Problem 51 - Chemistry The Science in Context

Question

Predict the rate law for the reaction \(2 \mathrm{Br} \mathrm{O}(g) \rightarrow \mathrm{Br}_{2}(g)+\) \(\mathrm{O}_{2}(g)\) if: a. The rate doubles when \([\mathrm{Br} \mathrm{O}]\) doubles. b. The rate quadruples when \([\mathrm{BrO}]\) doubles. c. The rate is halved when \([\mathrm{BrO}]\) is halved. d. The rate is unchanged when \([\mathrm{BrO}]\) is doubled.

Step-by-Step Solution

Answer

Question: For each of the following scenarios, determine the order of reaction with respect to BrO and write the rate law accordingly: a) The rate doubles when the concentration of BrO doubles. b) The rate quadruples when the concentration of BrO doubles. c) The rate is halved when the concentration of BrO is halved. d) The rate is unchanged when the concentration of BrO is doubled.

Step 1: Scenario a: Rate doubles when [BrO] doubles

If the rate doubles when the concentration of BrO doubles, it means that the reaction is first-order with respect to BrO. The rate law for this scenario can be written as: Rate \(= k [\mathrm{BrO}]^1\)

Step 2: Scenario b: Rate quadruples when [BrO] doubles

If the rate quadruples when the concentration of BrO doubles, it means that the reaction is second-order with respect to BrO. The rate law for this scenario can be written as: Rate \(= k [\mathrm{BrO}]^2\)

Step 3: Scenario c: Rate is halved when [BrO] is halved

If the rate is halved when the concentration of BrO is halved, it means that the reaction is first-order with respect to BrO. The rate law for this scenario can be written as: Rate \(= k [\mathrm{BrO}]^1\)

Step 4: Scenario d: Rate is unchanged when [BrO] is doubled

If the rate doesn't change when the concentration of BrO is doubled, it means that the reaction is zero-order with respect to BrO. The rate law for this scenario can be written as: Rate \(= k [\mathrm{BrO}]^0\) In this case, the rate only depends on the rate constant k, and not on the concentration of BrO.

Key Concepts

Chemical Kinetics

Chemical kinetics is an essential part of understanding how reactions occur. This branch of chemistry deals with the rates at which chemical processes take place and what factors affect these rates. A fundamental aspect of chemical kinetics is discerning how various conditions, like temperature, pressure, and concentration of reactants, influence the speed of a reaction.

For example, if we look at the reaction provided in the exercise, where \(2 \mathrm{Br} \mathrm{O}(g) \rightarrow \mathrm{Br}_{2}(g)+ \mathrm{O}_{2}(g)\), different scenarios illustrate how changes in the concentration of bromine monoxide (BrO) affect the reaction rate. Through these observations, we can begin to quantify the reaction's pace and better understand the behavior of the reacting molecules on a molecular level.

Reaction Order

• First-order: If a reaction's rate is directly proportional to the concentration of one reactant, we call it first-order with respect to that reactant. In scenario a, when the concentration of BrO doubles and the rate also doubles, this indicates a first-order reaction with respect to BrO, represented mathematically as \( \text{Rate} = k [\mathrm{BrO}]^1\).


• Second-order: For scenario b, the rate quadrupling when the concentration of BrO doubles is characteristic of a second-order reaction concerning BrO. This is depicted as \( \text{Rate} = k [\mathrm{BrO}]^2\).


• Zero-order: If the rate of a reaction remains unchanged irrespective of the concentration of a reactant, it is zero-order in terms of that reactant. In scenario d, the rate stays constant when the BrO concentration is doubled, which we represent by \( \text{Rate} = k [\mathrm{BrO}]^0\) or simply \( \text{Rate} = k\).

In essence, the reaction order tells us how the rate is affected by the concentration of reactants and is vital for plotting and interpreting rate laws for chemical reactions.

Rate Constant

The rate constant, symbolized by \(k\), is another important element in the study of chemical kinetics. It's a proportionality constant that links the reaction rate to the concentrations of reactants raised to the power indicated by the reaction order.

The value of the rate constant is specific to a particular reaction at a given temperature. It provides insights into the reaction’s inherent tendency to occur, independent of concentrations. Take scenario b for instance, where the reaction is second-order with respect to BrO. While the exponent tells us that doubling the concentration of BrO quadruples the rate, the actual quantitative increase in that rate is determined by the magnitude of the rate constant \(k\).

Therefore, whether we deal with a reaction that’s first, second, or zero-order, the rate constant serves as a crucial piece of the puzzle that helps us connect kinetic observations with theoretical modeling of reaction rates.