Problem 33 - Chemistry: The Central Science in SI Units, Global Edition

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: $$ \begin{array}{ccc} \hline\left[\mathrm{OCI}^{-}\right](M) & {\left[\mathrm{I}^{-}\right](M)} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\ \hline 1.5 \times 10^{-3} & 1.5 \times 10^{-3} & 1.36 \times 10^{-4} \\ 3.0 \times 10^{-3} & 1.5 \times 10^{-3} & 2.72 \times 10^{-4} \\ 1.5 \times 10^{-3} & 3.0 \times 10^{-3} & 2.72 \times 10^{-4} \\ \hline \end{array} $$ (a) Write the rate law for this reaction. (b) Calculate the rate constant with proper units. (c) Calculate the rate when \(\left[\mathrm{OCl}^{-}\right]=2.0 \times 10^{-3} \mathrm{M}\) and \(\left[\mathrm{I}^{-}\right]=5.0 \times 10^{-4} \mathrm{M}\)

Step-by-Step Solution

Answer

(a) The rate law for the reaction is: Rate = k[OCl⁻][I⁻] (b) The rate constant (k) is approximately 6.08 x 10² M⁻¹s⁻¹. (c) The reaction rate when [OCl⁻] = 2.0 x 10⁻³ M and [I⁻] = 5.0 x 10⁻⁴ M is approximately 6.08 x 10⁻⁵ M/s.

Step 1: Write the rate law for the reaction

To determine the rate law for the reaction, we want to find the orders of the reactants (OCl⁻ and I⁻). We can do this by examining how the initial concentrations affect the initial rate in the provided data. The rate law has the general form: Rate = k[OCl⁻]^m[I⁻]^n Where k is the rate constant, m and n are the orders of OCl⁻ and I⁻ respectively. Consider the two sets of initial concentrations: 1. [OCl⁻] = 1.5 x 10⁻³ M, [I⁻] = 1.5 x 10⁻³ M, Initial rate = 1.36 x 10⁻⁴ M/s 2. [OCl⁻] = 3.0 x 10⁻³ M, [I⁻] = 1.5 x 10⁻³ M, Initial rate = 2.72 x 10⁻⁴ M/s From these two sets of data, we can find the order m as follows: (2.72 x 10⁻⁴) / (1.36 x 10⁻⁴) = ([3.0 x 10⁻³]^m[I⁻]^n) / ([1.5 x 10⁻³]^m[I⁻]^n) 2 = (3.0 / 1.5)^m |_divide both sides by n as it cancels out_| 2 = 2^m Therefore, m = 1 Now, we look for the order n. Compare two other sets of initial concentrations: 1. [OCl⁻] = 1.5 x 10⁻³ M, [I⁻] = 1.5 x 10⁻³ M, Initial rate = 1.36 x 10⁻⁴ M/s 3. [OCl⁻] = 1.5 x 10⁻³ M, [I⁻] = 3.0 x 10⁻³ M, Initial rate = 2.72 x 10⁻⁴ M/s We find the order n as follows: (2.72 x 10⁻⁴) / (1.36 x 10⁻⁴) = ([OCl⁻]^m[3.0 x 10⁻³]^n) / ([OCl⁻]^m[1.5 x 10⁻³]^n) 2 = (3.0 / 1.5)^n |_divide both sides by m as it cancels out_| 2 = 2^n Therefore, n = 1 Now that we have determined the orders for both reactants, the rate law is: Rate = k[OCl⁻][I⁻]

Step 2: Calculate the rate constant with proper units

To find the rate constant (k), we can use any set of initial concentrations and their corresponding initial rate from the given data. Let's use the first set: 1.36 x 10⁻⁴ M/s = k(1.5 x 10⁻³ M)(1.5 x 10⁻³ M) k = ((1.36 x 10⁻⁴) M/s) / ((1.5 x 10⁻³)^2 M²) k ≈ 6.08 x 10² M⁻¹s⁻¹ Therefore, the rate constant (k) is approximately 6.08 x 10² M⁻¹s⁻¹.

