Problem 33 - Chemistry: The Central Science

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: (a) Write the rate law for this reaction. (b) Calculate the rate constant with proper units. (c) Calculate the rate when \(\left[\mathrm{OCI}^{-}\right]=2.0 \times 10^{-3} \mathrm{M}\) 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) cannot be calculated without more information. (c) The rate of the reaction is: \(Rate = 1.0 \times 10^{-6} k\)

Step 1: Write down the chemical reaction

: The chemical reaction between iodide ion (\(I^-\)) and hypochlorite ion (\(OCl^-\)) is given as: \[OCl^{-} + I^{-} \longrightarrow OI^{-} + Cl^{-}\]

Step 2: Identify the order of the reaction

: It is given that the reaction is rapid, meaning it is likely a first-order reaction. Therefore, the rate law can be written as: \[Rate = k[OCl^{-}][I^{-}]\] where k is the rate constant, and [\(OCl^-\)] and [\(I^-\)] are the concentrations of the reacting species.

Step 3: Calculate the rate constant (k)

: To calculate the rate constant, we need information about the rate and concentrations of the reactants at a given instant. As the information is not provided in the exercise, we cannot directly calculate k. Instead, we will use the given data in Step (c) to find the reaction rate and relate it to the rate constant.

Step 4: Calculate the reaction rate

: Given that [\(OCl^-\)] = \(2.0 \times 10^{-3}\,M\) and [\(I^-\)] = \(5.0 \times 10^{-4}\,M\), we can calculate the reaction rate using the rate law previously obtained: \[Rate = k[OCl^{-}][I^{-}] = k(2.0 \times 10^{-3}\,M)(5.0 \times 10^{-4}\,M)\] At this point, we cannot calculate the numerical value of the rate, as we were not given any information about the rate constant (k) in the exercise. However, we can find the rate in terms of the rate constant: \[Rate = 1.0 \times 10^{-6} k\] In summary: (a) The rate law for the reaction is given by: \(Rate = k[OCl^{-}][I^{-}]\) (b) The rate constant (k) was not attainable without more information. (c) The rate of the reaction can be expressed as: \(Rate = 1.0 \times 10^{-6} k\)

Key Concepts

Rate Law

In chemical kinetics, the rate law is an expression that relates the rate of a chemical reaction to the concentration of its reactants. It provides a mathematical equation that allows chemists to predict how fast a reaction will occur. The rate law typically takes the form:
\[Rate = k [A]^m [B]^n \ldots\]where:

• \(k\) is the rate constant.
• \([A]\) and \([B]\) are concentrations of reactants.
• \(m\) and \(n\) are the reaction order with respect to each reactant.

The overall order of the reaction is the sum of the powers of the concentration terms, \(m + n + \ldots\). This provides insights into the mechanism of the reaction by showing how different reactants influence its rate.
For the reaction between iodide and hypochlorite ions, the rate law is written as \(Rate = k[OCl^{-}][I^{-}]\), indicating it depends linearly on the concentration of both reactants.

Reaction Rate

The reaction rate defines how quickly a chemical reaction takes place. It is the change in concentration of a reactant or product per unit time. Reaction rates are usually expressed in molarity per second (M/s). Understanding the reaction rate provides valuable information about the dynamics of the reaction process.
In practice, the rate of a reaction is determined experimentally by measuring the concentration of a reactant or product at various times. This requires careful monitoring as conditions such as temperature and pressure can significantly affect the rate.
In our example regarding the iodide and hypochlorite ions, the reaction rate can be expressed using the equation:\[Rate = k [OCl^{-}] [I^{-}]\]This indicates the rate is directly proportional to the concentration of \([OCl^{-}]\) and \([I^{-}]\) at any given moment. By knowing this rate, scientists can predict the time required for a certain amount of reactant to be consumed.

Rate Constant

The rate constant, represented by the symbol \(k\), is a crucial parameter in a rate law equation. It serves as a proportionality constant specific to a particular reaction at a given temperature. Its value gives insights into the speed of the reaction. Higher values of \(k\) suggest a faster reaction, while lower values imply a slower one.
The units of the rate constant \(k\) depend on the overall order of the reaction:

• For a first-order reaction, \(k\) has units of s^{-1}.
• For a second-order reaction, \(k\) is expressed in M^{-1}s^{-1}.
• For a third-order reaction, k is written as M^{-2}s^{-1}.

Understanding the rate constant helps chemists understand the effect of concentration changes and conditions on the reaction rate. In our reaction between iodide and hypochlorite, \(k\) is the value we use to determine the rate at specific concentrations. However, without having its exact value, we cannot calculate the exact rate numerically.

First-Order Reaction

First-order reactions are characterized by their linear dependence on the concentration of one reactant. The rate law of a first-order reaction can be simplified to:
\[Rate = k[A]^1 = k[A]\]In such reactions, the rate constantly changes as the concentration of \([A]\) decreases over time. Because they only depend on a single reactant's concentration, it’s relatively straightforward to measure and understand their kinetics.
Key features of first-order reactions include:

• A constant half-life, independent of the starting concentration.
• An exponential decrease in the concentration of the reactant over time.
• Often observed in radioactive decay and some pharmacokinetic processes.

In our example, the reaction was presumed to be first-order based on its rapidity. However, if both \([OCl^-]\) and \([I^-]\) contribute equally to the rate in a practically identical fashion, the rate law becomes second-order instead. Therefore, understanding whether a reaction is first-order is crucial for accurately determining its kinetics.