Problem 22 - Introductory Chemistry Atoms First

Question

For the reaction $$\mathrm{BrO}_{3}^{-}+5 \mathrm{Br}^{-}+6 \mathrm{H}^{+} \rightarrow 3 \mathrm{Br}_{2}+3 \mathrm{H}_{2} \mathrm{O}$$ the experimentally determined rate law is: $$\text { Rate }=k\left[\mathrm{BrO}_{3}^{-}\right]\left[\mathrm{Br}^{-}\right]\left[\mathrm{H}^{+}\right]^{2}$$ (a) What is the order of this reaction with respect to \(\mathrm{Br}^{-}\) ? (b) What is the order of this reaction with respect to \(\mathrm{H}^{+} ?\) (c) What is the overall order of the reaction? (d) What happens to the rate of this reaction when you double the \(\mathrm{H}^{+}\) concentration?

Step-by-Step Solution

Answer

(a) The order of this reaction with respect to \(\mathrm{Br}^{-}\) is 1. (b) The order of this reaction with respect to \(\mathrm{H}^{+}\) is 2. (c) The overall order of the reaction is 4. (d) When the \(\mathrm{H}^{+}\) concentration is doubled, the rate of the reaction quadruples.

Step 1: (a) Order of reaction with respect to \(\mathrm{Br}^{-}\)

The rate law of the reaction is given by: $$\text{Rate} = k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][\mathrm{H}^{+}]^{2}$$ The order of the reaction with respect to \(\mathrm{Br}^{-}\) is represented by the exponent of \(\mathrm{Br}^{-}\) in the equation, which is 1.

Step 2: (b) Order of reaction with respect to \(\mathrm{H}^{+}\)

In the rate law, $$\text{Rate} = k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][\mathrm{H}^{+}]^{2}$$ The order of the reaction with respect to \(\mathrm{H}^{+}\) is represented by the exponent of \(\mathrm{H}^{+}\) in the equation, which is 2.

Step 3: (c) Overall order of the reaction

The overall order of the reaction is the sum of the exponents of all reactants in the rate law: Overall order = Order with respect to \(\mathrm{BrO}_{3}^{-}\) + Order with respect to \(\mathrm{Br}^{-}\) + Order with respect to \(\mathrm{H}^{+}\) In this case, the overall order is: Overall order = 1 (Order with respect to \(\mathrm{BrO}_{3}^{-}\), as it is not mentioned) + 1 (Order with respect to \(\mathrm{Br}^{-}\)) + 2 (Order with respect to \(\mathrm{H}^{+}\)) Overall order = \(1+1+2 = 4\)

Step 4: (d) Rate change upon doubling the \(\mathrm{H}^{+}\) concentration

When the \(\mathrm{H}^{+}\) concentration is doubled, let the new rate be represented as Rate': Original rate: $$\text{Rate} = k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][\mathrm{H}^{+}]^{2}$$ Increased rate: $$\text{Rate}' = k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][2\mathrm{H}^{+}]^{2}$$ Divide the increased rate by the original rate: $$\frac{\text{Rate}'}{\text{Rate}} = \frac{k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][2\mathrm{H}^{+}]^{2}}{k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][\mathrm{H}^{+}]^{2}}$$ After canceling out the constants and other reactants, we are left with: $$\frac{\text{Rate}'}{\text{Rate}} = \left(\frac{2\mathrm{H}^{+}}{\mathrm{H}^{+}}\right)^{2} = \left(2\right)^{2} = 4$$ So when the \(\mathrm{H}^{+}\) concentration is doubled, the rate of the reaction quadruples.

Key Concepts

Rate Law

The rate law of a chemical reaction provides a mathematical relationship between the concentration of reactants and the rate of the reaction. It is an experimental observation rather than a theoretical prediction. The general form is expressed as:

• \( \text{Rate} = k[A]^m[B]^n...[Z]^p \)

where \( k \) is the rate constant, \([A], [B], ...[Z]\) are the concentrations of the reactants, and \( m, n, ...p \) are the exponents representing the reaction order with respect to each reactant.

These exponents can be whole numbers, fractions, or even zero, and they are determined through experiments. For the given reaction, the rate law is:

\[ \text{Rate} = k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][\mathrm{H}^{+}]^{2} \]
It means the rate of reaction is dependent on the concentrations of \(\mathrm{BrO}_{3}^{-}\), \(\mathrm{Br}^{-}\), and \(\mathrm{H}^{+}\), with their respective orders indicated by their exponents.

Understanding the rate law is vital for predicting how changes in conditions can affect the speed of a reaction.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with understanding the speeds or rates of chemical reactions. It focuses on how different factors affect these rates and provides insight into the reaction mechanisms.

• **Reaction Rates**: Indicates how the concentration of a reactant or product changes with time. This gives an idea of how fast a reaction is proceeding.
• **Rate Constants and Temperature**: The rate constant \( k \), found in the rate law, changes with temperature. This relationship is often expressed with the Arrhenius equation.

Factors that influence reaction rates include:

• **Concentration**: Higher concentrations typically increase the rate.
• **Surface Area**: Greater surface area allows for more collisions of reactants.
• **Temperature**: Increasing temperature generally increases reaction rate.
• **Catalysts**: Substances that increase the reaction rate without undergoing permanent change themselves.

Understanding chemical kinetics is essential for various applications, including developing drugs and industrial chemical processes.

Rate of Reaction

The rate of reaction determines how quickly a reaction proceeds. It is usually measured as the change in concentration of reactants or products over time.

For the given reaction, it is expressed as:

\[ \text{Rate} = k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][\mathrm{H}^{+}]^{2} \]
This expression indicates that the rate depends directly on the concentrations of \( \mathrm{BrO}_{3}^{-} \), \( \mathrm{Br}^{-} \), and \( \mathrm{H}^{+} \). The specific orders (exponents) with respect to each reactant impact how changes in their concentrations influence the reaction rate.

• For instance, a 1st order reaction with respect to \( \mathrm{Br}^{-} \) means that if its concentration doubles, the rate doubles.
• Meanwhile, a 2nd order with respect to \( \mathrm{H}^{+} \) implies doubling its concentration increases the rate by a factor of four.

A clear understanding of the rate provides critical information for controlling reaction conditions to achieve desired outcomes.

Exponents in Rate Law

The exponents in a rate law expression indicate the reaction order with respect to each reactant. This information provides insight into how variations in reactant concentrations impact the reaction rate.

• **Order of Reaction**: Each reactant's exponent in the rate law indicates the order of reaction with respect to that reactant.

For our reaction:

\[ \text{Rate} = k[\mathrm{BrO}_{3}^{-}][\mathrm{Br}^{-}][\mathrm{H}^{+}]^{2} \]

• The order with regard to \( \mathrm{Br}^{-} \) is 1. This means that changes in \( \mathrm{Br}^{-} \) concentration will result in proportional changes in the rate.
• With respect to \( \mathrm{H}^{+} \), the order is 2. Doubling \( \mathrm{H}^{+} \) will quadruple the rate, as the rate depends on the square of the \( \mathrm{H}^{+} \) concentration.
• The order with \( \mathrm{BrO}_{3}^{-} \) is considered to be 1, implying a directly proportional relationship.

The overall reaction order is determined by summing these exponents, giving a comprehensive view of how the entire set of reactants influences the rate. This total gives insight into the complexities of the reaction mechanism itself.