Problem 86 - Chemistry: The Central Science

Question

Consider the following reaction between mercury(II) chloride and oxalate ion: $$ 2 \mathrm{HgCl}_{2}(a q)+\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}(a q) \longrightarrow{ }_{2 \mathrm{Cl}^{-}(a q)}+2 \mathrm{CO}_{2}(g)+\mathrm{Hg}_{2} \mathrm{Cl}_{2}(s) $$ The initial rate of this reaction was determined for several concentrations of \(\mathrm{HgCl}_{2}\) and \(\mathrm{C}_{2} \mathrm{O}_{4}^{2-}\), and the following rate data were obtained for the rate of disappearance of \(\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}\) \begin{tabular}{llll} \hline Experiment & {\(\left[\mathrm{HgCl}_{2}\right](M)\)} & {\(\left[\mathbf{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}\right](M)\)} & 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{tabular} (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 \(\left(\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}\right)\) 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 \(= k[\mathrm{HgCl}_{2}][\mathrm{C}_{2}\mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}]^{3}\) (b) The value of the rate constant is \(3.674 \times 10^{-2} \mathrm{M^{-3}s^{-1}}\). (c) The reaction rate when the concentration of \(\mathrm{HgCl}_{2}\) is \(0.100 \mathrm{M}\) and that of \((\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-})\) is \(0.25 \mathrm{M}\) is \(5.7 \times 10^{-5} \mathrm{M \cdot s^{-1}}\).

Step 1: Determine the order with respect to HgCl2

We can determine the order with respect to each reactant by comparing initial rates in different experiments. Let's first determine the order of the reaction with respect to [\(\mathrm{HgCl}_{2}\)]: Compare Experiment 1 to Experiment 4: In both experiments, the concentration of \(C_{2}O_{4}^{2-}\) remains the same. This allows us to see the effect of changing the concentration of \(\mathrm{HgCl}_{2}\) on the reaction rate. \[\frac{Rate_{4}}{Rate_{1}} = \frac{4.8 \times 10^{-5}}{3.2 \times 10^{-5}} = 1.5\] \[\frac{[\mathrm{HgCl}_{2}]_{4}}{[\mathrm{HgCl}_{2}]_{1}} = \frac{0.246}{0.164} = 1.5\] Since the ratio of the rates is equal to the ratio of concentrations, we can conclude that the reaction is first order with respect to \(\mathrm{HgCl}_{2}\).

Step 2: Determine the order with respect to C2O4^2-

Now, let's determine the order of the reaction with respect to \(C_{2}O_{4}^{2-}\): Compare Experiment 1 to Experiment 2: In both experiments, the concentration of \(\mathrm{HgCl}_{2}\) remains the same. This allows us to see the effect of changing the concentration of \(C_{2}O_{4}^{2-}\) on the reaction rate. \[\frac{Rate_{2}}{Rate_{1}} = \frac{2.9 \times 10^{-4}}{3.2 \times 10^{-5}} = 9\] \[\frac{[C_{2}O_{4}^{2-}]_{2}}{[C_{2}O_{4}^{2-}]_{1}} = \frac{0.45}{0.15} = 3\] Since the ratio of the rates equals the cube of the ratio of the concentrations, we can conclude that the reaction is third order with respect to \(C_{2}O_{4}^{2-}\).

Step 3: Determine the rate law and rate constant

Now that we know that the reaction is first order with respect to \(\mathrm{HgCl}_{2}\) and third order with respect to \(C_{2}O_{4}^{2-}\), we can write the rate law: Rate \(= k[\mathrm{HgCl}_{2}][\mathrm{C}_{2}\mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-}]^{3}\) To find the rate constant \(k\), we can use the data from any of the experiments. Let's use Experiment 1: \(3.2 \times 10^{-5} = k(0.164)(0.15)^{3}\) Solving for \(k\): \(k = \frac{3.2 \times 10^{-5}}{(0.164)(0.15)^{3}} = 3.674 \times 10^{-2} \mathrm{M^{-3}s^{-1}}\)

