Question

The iodide ion reacts with hypochlorite ion (the active ingredient in chlorine bleaches) in the following way: \(\mathrm{OCl}^{-}+\mathrm{I}^{-} \longrightarrow \mathrm{OI}^{-}+\mathrm{Cl}^{-} .\) This rapid reaction gives the following rate data: \begin{tabular}{lll} \hline \(\left.\mathrm{OCl}^{-}\right](\mathrm{M})\) & {\(\left[\mathrm{I}^{-}\right](M)\)} & Rate \((M / \mathrm{s})\) \\ \hline \(1.5 \times 10^{-3}\) & \(1.5 \times 10^{-3}\) & \(1.36 \times 10^{-4}\) \\ \(3.0 \times 10^{-3}\) & \(1.5 \times 10^{-3}\) & \(2.72 \times 10^{-4}\) \\ \(1.5 \times 10^{-3}\) & \(3.0 \times 10^{-3}\) & \(2.72 \times 10^{-4}\) \\ \hline \end{tabular} (a) Write the rate law for this reaction. (b) Calculate the rate constant. (c) Calculate the rate when \(\left[\mathrm{OCl}^{-}\right]=\) \(2.0 \times 10^{-3} M\) and \(\left[1^{-}\right]=5.0 \times 10^{-4} M\)

Short answer

(a) The rate law for this reaction is: \(rate = k[\mathrm{OCl}^{-}][\mathrm{I}^{-}]\) (b) The rate constant (k) is \(60.44 M^{-1}s^{-1}\) (c) The rate when \([\mathrm{OCl}^{-}]= 2.0 \times 10^{-3} M\) and \([\mathrm{I}^{-}]= 5.0 \times 10^{-4} M\) is \(6.044 \times 10^{-5} M/s\).

Step-by-step solution

Determine the order of the reaction with respect to each reactant

To determine the order of the reaction, we need to study how the rate changes with the reactant concentrations. By comparing the data given in the table, we can determine the order of the reaction for each reactant. Let the rate law for the reaction be: \[rate = k[\mathrm{OCl}^{-}]^m[\mathrm{I}^{-}]^n\] Comparing the first and second rows of the table, we can observe that when the concentration of both reactants doubles, but the concentration of iodide ion is the same, the rate also doubles. Therefore, \(\frac{2.72\times 10^{-4}}{1.36\times 10^{-4}}=2\) \(\frac{k(3.0\times10^{-3})^m(1.5\times10^{-3})^n}{k(1.5\times 10^{-3})^m(1.5\times 10^{-3})^n}=2\) \(2^m=2\) From this equation, we can conclude that \(m=1\). Now let's compare the first row and third row. We can see that when the concentration of hypochlorite ion remains constant and the concentration of iodide ion doubles, the rate also doubles: \(\frac{2.72\times 10^{-4}}{1.36\times 10^{-4}}=2\) \(\frac{k(1.5\times 10^{-3})^m(3.0\times 10^{-3})^n}{k(1.5\times 10^{-3})^m(1.5\times 10^{-3})^n}=2\) \(2^n=2\) From this equation, we can conclude that \(n=1\).

Write the rate law

Now that we have determined the order of the reaction with respect to each reactant, we can write the rate law: rate = k[\(\mathrm{OCl}^{-}\)]^{1}[\(\mathrm{I}^{-}\)]^{1}

Calculate the rate constant (k) from given data

To find the rate constant (k), we can use any row of the given data. Let's use the first row. Rearranging the rate law formula and plugging in the values: \(k=\frac{rate}{[\mathrm{OCl}^{-}][\mathrm{I}^{-}]}\) \(k=\frac{1.36 \times 10^{-4}}{[(1.5 \times 10^{-3})(1.5 \times 10^{-3})]}\) \(k=60.44 M^{-1}s^{-1}\)

Calculate the rate when \({[\mathrm{OCl}^{-}]= 2.0 \times 10^{-3} M}\) and \({[\mathrm{I}^{-}] = 5.0 \times 10^{-4} M}\)

