Problem 33 - Chemistry The Central Science In Si Units

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}{lcc} \hline\(\left[0 \mathrm{Cl}^{-}\right](M)\) & {\(\left[I^{-}\right](M)\)} & Initial Rate \((M / s)\) \\ \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 with proper units. (c) Calculate the rate when \(\left[\mathrm{OCl}^{-}\right]=2.0 \times 10^{-3} \mathrm{M}\) and \(\left[1^{-}\right]=5.0 \times 10^{-4} \mathrm{M}\).

Step-by-Step Solution

Answer

(a) Rate = k[OCl⁻][I⁻]. (b) k = 60.44 M⁻¹s⁻¹. (c) Rate = 6.044 × 10⁻⁵ M/s.

Step 1: Determine the Order with Respect to Each Reactant

To find the order of the reaction with respect to each reactant, compare the rates from trials where only one reactant concentration changes while the other remains constant. Comparing the first two trials, when \([\mathrm{OCl}^{-}]\) doubles and \([I^-]\) is constant, the rate also doubles. So, the reaction is first order with respect to \([\mathrm{OCl}^{-}]\). In the first and third trials, doubling \([I^-]\) while keeping \([\mathrm{OCl}^{-}]\) constant also results in the rate doubling, indicating first order with respect to \([I^-]\).

Step 2: Write the Rate Law

Since the reaction is first order with respect to both \([\mathrm{OCl}^{-}]\) and \([I^-]\), the rate law can be written as: \( \text{Rate} = k[\mathrm{OCl}^{-}]^1[I^-]^1 \) or simply \( \text{Rate} = k[\mathrm{OCl}^{-}][I^-] \).

Step 3: Calculate the Rate Constant (k)

Using the rate law \( \text{Rate} = k[\mathrm{OCl}^{-}][I^-] \) and the data from the first trial \( \text{Rate} = 1.36 \times 10^{-4} \, M/s\), substitute the concentrations: \( k = \frac{1.36 \times 10^{-4}}{1.5 \times 10^{-3} \times 1.5 \times 10^{-3}} = \frac{1.36 \times 10^{-4}}{2.25 \times 10^{-6}} = 60.44 \). The units of k will be \( M^{-1}s^{-1} \) because the reaction is overall second order.

Step 4: Calculate the Rate for Given Concentrations

To find the rate when \([\mathrm{OCl}^{-}] = 2.0 \times 10^{-3} \, M \) and \([I^-] = 5.0 \times 10^{-4} \text{ M} \), use the previously calculated rate constant \( k = 60.44 \text{ M}^{-1}\text{s}^{-1} \): \( \text{Rate} = 60.44 \times (2.0 \times 10^{-3}) \times (5.0 \times 10^{-4}) = 60.44 \times 1.0 \times 10^{-6} = 6.044 \times 10^{-5} \text{ M/s} \).

Key Concepts

Rate Law

In chemical kinetics, the rate law is an equation that relates the rate of a chemical reaction to the concentration of its reactants. For the reaction between iodide ions and hypochlorite ions, the rate law shows how changes in the concentrations of these ions affect the reaction rate. It expresses the general form as:

• Rate = k [Reactant1]^a [Reactant2]^b

Here, 'k' represents the rate constant, while 'a' and 'b' are the orders with respect to each reactant, determined experimentally. The coefficients 'a' and 'b' specify how sensitive the reaction rate is to changes in the concentration of the reactants. In our specific reaction, since both iodide ( [I^-] ) and hypochlorite ( [OCl^-] ) show a first-order dependence, the rate law simplifies to:

• Rate = k[ [OCl^-] ][ [I^-] ]

Determining the rate law allows chemists to predict the behavior of the reaction under various conditions.

Reaction Order

Reaction order refers to the power to which the concentration of a reactant is raised in the rate law equation. It reveals how sensitive the rate of reaction is to changes in concentration.

• First-order: The rate doubles if the concentration of the reactant is doubled.
• Second-order: The rate quadruples if the concentration is doubled.

Understanding reaction order is crucial as it helps in predicting how the concentration affects the speed of the reaction. In the example provided, the reaction is first order with respect to both OCl^- and I^-. Therefore, the overall order of the reaction is second order, calculated by adding up the individual orders: 1 (for OCl^-) + 1 (for I^-) = 2. Knowing the reaction order helps in devising methods to control the reaction speed effectively in industrial and laboratory settings.

Rate Constant

The rate constant, denoted as k, is a crucial component of the rate law equation, representing the rate of reaction when the concentrations of the reactants are at unity. It is a constant value unique to each reaction and is independent of the concentrations of reactants but can be affected by temperature and catalysts.

• Units of k vary: For a second-order reaction, like the example, the units are M^-1s^-1.
• Calculated by rearranging the rate law: k = Rate / ([OCl^-] [I^-])

In the exercise provided, the rate constant is determined using the experimental data: k = 60.44 M^-1s^-1. This tells us about the proportionality and how fast the reaction proceeds under the measured conditions. Knowing k helps chemists understand the reaction's pace and design experiments or industrial processes to optimize reaction speeds.