Step 4: Find the reaction rate under specific conditions

Now we can use our rate law and rate constant to find the reaction rate when the concentration of \(\mathrm{HgCl}_{2}\) is \(0.100 \mathrm{M}\) and that of \((\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-})\) is \(0.25 \mathrm{M}\): Rate \(= (3.674 \times 10^{-2}\mathrm{ M^{-3}s^{-1}})(0.100\mathrm{M})(0.25\mathrm{M})^{3}\) Rate \(= 5.7 \times 10^{-5} \mathrm{M \cdot s^{-1}}\) The reaction rate when the concentration of \(\mathrm{HgCl}_{2}\) is \(0.100 \mathrm{M}\) and that of \((\mathrm{C}_{2} \mathrm{O}_{4}{\underline{\phantom{xx}}}^{2-})\) is \(0.25 \mathrm{M}\) is \(5.7 \times 10^{-5} \mathrm{M \cdot s^{-1}}\).

Key Concepts

Reaction Rate

The reaction rate is a measure of how quickly a chemical reaction proceeds. Specifically, it refers to the change in concentration of a reactant or product per unit time. For example, in our given reaction involving mercury(II) chloride and oxalate ion, reaction rate is examined as the rate of disappearance of the oxalate ion (\(C_2O_4^{2-}\)). This can be generally expressed as a negative change in the concentration of a reactant divided by the change in time, \(-\Delta[Reactant]/\Delta t\), or the change in concentration of a product over the change in time, \(\Delta[Product]/\Delta t\). In practice, reaction rates can be affected by various factors, such as the concentrations of reactants, temperature, presence of a catalyst, and surface area of reactants. This is important in understanding the kinetics of a reaction, as it dictates how fast a product can be formed from the reactants.

Rate Law

The rate law is an equation that relates the reaction rate to the concentrations of reactants. It's a crucial aspect of chemical kinetics because it allows chemists to predict the speed of a reaction under different conditions. The general form of a rate law is \(Rate = k[A]^{m}[B]^{n}\), where \(k\) is the rate constant, and \(m\) and \(n\) are the orders of the reaction with respect to reactants A and B, respectively. The exponents represent the influence of each reactant's concentration on the rate of the reaction.For instance, the rate law for our reaction, determined experimentally, turned out to be \(Rate = k[HgCl_{2}][C_{2}O_{4}^{2-}]^{3}\). This indicates that the reaction rate is directly proportional to the concentration of mercury(II) chloride and to the cube of the concentration of oxalate ion. Accurate determination of the rate law is important, as it provides insights into the mechanism of the reaction.

Rate Constant

The rate constant, designated as \(k\), is the proportionality factor in the rate law equation. Its value is specific to each chemical reaction and can be influenced by various external conditions, most notably temperature. The value of the rate constant offers information about the reaction's speed — a larger \(k\) signifies a faster reaction under the same conditions.In our exercise, the rate constant is determined after establishing the orders of the reactants. By rearranging the rate law and inserting the values from one of the experiments, we can calculate \(k\). It's interesting to note that while the units of \(k\) in a rate law depend on the overall reaction order, in this case, with a fourth-order reaction, its units are precisely \(M^{-3}s^{-1}\). Whenever experimental conditions change, for example with temperature, the value of \(k\) would also change, thus affecting the reaction rate.

Reaction Order

Reaction order is a term in chemical kinetics that refers to the power to which the concentration of a reactant is raised in the rate law. It indicates the dependency of the reaction rate on the concentration of that specific reactant. The overall reaction order is the sum of the orders with respect to each of the reactants involved.In the provided experiment, we found that the reaction is first order with respect to \(HgCl_{2}\) and third order with respect to \(C_2O_4^{2-}\), making the overall reaction order fourth. This overall order is not simply determined by stoichiometry of the reaction equation but through experimental data. The concept of reaction order is vital in understanding reaction mechanisms and designing chemical processes because it helps predict how changes in concentrations will affect the reaction rate.