Question

Data for the following reaction are given in the table. $$2 \mathrm{NO}(\mathrm{g})+\mathrm{Br}_{2}(\mathrm{g}) \longrightarrow 2 \mathrm{NOBr}(\mathrm{g})$$ $$\begin{array}{llll}\hline \text { Experiment } & \begin{array}{l}{[\mathrm{NO}]} \\\\(\mathrm{M})\end{array} & \begin{array}{l}{\left[\mathrm{Br}_{2}\right]} \\\\(\mathrm{M})\end{array} & \begin{array}{l}\text { Initial Rate } \\\\(\mathrm{mol} / \mathrm{L} \cdot \mathrm{s})\end{array} \\\\\hline 1 & 1.0 \times 10^{-2} & 2.0 \times 10^{-2} & 2.4 \times 10^{-2} \\\2 & 4.0 \times 10^{-2} & 2.0 \times 10^{-2} & 0.384 \\\3 & 1.0 \times 10^{-2} & 5.0 \times 10^{-2} & 6.0 \times 10^{-2} \\\\\hline\end{array}$$ (a) What is the order of the reaction with respect to \([\mathrm{NO}] ?\) (b) What is the order with respect to \(\left[\mathrm{Br}_{2}\right] ?\) (c) What is the overall order of the reaction?

Short answer

(a) The order with respect to [NO] is 2. (b) The order with respect to [Br2] is 1. (c) The overall order of the reaction is 3.

Step-by-step solution

Identify Rate Law

The rate law for a reaction is given by: \( ext{Rate} = k[ ext{NO}]^m[ ext{Br}_2]^n \), where \(m\) is the order with respect to NO, \(n\) is the order with respect to \(\text{Br}_2\), and \(k\) is the rate constant.

Determine Order with Respect to [NO]

Compare experiments 1 and 2, where \([\text{Br}_2]\) is constant. - For experiment 1, \([\text{NO}] = 1.0 \times 10^{-2}\) M, rate is \(2.4 \times 10^{-2}\) mol/L·s.- For experiment 2, \([\text{NO}] = 4.0 \times 10^{-2}\) M, rate is \(0.384\) mol/L·s.- The ratio \(\frac{0.384}{2.4 \times 10^{-2}} = 16\), and the ratio \(\frac{4.0 \times 10^{-2}}{1.0 \times 10^{-2}} = 4\).- Since \(4^m = 16\), \(m = 2\). Thus, the order with respect to \([\text{NO}]\) is 2.

Determine Order with Respect to [Br2]

Compare experiments 1 and 3, where \([\text{NO}]\) is constant. - For experiment 1, \([\text{Br}_2] = 2.0 \times 10^{-2}\) M, rate is \(2.4 \times 10^{-2}\) mol/L·s.- For experiment 3, \([\text{Br}_2] = 5.0 \times 10^{-2}\) M, rate is \(6.0 \times 10^{-2}\) mol/L·s.- The ratio \(\frac{6.0 \times 10^{-2}}{2.4 \times 10^{-2}} = 2.5\), and the ratio \(\frac{5.0 \times 10^{-2}}{2.0 \times 10^{-2}} = 2.5\) as well.- Since \(2.5^n = 2.5\), \(n = 1\). Thus, the order with respect to \([\text{Br}_2]\) is 1.

Calculate Overall Order of Reaction

The overall order of the reaction is the sum of the orders with respect to each reactant, which is \(m + n = 2 + 1 = 3\).

Key concepts

Rate Law

The rate law provides a mathematical description of how the concentration of reactants affects the reaction rate. In a reaction, not all reactants influence the rate equally. The rate law helps elucidate this by presenting a formula: \( \text{Rate} = k [\text{A}]^m [\text{B}]^n \). Here, \( k \) is the rate constant, and \( m \) and \( n \) are the reaction orders with respect to reactants A and B, respectively. These parameters must be determined experimentally.
To find the rate law, it's crucial to examine how the concentration of each reactant separately influences the rate of reaction. By comparing different experimental setups where one reactant’s concentration changes while the other's remains the same, we can deduce the order for each reactant.

• If doubling a reactant’s concentration doubles the rate, the order is 1 (first order).
• If doubling the concentration quadruples the rate, the order is 2 (second order).
• The order can also be zero, indicating no change in rate regardless of concentration change.

Chemical Kinetics

Chemical kinetics is the branch of chemistry focusing on the speed, or rate, of a chemical reaction and the factors affecting it. Understanding kinetics allows chemists to predict how different conditions might speed up or slow down reactions.
As part of chemical kinetics, the following factors are scrutinized:

• Concentration of reactants: As already mentioned, the change in concentration can influence the rate at which products are formed.
• Temperature: Generally, increasing the temperature increases the rate of a reaction.
• Catalysts: These are substances that increase the reaction rate without being consumed.
• Surface Area: This is particularly relevant if a reactant is in a different phase (like a solid in a liquid), as a larger surface area can increase reaction rates.

By applying chemical kinetics, scientists can formulate strategies to control reactions, whether speeding them up for industrial purposes or slowing them down for safety reasons.

Rate Equation

The rate equation ties in the concentration of reactants to the reaction rate in a precise way. For the given reaction example, the rate equation can be expressed as: \[ \text{Rate} = k [\text{NO}]^2 [\text{Br}_2]^1 \].
This equation illustrates several critical insights:

• \([\text{NO}]^2\) implies that the reaction rate is highly sensitive to changes in the concentration of NO. Doubling the NO concentration increases the rate by a factor of four, representing a second-order dependence.
• \([\text{Br}_2]^1\) means that the reaction rate is directly proportional to the concentration of \(\text{Br}_2\), indicating a first-order dependence.
• The molecularity of the rate equation is suspended by the physical occurrences that aren't dictated purely by stoichiometry but by the data collected.

The overall reaction order is 3, which is the sum of the individual orders (2 for NO and 1 for \(\text{Br}_2\)). Understanding the rate equation is essential for predicting reaction behavior under various conditions.

Initial Rate Method

The initial rate method is an important experimental technique used in determining reaction order. It involves measuring the reaction rate just after the reaction begins, minimizing the effects of product buildup.
In practice, the initial rate method consists of these steps:

• Run multiple experiments, altering the concentration of one reactant while keeping others constant. This isolates the effect of that single reactant on the rate.
• Measure the initial rate of reaction for each experimental setup.
• Analyze how changes in concentration affect the rate to deduce the order of the reaction with respect to each reactant.

This method is particularly useful because it provides clear insights into the role of each reactant in the rate equation and helps to establish an empirical rate law. By focusing on initial rates, researchers can avoid complexities introduced by intermediate products or reversible reactions as the reaction progresses.