Question

You are studying the reaction \(\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \longrightarrow 2 \mathrm{AB}(g)\) to determine its rate law. Assuming that you have a valid experimental procedure for obtaining \(\left[\mathrm{A}_{2}\right]\) and \(\left[\mathrm{B}_{2}\right]\) at various times, explain how you determine (a) the initial rate, (b) the reaction orders, and (c) the rate constant.

Short answer

Measure initial rates, determine reaction orders using log-log plots, and calculate the rate constant using the rate law equation.

Step-by-step solution

- Calculate Initial Rate

Measure the initial concentrations of \( \text{A}_{2} \) and \( \text{B}_{2} \). Monitor the concentration of \( \text{AB} \) at different time intervals. Use the initial slope of the concentration vs. time graph of \( \text{AB} \) to find the initial rate. The formula for the initial rate is \( \text{rate} = \frac{\text{d}[\text{AB}]}{\text{d}t} \text{ at } t = 0 \).

- Determine Reaction Order with Respect to \( \text{A}_{2} \)

Keep the concentration of \( \text{B}_{2} \) constant. Vary the concentration of \( \text{A}_{2} \) and measure the initial rate for each concentration. Plot \( \text{log}(\text{rate}) \) vs. \( \text{log}([ \text{A}_{2} ]) \). The slope of the line will give the order of the reaction with respect to \( \text{A}_{2} \), denoted as \( n \).

- Determine Reaction Order with Respect to \( \text{B}_{2} \)

Keep the concentration of \( \text{A}_{2} \) constant. Vary the concentration of \( \text{B}_{2} \) and measure the initial rate for each concentration. Plot \( \text{log}(\text{rate}) \) vs. \( \text{log}([ \text{B}_{2} ]) \). The slope of the line will give the order of the reaction with respect to \( \text{B}_{2} \), denoted as \( m \).

- Calculate Rate Constant

Use the determined reaction orders \( n \) and \( m \). Write the rate law as \( \text{rate} = k [ \text{A}_{2} ]^n [ \text{B}_{2} ]^m \). Use one set of experimental data (initial concentrations and initial rate) to solve for the rate constant \( k \).

Key concepts

Initial Rate Calculation

Understanding the initial rate of a chemical reaction is crucial. To determine it, you need to follow a few steps. Start by measuring the initial concentrations of both reactants, in this case, \(\text{A}_{2}\) and \(\text{B}_{2}\). You can then monitor the concentration of the product \(\text{AB}\) at different time intervals. Typically, this is done using experiments with specialized equipment that measures concentration changes over time.
If you plot the concentration of \(\text{AB}\) against time, the initial rate can be found by calculating the slope of the line at the very beginning of the reaction (when time, \(t\), is zero). The mathematical expression for the initial rate is: \[ \text{rate} = \frac{\text{d}[\text{AB}]}{\text{d}t} \text{ at } t = 0 \] This means you are looking at how quickly the concentration of \(\text{AB}\) is changing at the start. Measuring this accurately will give you the initial rate.
The initial rate helps you understand how quickly a reaction begins and is essential for determining other key aspects of the reaction.

Reaction Order

The reaction order gives insight into how the concentration of reactants affects the rate of reaction. For the reaction \( \text{A}_{2}(g) + \text{B}_{2}(g) \rightarrow 2 \text{AB}(g) \), you determine the reaction order with respect to each reactant separately.
Determining Reaction Order with Respect to \( \text{A}_{2} \): Keep the concentration of \( \text{B}_{2} \) constant. Vary the concentration of \( \text{A}_{2} \) and measure the initial rate for each concentration.

• Plot a graph of \(\text{log}(\text{rate})\) against \(\text{log}([ \text{A}_{2} ])\).
  
• The slope of this line will give you the reaction order with respect to \(\text{A}_{2}\), denoted as \(\text{n}\).

Determining Reaction Order with Respect to \(\text{B}_{2}\): Keep the concentration of \(\text{A}_{2}\) constant. Vary the concentration of \(\text{B}_{2}\) and measure the initial rate for each concentration.

• Similarly, plot \( \text{log}(\text{rate}) \) against \( \text{log}([ \text{B}_{2} ]) \).
  
• The slope of this line will give you the reaction order with respect to \(\text{B}_{2}\), denoted as \(\text{m}\).

Understanding these reaction orders helps you build the complete rate law for the reaction.

Rate Constant

Once you have the reaction orders, you can determine the rate constant, \( k \). The rate constant is a proportionality factor in the rate law that is specific to a particular reaction at a given temperature.
First, combine the reaction orders \( n \) and \( m \) to write the overall rate law for the reaction. For the reaction \( \text{A}_{2}(g) + \text{B}_{2}(g) \rightarrow 2 \text{AB}(g) \), the rate law can be written as: \[ \text{rate} = k [ \text{A}_{2} ]^n [ \text{B}_{2} ]^m \]
To find the rate constant \( k \), you need one set of experimental data, i.e., the initial concentrations of \( \text{A}_{2} \) and \( \text{B}_{2} \), and the corresponding initial rate. Plug these values into the rate law and solve for \( k \).

• Use your chosen experimental data set to substitute into the rate law equation.
• Rearrange the equation to solve for \( k \).

The value of \( k \) gives you a deeper understanding of the reaction's dynamics and how fast it proceeds under specific conditions.