Question

The following set of data was obtained by the method of initial rates for the reaction: $$ \begin{array}{r} \mathrm{BrO}_{3}^{-}(\mathrm{aq})+5 \mathrm{Br}(\mathrm{aq})+6 \mathrm{H}^{+}(\mathrm{aq}) \rightarrow \\ 3 \mathrm{Br}_{2}(\mathrm{aq})+3 \mathrm{H}_{2} \mathrm{O} \text { (1) } \end{array} $$ Calculate the initial rate when \(\mathrm{BrO}_{3}^{-}\)is \(0.30 \mathrm{M}\), \(\mathrm{Br}\) is \(0.050 \mathrm{M}\) and \(\mathrm{H}^{+}\)is \(0.15 \mathrm{M}\). $$ \begin{array}{llll} \hline\left[\mathrm{BrO}_{3}^{-}\right], \mathrm{M} & {[\mathrm{Br}], \mathrm{M}} & {\left[\mathrm{H}^{+}\right], \mathrm{M}} & \text { Rate, } \mathrm{M} / \mathrm{s} \\ \hline 0.10 & 0.10 & 0.10 & 8.0 \times 10^{-4} \\ 0.20 & 0.10 & 0.10 & 1.6 \times 10^{-3} \\ 0.20 & 0.15 & 0.10 & 2.4 \times 10^{-3} \\ 0.10 & 0.10 & 0.25 & 5.0 \times 10^{-3} \\ \hline \end{array} $$ a. \(3.17 \times 10^{-4} \mathrm{M} / \mathrm{s}\) b. \(6.7 \times 10^{-3} \mathrm{M} / \mathrm{s}\) c. \(2.7 \times 10^{-3} \mathrm{M} / \mathrm{s}\) d. \(1.71 \times 10^{-3} \mathrm{M} / \mathrm{s}\)

Short answer

The initial rate is \(2.7 \times 10^{-3} \mathrm{M/s}\) (Option c).

Step-by-step solution

Identifying the Rate Law

Given the reaction, the rate law can generally be written as: \( \text{Rate} = k[\text{BrO}_3^-]^x[\text{Br}^-]^y[\text{H}^+]^z \), where \( k \) is the rate constant and \( x \), \( y \), and \( z \) are the orders with respect to \( \text{BrO}_3^- \), \( \text{Br}^- \), and \( \text{H}^+ \), respectively.

Determine the Order of Reaction for Each Reactant

By examining the changes in concentration and their effects on the rate:1. Compare Experiments 1 and 2: ([\text{BrO}_3^-]) is doubled, other concentrations constant, rate doubles. \( x = 1 \).2. Compare Experiments 2 and 3: ([\text{Br}^-]) increases 50%, other concentrations remain constant, rate increases by 50%. \( y = 1 \).3. Compare Experiments 1 and 4: ([\text{H}^+]) increases 2.5 times, other concentrations constant, rate increases 6.25 times. \(1.25 = 2.5^z\), so \( z = 2 \).

Calculate the Rate Constant \( k \)

Using Experiment 1 data: \( \text{Rate} = k[\text{BrO}_3^-]^1[\text{Br}^-]^1[\text{H}^+]^2 \)\( 8.0 \times 10^{-4} = k(0.10)(0.10)(0.10)^2 \)\( k = \frac{8.0 \times 10^{-4}}{0.10 \times 0.10 \times (0.10)^2} = 8 \).

Calculate Initial Rate for Given Concentrations

Substitute the given concentrations into the rate law:\( [\text{BrO}_3^-] = 0.30, [\text{Br}^-] = 0.050, [\text{H}^+] = 0.15 \).\( \text{Rate} = 8(0.30)(0.050)(0.15)^2 \).Calculate:\( \text{Rate} = 8 \times 0.30 \times 0.050 \times 0.15^2 \).

Simplifying the Expression

Calculate each part separately:\( 0.15^2 = 0.0225 \).\( 8 \times 0.30 = 2.4 \).\( 2.4 \times 0.050 = 0.12 \).\( 0.12 \times 0.0225 = 0.0027 \).Thus, the rate is \( 2.7 \times 10^{-3} \text{ M/s} \).

Key concepts

Rate Equation

In chemical kinetics, the rate equation is key to understanding how reactant concentrations affect the speed of a reaction. It is also known as the rate law. For a given reaction, the rate equation takes the form: \[ \text{Rate} = k[A]^x[B]^y[C]^z \] - Here, \(k\) is the rate constant, which is unique for each reaction and varies with temperature. - \( [A] \, [B] \, [C] \) represent the molar concentrations of the reactants involved in the chemical reaction.- The exponents \(x, y, z\) are the orders of reaction with respect to each reactant and must be determined experimentally. These exponents reveal how changes in concentration affect the reaction rate. If an exponent, like \(x\), is one, doubling the concentration of that reactant doubles the rate, assuming other factors are constant. Understanding the rate equation helps us predict how certain conditions will change the reaction's pace.

Order of Reaction

The order of reaction refers to the powers to which the concentrations of reactants are raised in the rate equation. It summarizes how changes in concentration impact the rate of reaction. Each reactant has its own order, which need not be an integer, and they can be summed to give the overall reaction order. For the given reaction, - Considering \( BrO_3^- \, Br^- \, ext{and} \, H^+ \, \) the orders are found to be 1, 1, and 2 respectively. - Therefore, the reaction is first-order with respect to \( BrO_3^- \) and \( Br^- \, \) and second-order with respect to \( H^+ \). - The overall order is the sum of these individual orders: \( 1 + 1 + 2 = 4 \).Understanding the order of reaction gives insight into the mechanism by which a reaction proceeds. It's crucial for calculating reaction rates especially in complex reactions where multiple steps are involved.

Rate Constant

The rate constant, represented by \( k \, \) in the rate equation, is a crucial factor in determining reaction speed. Its value depends on the specific reaction and conditions like temperature. The rate constant remains unchanged as long as temperature and catalyst remain the same.During the calculation using trial data: - \( k \) was determined using initial concentrations and observed rate for one of the experiments. - In the example, by substituting known values from Experiment 1 into the rate equation, we found: \[ k = \frac{8.0 \times 10^{-4} \, \text{M/s}}{(0.10)^1 (0.10)^1 (0.10)^2} = 8 \]Once known, \( k \) can be used to calculate rates under different conditions or even predict unknown data points. Its understanding helps in designing reactions that require specific timing or rates for industrial processes.

Reaction Kinetics

Reaction kinetics is the study of rates at which chemical processes occur. It involves examining different factors and conditions that influence the reaction's speed. Key factors include: - **Concentration:** As shown, increasing concentration of reactants often increases rate. - **Temperature:** Usually, raising temperature increases reaction rates, often exponentially. - **Catalysts:** They alter reaction pathways to lower energy barriers, increasing reaction rate without being consumed. The initial rate method is frequently used in kinetics. It involves measuring how instantaneous rate changes when initial concentrations of reactants are altered. This method was used in the example provided, revealing how each reactant’s concentration specifically affects rate. Understanding reaction kinetics is crucial for controlling chemical processes in industry, biology, and environmental systems. It enables the drawing of connections between microscopic molecular movements and macroscopic observable behavior.