Problem 34 - Chemistry: The Central Science in SI Units, Global Edition

Question

The reaction \(2 \mathrm{ClO}_{2}(a q)+2 \mathrm{OH}^{-}(a q) \longrightarrow \mathrm{ClO}_{3}^{-}(a q)+\) \(\mathrm{ClO}_{2}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\) was studied with the following results: $$ \begin{array}{lccc} \hline \text { Experiment } & {\left[\mathrm{CIO}_{2}\right](M)} & {\left[\mathrm{OH}^{-}\right](M)} & \text { Initial Rate }(M / s) \\ \hline 1 & 0.060 & 0.030 & 0.0248 \\ 2 & 0.020 & 0.030 & 0.00276 \\ 3 & 0.020 & 0.090 & 0.00828 \\ \hline \end{array} $$ (a) Determine the rate law for the reaction. (b) Calculate the rate constant with proper units. (c) Calculate the rate when \(\left[\mathrm{ClO}_{2}\right]=0.100 \mathrm{M}\) and \(\left[\mathrm{OH}^{-}\right]=0.050 \mathrm{M}\)

Step-by-Step Solution

Answer

(a) The rate law for the reaction is Rate = k[ClO2]^2[OH-]^1. (b) The rate constant (k) is 9.2 M^(-2)s^(-1). (c) The rate of the reaction when [ClO2] = 0.100 M and [OH-] = 0.050 M is 0.046 M/s.

Step 1: Determine the order of the reaction with respect to each reactant

From the given table, we can compare the rates of the different experiments to determine the order of the reaction with respect to each reactant. Consider experiments 1 and 2. The [ClO2] in experiment 1 is 3 times that of experiment 2, while [OH-] remains the same. Comparing the rates of these two experiments, we get: \(\frac{0.0248}{0.00276} = 9\) Since the [ClO2] increased by a factor of 3, and the rate increased by a factor of 9, we can deduce that the order of the reaction with respect to ClO2 is 2 (first order: rate increase would be 3; second order: rate increase would be \(3^2 = 9\); etc.) Now, let's consider experiments 2 and 3. The [OH-] in experiment 3 is 3 times that of experiment 2 while the [ClO2] remains the same. Comparing the rates of these two experiments, we get: \(\frac{0.00828}{0.00276} = 3\) Since the [OH-] increased by a factor of 3, and the rate increased by a factor of 3 as well, we can deduce that the order of the reaction with respect to OH- is 1 (first-order reaction since rate increase is same as concentration increase). Thus, the rate law for the reaction can be written as: Rate = k[ClO2]^2[OH-]^1

Step 2: Calculate the rate constant

Now that we have determined the rate law, we can calculate the rate constant 'k' using the concentration and rate data from any of the experiments. We will use the data from experiment 1. Rate = k[ClO2]^2[OH-]^1 0.0248 = k(0.060)^2(0.030) Solving for k: k = \(\frac{0.0248}{(0.060)^2(0.030)}\) = 9.2 M^(-2)s^(-1) The rate constant (k) is 9.2 M^(-2)s^(-1).

Step 3: Calculate the rate for given concentrations

Now that we have the rate law and the rate constant, we can calculate the rate of the reaction when [ClO2] = 0.100 M and [OH-] = 0.050 M. Rate = k[ClO2]^2[OH-]^1 Rate = (9.2 M^(-2)s^(-1))(0.100 M)^2(0.050 M) Rate = 0.046 M/s The rate of the reaction when [ClO2] = 0.100 M and [OH-] = 0.050 M is 0.046 M/s.

Key Concepts

Reaction Order

The reaction order tells us how a reaction rate is affected by the concentration of each reactant. In this exercise, we determine the reaction order by comparing the initial rates of different experiments while altering reactant concentrations.

By examining how changes in the concentration of reactants affect the rate, we deduce whether the reaction is first order, second order, or another type.

For instance, holding the concentration of OH- constant while varying ClO2 helps us observe the direct impact of ClO2 on the reaction rate.

• If the rate changes in direct proportion to a reactant's concentration change, it indicates a first-order reaction concerning that reactant.
• If the rate changes as the square (or some other power) of the concentration change, it indicates a second (or higher) order reaction.

In this exercise, the data shows that the reaction is second order with respect to ClO2 (since the rate triples when the concentration triples) and first order with respect to OH-.

Rate Law

The rate law is an equation that relates the reaction rate to the concentration of reactants. This model allows us to predict the rate under different conditions.

In our example, the rate law was determined based on the reaction orders derived. The general form is:

• Rate = k[A]^m[B]^n

where m and n are the orders of the reaction with respect to reactants A and B, respectively, and k is the rate constant. Using the reaction orders determined earlier, our rate law is expressed as:

• Rate = k[ClO2]^2[OH-]^1

The exponents reflect the reaction order for ClO2 and OH-. This expression tells us how the rate is affected by each reactant. Knowing the rate law is crucial for calculating the reaction rate under different concentration scenarios.

Rate Constant

The rate constant, denoted as k, is a crucial part of the rate law that combines with the reactant concentrations to predict the reaction rate. It is notably affected by factors such as temperature and the presence of a catalyst.

Its units depend on the overall order of the reaction. For example, in our second-order reaction with respect to ClO2 and first-order with respect to OH-, the rate constant has units of M^(-2)s^(-1).

• Rate constants help in comparing the speeds of different reactions under similar conditions.
• They're determined experimentally, as shown through calculations using data from one of the experiments.

In this exercise, using the given concentration data from Experiment 1, the rate constant is calculated to be 9.2 M^(-2)s^(-1). Understanding the rate constant is key for accurately modeling the kinetics of chemical reactions.