In aqueous solution, hypobromite ion, BrO-, reacts to produce bromate ion, BrO3-, and bromide ion, Br-, according to the following chemical equation.
3 BrO-(aq) → BrO3-(aq) + 2 Br-(aq)
A plot of 1/[BrO-] vs. time is linear and the slope is equal to 0.056 M-1s-1. If the initial concentration of BrO- is 0.80 M, how long will it take one-half of the BrO- ion to react?
A) 4.5 × 10-2 s
B) 7.1 s
C) 12 s
D) 22 s
The correct answer and explanation is :
To solve this problem, we need to determine the time it takes for half of the BrO- ions to react, which corresponds to the time required for the concentration of BrO- to drop to half of its initial value.
Given:
- The rate law is of second-order with respect to BrO- (since the plot of $1/[BrO^-]$ vs. time is linear, indicating a second-order reaction).
- The slope of the plot is $0.056 \, \text{M}^{-1}\text{s}^{-1}$.
- The initial concentration of BrO- is $0.80 \, \text{M}$.
Second-order reaction integrated rate law:
For a second-order reaction, the integrated rate law is:
$$
\frac{1}{[BrO^-]} = \frac{1}{[BrO^-]_0} + kt
$$
Where:
- $[BrO^-]_0$ is the initial concentration of BrO-,
- $[BrO^-]$ is the concentration of BrO- at time $t$,
- $k$ is the rate constant (in M⁻¹s⁻¹),
- $t$ is time.
When half of the BrO- reacts, the concentration of BrO- will be half of its initial concentration:
$$
[BrO^-] = \frac{[BrO^-]_0}{2}
$$
Solve for time when $[BrO^-]$ is half of its initial concentration:
Substitute into the integrated rate law:
$$
\frac{1}{[BrO^-]_0/2} = \frac{1}{[BrO^-]_0} + k \cdot t
$$
This simplifies to:
$$
\frac{2}{[BrO^-]_0} = \frac{1}{[BrO^-]_0} + k \cdot t
$$
Now subtract $\frac{1}{[BrO^-]_0}$ from both sides:
$$
\frac{1}{[BrO^-]_0} = k \cdot t
$$
Solving for time:
$$
t = \frac{1}{k \cdot [BrO^-]_0}
$$
Substituting the values:
- $k = 0.056 \, \text{M}^{-1}\text{s}^{-1}$,
- $[BrO^-]_0 = 0.80 \, \text{M}$,
$$
t = \frac{1}{0.056 \times 0.80} = \frac{1}{0.0448} \approx 22.3 \, \text{s}
$$
Answer:
The time required for half of the BrO- ions to react is approximately 22 seconds, so the correct answer is:
D) 22 s.
Explanation:
This problem involves understanding the kinetics of a second-order reaction. In second-order reactions, the concentration decreases more gradually as compared to first-order reactions. The relationship between the concentration of reactants and time follows the integrated rate law for second-order reactions, which is a hyperbolic function. By using the given information and the rate law, we can calculate the time required for the concentration of BrO- to reach half of its initial value.