Finally, we can use the calculated rate constant k, and the given concentrations of reactants to compute the rate: rate = \(60.44 M^{-1}s^{-1}\)(\(2.0 \times 10^{-3} M\))(\(5.0 \times 10^{-4} M\)) rate = \(6.044 \times 10^{-5} \frac{M}{s}\) Thus, the rate when \({[\mathrm{OCl}^{-}]= 2.0 \times 10^{-3} M}\) and \({[\mathrm{I}^{-}]= 5.0 \times 10^{-4} M}\) is \(6.044 \times 10^{-5} M/s\).

Key concepts

Understanding Rate Laws in Chemical Kinetics

The rate law is a mathematical equation that describes the relationship between the rate of a chemical reaction and the concentration of its reactants. It's an essential component of chemical kinetics, the branch of physical chemistry that deals with understanding the rates of chemical processes.

A rate law can typically be expressed in the form: \[rate = k[\text{A}]^{m}[\text{B}]^{n}\]where \(rate\) is the reaction rate, \(k\) is the rate constant, \(m\) and \(n\) are the orders of the reaction with respect to reactants A and B, and \([\text{A}]\) and \([\text{B}]\) are their respective concentrations.

The orders of the reaction (the exponents \(m\) and \(n\)) are determined experimentally and indicate how sensitive the rate is to changes in reactant concentrations. In some cases, like in our example problem with the iodide ion and hypochlorite ion, these orders are integers, but they can also be zero, fractional, or negative, depending on the reaction mechanism. Understanding how to derive a rate law from experimental data, like the given concentration and rate pairs, is crucial for studying reaction kinetics and predicting how a reaction will proceed under different conditions.

Deciphering Reaction Order

The reaction order is a key concept in chemical kinetics that indicates the dependency of the reaction rate on the concentration of each reactant. In our textbook exercise, we've seen this dependence characterized by exponents in the rate law equation.

In a rate law like \(rate = k[\text{A}]^{m}[\text{B}]^{n}\), we would say the reaction is of order \(m\) with respect to reactant A and order \(n\) with respect to reactant B. The overall reaction order is the sum of these individual orders, in this case, \(m + n\).

For instance, if we determine that both \(m\) and \(n\) are 1, as was done in our exercise, the reaction is first-order with respect to both A and B and second-order overall. The reaction order helps predict how a change in concentration affects the reaction rate, which is indispensable for controlling chemical processes in industrial applications and research.

Understanding the Rate Constant

The rate constant, symbolized as \(k\), is a proportionality factor in the rate law equation that provides the rate of a reaction when all reactant concentrations are at one molar. Importantly, the value of the rate constant is specific to a particular reaction at a given temperature and does not change with the concentrations of reactants. It only changes with temperature or if a catalyst is added to the reaction.

In our exercise, by manipulating the provided data, we calculated the rate constant for the iodide and hypochlorite ion reaction. This constant, once determined, is a powerful piece of information. It allows us to predict the rate of reaction for any combination of reactant concentrations at the same temperature, making the rate constant a fundamental figure in chemical kinetics for designing and optimizing chemical reactions in both laboratory and industrial settings. The units of the rate constant also offer insight into the reaction order, as they change depending on the overall order of the reaction.

Significance of Reactant Concentration

Reactant concentration is a term denoting the amount of reactant present in a given volume of solution, typically expressed in molarity (moles per liter). It plays a crucial role in chemical kinetics as it directly influences the rate of reaction; higher reactant concentrations generally lead to faster reactions.

In our example problem, we saw that the rate of reaction depends linearly on the concentrations of both iodide and hypochlorite ions, reflecting their first-order dependence. The ability to manipulate and measure reactant concentration gives chemists and chemical engineers the ability to control the rate of reactions, which is essential for procedures ranging from chemical synthesis to pharmaceutical drug development. The relationship between reactant concentration and reaction rate is core to understanding how reactions occur and how to effectively influence them.