Question

The iodide ion reacts with hypochlorite ion (the active ingredient in chlorine bleaches) in the following way: ${\mathrm{OCl}}^{-}+{\mathrm{I}}^{-}⟶{\mathrm{OI}}^{-}+{\mathrm{Cl}}^{-}$. This rapid reaction gives the following rate data: $$\overline{)\begin{array}{lll}\left[{\mathrm{OCl}}^{-}\right](M)& \left[I^{-}\right](M)& \text{ Initial Rate }(M/s)\\ 1.5\times {10}^{-3}& 1.5\times {10}^{-3}& 1.36\times {10}^{-4}\\ 3.0\times {10}^{-3}& 1.5\times {10}^{-3}& 2.72\times {10}^{-4}\\ 1.5\times {10}^{-3}& 3.0\times {10}^{-3}& 2.72\times {10}^{-4}\end{array}}$$ (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 $\left[{\mathrm{I}}^{-}\right]=5.0\times {10}^{-4}\mathrm{M}$

Short answer

(a) The rate law for this reaction is: Rate = k[OCl^-][I^-] (b) The rate constant k = 60.4 M^-1 s^-1 (c) The rate when [OCl^-]=2.0×10^-3 M and [I^-]=5.0×10^-4 M is 6.04×10^-5 M/s.

Step-by-step solution

a) Write the rate law for this reaction.

To write the rate law for this reaction, we need to determine the order of reaction with respect to the iodide ion (I^-) and the hypochlorite ion (OCl^-). The general rate law for a reaction aA + bB -> cC + dD is: Rate = k[A]^m[B]^n, where m and n are the orders of the reaction with respect to A and B, respectively, and k is the rate constant. Taking the ratios of initial rates and concentrations, we can find the reaction orders (m and n): From the first two experimental data sets: $\frac{2.72\times {10}^{-4}}{1.36\times {10}^{-4}}=\frac{(3.0\times {10}^{-3}{)}^m(1.5\times {10}^{-3}{)}^n}{(1.5\times {10}^{-3}{)}^m(1.5\times {10}^{-3}{)}^n}$ Simplifying, we get: $2=(2{)}^m$ So, m = 1 Now, from the first and the third experimental data sets: $\frac{2.72\times {10}^{-4}}{1.36\times {10}^{-4}}=\frac{(1.5\times {10}^{-3}{)}^m(3.0\times {10}^{-3}{)}^n}{(1.5\times {10}^{-3}{)}^m(1.5\times {10}^{-3}{)}^n}$ Simplifying, we get: $2=(2{)}^n$ So, n = 1 Therefore, the rate law for this reaction is: Rate = k[OCl^-][I^-]

b) Calculate the rate constant with proper units.

To calculate the rate constant (k), we will use the rate law and the initial rate and concentration values from any of the given data sets. Let's use the first set of data: Rate = 1.36×10^-4 M/s [OCl^-] = 1.5×10^-3 M [I^-] = 1.5×10^-3 M Using the rate law, Rate = k[OCl^-][I^-], we have: $1.36\times {10}^{-4}=k(1.5\times {10}^{-3})(1.5\times {10}^{-3})$ Solving for k, we get: $k=\frac{1.36\times {10}^{-4}}{(1.5\times {10}^{-3}{)}^2}=60.4$ The rate constant k = 60.4 M^-1 s^-1

c) Calculate the rate when [OCl^-]=2.0×10^-3 M and [I^-]=5.0×10^-4 M

Now, let's find the rate for the given concentrations of OCl^- and I^- using the determined rate constant and rate law: [OCl^-] = 2.0×10^-3 M [I^-] = 5.0×10^-4 M k = 60.4 M^-1 s^-1 Rate = k[OCl^-][I^-] Rate = (60.4)(2.0×10^-3)(5.0×10^-4) After calculating the product, we get: Rate = 6.04×10^-5 M/s Hence, the rate when [OCl^-]=2.0×10^-3 M and [I^-]=5.0×10^-4 M is 6.04×10^-5 M/s.

Key concepts

Rate Law

The rate law is a mathematical representation of the relationship between the concentration of reactants and the rate of a chemical reaction. It showcases how the rate depends on the concentration of each reactant raised to a certain power, which represents the order of the reaction for that reactant. Understanding the rate law is critical because it helps us predict how changing the concentrations of reactants affects the speed of the reaction.

For the given reaction where the iodide ion reacts with the hypochlorite ion, the rate law was determined using the principle of initial rates. The analysis of the provided experimental data allowed the derivation of the rate law: Rate = k[OCl^-][I^-], where k is the rate constant, and the exponents 1 for both OCl^- and I^- indicate that the reaction is first order with respect to both ions. This implies that doubling the concentration of either ion will double the reaction rate.

Incorporating exercise improvement, it's vital to highlight that when determining the rate law, one must not assume the stoichiometry of the overall reaction corresponds to the order with respect to each reactant. Reaction order must be empirically determined, as it has been in this exercise.

Rate Constant Calculation

Once the rate law is established, the next step is to calculate the rate constant, which is denoted by the symbol 'k'. The rate constant is unique for every reaction at a given temperature and represents the proportionality factor in the rate law equation.

In our exercise, the rate constant was calculated using the equation derived from the rate law and the given initial concentrations and rates. The use of units is essential to ensure the correct form of the rate constant is obtained. For a first-order reaction with respect to two reactants, as we have here, the units for the rate constant are M^-1s^-1, reflecting that the rate is in M/s and concentration in M (molarity).

To calculate 'k', we rearrange the rate law to solve for the rate constant, using known initial rates and concentrations from the experimental data. For example, using the first set of data, we can solve for 'k' with the formula: k = Rate/([OCl^-][I^-]). It's crucial to always pay attention to the units throughout the calculation to arrive at the correct rate constant value.

Reaction Order

The reaction order is an integral part of the rate law that signifies how the rate of reaction depends on the concentrations of reactants. Each reactant can have a different reaction order, which typically is an integer, but it can also be a fraction in some reactions. The overall reaction order is the sum of the orders with respect to all reactants.

In our case, the reaction is first order with respect to both the iodide ion and the hypochlorite ion, leading to an overall second-order reaction. The individual orders—1 for [OCl^-] and 1 for [I^-]—were deduced by observing how changing the concentrations impacted the initial rate of reaction. Understanding reaction order is paramount for determining how a reaction rate will change under various concentration conditions.

To truly grasp this concept, a student may perform dimensional analysis on the rate law and compare the exponents to the units of the rate constant to confirm the reaction orders. For instance, in a second-order reaction like our example, the units of 'k' (M^-1s^-1) reflect that when you multiply the concentration terms (each with units of M) by 'k', you get a rate with the correct units (M/s).