therefore, the rate law is: \[ Rate=k[HgCl_{2}][C_{2}O_{4}^{2-}]^{2} \] #Calculate the Rate Constant#

Use the Rate Law and Experimental Data

Now that we have determined the rate law, we can use the experimental data to calculate the rate constant, k. Choose any of the experiments and substitute the initial concentrations and rate values into the rate law. Then, solve for k. Using experiment 1: \[ Rate_{1} = k[HgCl_{2}]_{1}[C_{2}O_{4}^{2-}]_{1}^2 \] \[ 3.2 \times 10^{-5} M/s = k(0.164 M)(0.15 M)^2 \]

Solve for k

Now, let's solve for k: \[ k = \frac{3.2 \times 10^{-5} M/s}{(0.164 M)(0.15 M)^2} \] \[ k = 7.16 \times 10^{-3} M^{-2}s^{-1} \] So, the rate constant, k, is approximately 7.16 x 10^{-3} M^{-2}s^{-1}. #Reaction Rate with Given Concentrations#

Use the Rate Law and Calculated k

Now, we can use the calculated rate constant to find the reaction rate when the HgCl2 concentration is 0.100 M and the C2O42- concentration is 0.25 M: \[ Rate = k[HgCl_{2}][C_{2}O_{4}^{2-}]^2 \]

Calculate the Reaction Rate

Substitute the given concentrations and the calculated k value into the rate law: \[ Rate = (7.16 \times 10^{-3} M^{-2}s^{-1})(0.100 M)(0.25 M)^2 \] \[ Rate = 8.95 \times 10^{-5} M/s \] Therefore, the reaction rate with the given concentrations is approximately 8.95 x 10^{-5} M/s.

Key concepts

Rate Law

Understanding the rate law involves identifying how the rate of a chemical reaction is affected by the concentration of reactants. The rate law for a reaction cannot be deduced from the stoichiometry of the reaction; it must be determined experimentally.

The general form of the rate law can be expressed as: \[ Rate = k[A]^{m}[B]^{n} \] where \( k \) is the rate constant, \( [A] \) and \( [B] \) are the concentrations of reactants, and \( m \) and \( n \) are the reaction orders with respect to each reactant. When calculating the rate law from experimental data, as seen in the provided step-by-step solution, comparing rates under different concentrations reveals the order of the reaction with respect to each reactant.

Reaction Order

The reaction order indicates how the rate is affected by the concentration of reactants. If the rate changes proportionally with the concentration, the reaction is first-order. If it changes quadratically, it's second-order. In more complex scenarios, orders can be fractional or even zero. As detailed in the solution, when the concentration of \( \text{HgCl}_2 \) was changed by a factor of 1.5 and the rate also changed by the same factor, it indicated a first-order reaction with respect to \( \text{HgCl}_2 \). However, when the concentration of \( \text{C}_2\text{O}_4^{2-} \) was tripled and the rate increased by a factor of 9 (3 squared), it suggested a second-order reaction with respect to \( \text{C}_2\text{O}_4^{2-} \).

Rate Constant

The rate constant \( k \) is a proportionality factor that relates the rate of a reaction to the concentrations of reactants. It is determined experimentally and can be influenced by temperature, but it is independent of reactant concentrations. The units of the rate constant vary with the overall reaction order. Our step-by-step calculation used the initial rates and concentrations from an experiment to find the rate constant, which is crucial in predicting reaction rates under different conditions.

To emphasize the units, for a reaction with an overall second-order like ours, the rate constant's units will be \( M^{-1}s^{-1} \). However, since the reaction is third order in its entirety (first order in \( \text{HgCl}_2 \) and second order in \( \text{C}_2\text{O}_4^{2-} \)), the unit is \( M^{-2}s^{-1} \), as calculated in the provided solution.

Half-Life of a Reaction

The half-life of a reaction is the time taken for the concentration of a reactant to decrease by half its initial value. It is particularly important in reactions of first-order, where the half-life is constant and independent of the initial concentration. For more complex reactions, like the one considered in this exercise which is not first-order, the half-life would depend on the initial concentration and the rate constant. While the exercise asked for calculating half-life, it is essentially a practice of connecting the dots between concentration, rate constant, and time.

Concentration vs. Rate Analysis

To understand the rate at which a chemical reaction proceeds, one must analyze how varying concentrations of reactants influence the reaction rate. This is achieved by comparing experimental data where one reactant's concentration is varied while the others are held constant. The step-by-step solution for our exercise shows that as the concentration of \( \text{HgCl}_2 \) and \( \text{C}_2\text{O}_4^{2-} \) changes, the rate reacts differently. This linkage between concentration changes and rate changes provides insights into the mechanism of the reaction, driving us toward knowing precisely how to control and predict the speed of chemical processes.