Question

For the simple decomposition reaction \(\mathrm{AB}(g) \longrightarrow \mathrm{A}(g)+\mathrm{B}(g)\) rate \(=k[\mathrm{AB}]^{2}\) and \(k=0.2 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{s} .\) How long will it take for \([\mathrm{AB}]\) to reach one-third of its initial concentration of \(1.50 \mathrm{M}\) ?

Short answer

The time is \ \frac{20}{3} \ \text{s} or approximately 6.67 seconds.

Step-by-step solution

- Identify the Reaction Order

The given rate law is \[ \text{rate} = k[\text{AB}]^2 \]. Since the rate depends on the concentration to the second power, it is a second-order reaction.

- Write the Integrated Rate Law for Second-Order Reactions

For a second-order reaction, the integrated rate law is \[ \frac{1}{[\text{AB}]} = kt + \frac{1}{[\text{AB}_0]} \], where \([\text{AB}_0]\) is the initial concentration and \(k\) is the rate constant.

- Plug in Initial and Final Concentrations

In this problem, \([\text{AB}_0] = 1.50 \text{ M}\) and \([\text{AB}] = \frac{1.50}{3} = 0.50 \text{ M}\).

- Substitute Known Values into the Integrated Rate Law

Substitute \(k = 0.2 \text{ L/mol·s}\), \([\text{AB}_0] = 1.50 \text{ M}\), and \([\text{AB}] = 0.50 \text{ M}\) into the integrated rate law \[ \frac{1}{[\text{AB}]} = kt + \frac{1}{[\text{AB}_0]} \].

- Solve for Time \(t\)

Plug in the values: \[ \frac{1}{0.50 \text{ M}} = 0.2 \text{ L/mol·s} \times t + \frac{1}{1.50 \text{ M}} \]. This simplifies to \[ 2 = 0.2t + \frac{2}{3} \]. Subtract \(\frac{2}{3}\) from both sides to get \[ \frac{4}{3} = 0.2t \]. Solving for \(t\), we get \[ t = \frac{4}{3 \times 0.2} = \frac{4}{0.6} = \frac{40}{6} = \frac{20}{3} \text{ s} \].

Key concepts

Integrated Rate Law

The integrated rate law is a mathematical expression that connects the concentration of reactants to time.
For second-order reactions, the integrated rate law is given as:
\( \frac{1}{[AB]} = kt + \frac{1}{[AB_0]} \).
Here, \( [AB] \) is the concentration at time \( t \); \( k \) is the rate constant, and \( [AB_0] \) is the initial concentration.
This law helps us understand how the concentration of the reactant changes over time and is a powerful tool in reaction kinetics.

Reaction Order

Knowing the reaction order is crucial in determining how a reactant's concentration affects the rate of reaction.
In our example, the given rate law is \( \text{rate} = k[AB]^2 \). This shows a second-order reaction because the exponent of \( [AB] \) is 2.

It's essential to identify the reaction order to use the correct integrated rate law.
Reaction orders can be zero, first, or any higher integer, and they significantly impact the kinetics analysis.

Rate Constant Calculation

The rate constant \( k \) is a crucial parameter in reaction kinetics.
For our reaction, \( k = 0.2 \text{ L/mol·s} \). To calculate the time it takes for \( [AB] \) to reach a certain concentration, we substitute the values in the integrated rate law.
This shows the direct relationship between reaction rate and concentration.
The rate constant also helps predict how fast a reaction progresses under various conditions.

Concentration Change Over Time

To understand how concentration changes over time, we use the integrated rate law with specific initial and final concentrations.
In our scenario, we want to find the time it takes for \( [AB] \) to drop to one-third of its initial concentration (1.50 M to 0.50 M).
Substituting these values into the rate law: \( \frac{1}{0.50} = 0.2t + \frac{1}{1.50} \).
We solve this equation to find \( t \). Details like these demonstrate the impact of time and conditions on concentration, helping us make predictions about the reaction's future behavior.