Question

Consider the gas-phase reaction between nitric oxide and bromine at \(273^{\circ} \mathrm{C}: 2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g) \longrightarrow 2 \mathrm{NOBr}(g) .\) The following data for the initial rate of appearance of NOBr were obtained: (a) Determine the rate law. (b) Calculate the average value of the rate constant for the appearance of NOBr from the four data sets. (c) How is the rate of appearance of NOBr related to the rate of disappearance of \(\mathrm{Br}_{2}\) ? (d) What is the rate of disappearance of \(\mathrm{Br}_{2}\) when \([\mathrm{NO}]=0.075 \mathrm{M}\) and \(\left[\mathrm{Br}_{2}\right]=0.25 \mathrm{M} ?\)

Short answer

In summary, the rate law for the given reaction is Rate = \(k[\mathrm{NO}]^2[\mathrm{Br}_2]\), with an average rate constant k of 1200 M^{-2}s^{-1}. The rate of disappearance of Br2 is -0.5 times the rate of appearance of NOBr. Finally, when [NO] = 0.075 M and [Br2] = 0.25 M, the rate of disappearance of Br2 is approximately -8.4375 M/s.

Step-by-step solution

Determine the order with respect to NO and Br2

We will compare Experiments 1 and 2 to find the order with respect to NO. In Experiment 1, [NO] = 0.10 M and the initial rate is 24 M/s. In Experiment 2, [NO] = 0.25 M and the initial rate is 150 M/s. Hence, we have: \( \frac{150}{24} = \frac{0.25^n}{0.10^n} \) Solving for n with trial and error, we find its value to be 2 (approximately). Thus, the reaction is second order with respect to NO. Next, we will compare Experiments 1 and 3 to find the order with respect to Br2. In Experiment 1, [Br2] = 0.20 M and the initial rate is 24 M/s. In Experiment 3, [Br2] = 0.50 M and the initial rate is 60 M/s. Now we get: \( \frac{60}{24} = \frac{0.50^m}{0.20^m} \) Solving for m with trial and error, we find its value to be 1 (approximately). Thus, the reaction is first order with respect to Br2.

Determine the rate law

Now that we know the orders for NO and Br2, we can determine the rate law. The rate law is given by: Rate = \(k[\mathrm{NO}]^2[\mathrm{Br}_2]\)

Calculate the average rate constant

Using the rate law from Step 2, we can solve for the rate constant, k, for each data set. Experiment 1: \( k = \frac{24}{(0.10)^2(0.20)} = 1200 \, M^{-2}s^{-1} \) Experiment 2: \( k = \frac{150}{(0.25)^2(0.20)} = 1200 \, M^{-2}s^{-1} \) Experiment 3: \( k = \frac{60}{(0.10)^2(0.50)} = 1200 \, M^{-2}s^{-1} \) Experiment 4: \( k = \frac{735}{(0.35)^2(0.50)} \approx 1200 \, M^{-2}s^{-1} \) The average rate constant is 1200 M^{-2}s^{-1}.

Relate the rate of appearance of NOBr to the rate of disappearance of Br2

Looking at the balanced equation, we can see that for every 2 moles of NOBr formed, 1 mole of Br2 is consumed. Hence, we can write, Rate of disappearance of Br2 = -0.5 × Rate of appearance of NOBr

Calculate the rate of disappearance of Br2 for given concentrations

Now, we can use the rate law, average rate constant, and the given concentrations of NO and Br2 to find the rate of disappearance of Br2. Rate of appearance of NOBr = 1200 × \( (0.075)^2 \) × (0.25) ≈ 16.875 M/s Now using the relationship derived in Step 4, Rate of disappearance of Br2 = -0.5 × 16.875 ≈ -8.4375 M/s.

Key concepts

Reaction Order

The concept of reaction order refers to the power to which the concentration of a reactant is raised in the rate law expression of a chemical reaction. Determining the reaction order involves understanding how changes in reactant concentrations affect the rate of the reaction.
The general form is:
\[ ext{Rate} = k[ ext{A}]^m[ ext{B}]^n \]
where \(m\) and \(n\) are the orders of the reaction with respect to reactants A and B, respectively.

• First-order reactions imply the rate is directly proportional to the concentration of one reactant (exponent of 1).
• Second-order reactions indicate the rate is proportional to the square of the concentration of one reactant or the product of the concentrations of two different reactants.
• Zeroth-order reactions suggest the rate is independent of the concentration of one of the reactants.

In the given reaction between NO and Br\(_2\), it was determined that the reaction is second order with respect to NO and first order with respect to Br\(_2\). This means that the rate of reaction quadruples when the concentration of NO is doubled, and it doubles when the concentration of Br\(_2\) is doubled.

Rate Constant

The rate constant, \(k\), is a crucial part of the rate law. It serves as a proportionality factor that links the reaction rate to the concentrations of reactants raised to their respective orders.
The specific value of \(k\) is determined experimentally and can vary with temperature and the presence of a catalyst.
For the sample problem, the rate constant was calculated using the reaction order determined for the given reaction. The formula used was:
\[ k = \frac{\text{Rate}}{[ ext{NO}]^2[ ext{Br}_2]} \]
Experimentally, it was found to be 1200 M\(^{-2}\)s\(^{-1}\) on average, regardless of the different concentration conditions in each experiment.
This indicates a consistent rate law and confirms the reaction follows the given mechanism under the tested conditions.

Chemical Kinetics

Chemical kinetics focuses on understanding the speed or rate of chemical reactions and the factors affecting them. It involves studying how different conditions such as concentration, temperature, and the presence of catalysts can influence the rate.
Understanding kinetics helps chemists to manipulate reaction conditions to optimize or control the reaction rate, making it crucial in industrial applications and research.

• Concentration: Higher concentrations lead to more frequent collisions between reactant molecules, thus affecting the rate.
• Temperature: An increase in temperature generally increases reaction rates as it gives molecules more energy to collide.
• Catalysts: These substances alter the rate without being consumed in the reaction, often by providing an alternative reaction pathway with a lower activation energy.

For the NO and Br\(_2\) reaction, kinetics is evaluated by analyzing how the reaction rate varies with different reactant concentrations, as seen in the experiments.

Reaction Rate

A reaction rate is a measure of how quickly products are formed or reactants are consumed in a chemical reaction. It can be expressed in terms of the change in concentration of a reactant or product per unit of time.
The rate of a reaction provides insight into the efficiency and speed at which a chemical process takes place, making it a key focus in reactions involving synthesis, energy production, or decomposition.
In practice, the reaction rate for a given concentration setup can be calculated using the rate law, which includes the concentration of reactants with their respective orders and the rate constant \(k\).
For instance, in the provided exercise, the relationship between the rate of formation of NOBr and the rate of consumption of Br\(_2\) is derived by analyzing the stoichiometry of the chemical equation. The rate of disappearance of Br\(_2\) was deduced to be half the rate of appearance of NOBr. This illustrates the stoichiometric relationship between reactant and product in the balanced chemical reaction.