Step 3: Calculate the reaction rate with given concentrations

Now that we have the rate law and the rate constant, we can calculate the reaction rate for the given concentrations: [OCl⁻] = 2.0 x 10⁻³ M [I⁻] = 5.0 x 10⁻⁴ M Rate = k[OCl⁻][I⁻] Rate = (6.08 x 10² M⁻¹s⁻¹)(2.0 x 10⁻³ M)(5.0 x 10⁻⁴ M) Rate ≈ 6.08 x 10⁻⁵ M/s Therefore, the reaction rate when [OCl⁻] = 2.0 x 10⁻³ M and [I⁻] = 5.0 x 10⁻⁴ M is approximately 6.08 x 10⁻⁵ M/s.

Key Concepts

Rate Law

The rate law is a mathematical equation that describes how the concentration of reactants affects the speed or rate of a chemical reaction. The rate law is expressed with the equation: \[ \text{Rate} = k [\text{Reactant}_1]^m [\text{Reactant}_2]^n \] where:

• Rate is the speed of the reaction.
• k is the rate constant, specific to every reaction.
• m and n are the orders of the reaction with respect to each reactant.

In our reaction between iodide ions and hypochlorite ions, the rate law is found by observing the change in initial rates with different reactant concentrations. By examining the given data, we find that the reaction is first order in both OCl⁻ and I⁻, indicated by: \[ \text{Rate} = k [\text{OCl}^-][\text{I}^-] \] This means doubling the concentration of either reactant will double the reaction rate.

Reaction Rate

Reaction rate refers to the speed at which reactants are converted to products in a chemical reaction. It is typically measured as the change in concentration of a reactant or product per unit time. Units are often in \( \text{moles per liter per second (M/s)} \). In the iodide and hypochlorite reaction we consider, the reaction rate can be explicitly calculated if the rate constant and correct concentrations are provided. For example, if given concentrations are \([\text{OCl}^-] = 2.0 \times 10^{-3} \text{ M}\) and \([\text{I}^-] = 5.0 \times 10^{-4} \text{ M}\), and a known rate constant \(k = 6.08 \times 10^2 \text{ M}^{-1}\text{s}^{-1}\), you would substitute into the rate law formula to find: \[ \text{Rate} = (6.08 \times 10^2 \text{ M}^{-1}\text{s}^{-1})(2.0 \times 10^{-3} \text{ M})(5.0 \times 10^{-4} \text{ M}) \approx 6.08 \times 10^{-5} \text{ M/s} \] By understanding how to apply these calculations, students can predict changes in reaction conditions.

Rate Constant

The rate constant \(k\) is a proportionality factor in the rate law of a reaction that quantifies the speed of a reaction at a given temperature. Each reaction has its own unique rate constant, dependent on specific conditions like temperature and the presence of catalysts. To calculate the rate constant, we rearrange the rate law to solve for \(k\): \[ k = \frac{\text{Rate}}{[\text{Reactant}_1]^m [\text{Reactant}_2]^n} \] In our iodide and hypochlorite reaction, using the provided initial data: \[ 1.36 \times 10^{-4} \text{ M/s} = k (1.5 \times 10^{-3} \text{ M})(1.5 \times 10^{-3} \text{ M}) \] Solving this equation gives: \[ k \approx 6.08 \times 10^2 \text{ M}^{-1}\text{s}^{-1} \] This calculation shows the speed of converting reactants to products goes quite fast under given conditions.

Reaction Order

The reaction order refers to the power to which the concentration of a reactant is raised in the rate law equation. Identifying the reaction order for each reactant helps in understanding the influence each has on the overall reaction rate. In the iodide ion and hypochlorite ion reaction:

• The reaction is first order with respect to both \([\text{OCl}^-]\) and \([\text{I}^-]\) as determined through experimental data comparison.
• The overall reaction order is the sum of the individual orders, here: \[ m + n = 1 + 1 = 2 \]

This means the reaction rate is proportional to the square of the concentration of the reactants. Therefore, if the concentrations of both reactants double, the reaction rate quadruples.