Question

The reaction rate as a function of initial reactant pressures was investigated for the reaction \(2 \mathrm{NO}(g)+\) \(2 \mathrm{H}_{2}(g) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g),\) and the following data were obtained: $$\begin{array}{cccc}\text { Run } & P_{o} \mathrm{H}_{2}(\mathrm{kPa}) & P_{o} \mathrm{NO}(\mathrm{kPa}) & \text { Rate }\left(\mathrm{kPa} \mathrm{s}^{-1}\right) \\\\\hline 1 & 53.3 & 40.0 & 0.137 \\\2 & 53.3 & 20.3 & 0.033 \\\3 & 38.5 & 53.3 & 0.213 \\\4 & 19.6 & 53.3 & 0.105\end{array}$$ What is the rate law expression for this reaction?

Short answer

The rate law expression for this reaction is: Rate = k * [NO]² * [H₂]

Step-by-step solution

Examine the change on one reactant's pressure while keeping the other constant

Let's examine Runs 1 and 2. In these runs, the initial pressure of H₂ gas is kept constant while the initial pressure of NO gas changes. From Run 1 to Run 2: \(P_{o} \mathrm{H}_{2}\) is constant. \(P_{o} \mathrm{NO}\) changes from 40.0 kPa to 20.3 kPa (approximately a factor of 0.5) Rate changes from 0.137 kPa s⁻¹ to 0.033 kPa s⁻¹ (approximately a factor of 0.24)

Determine the effect of the change in NO pressure on the reaction rate

Divide the rate from Run 2 by the rate from Run 1 and compare it to the change in NO pressure: \[\frac{\text{Rate}_2}{\text{Rate}_1} = \frac{0.033}{0.137} \approx 0.24 \] Now divide the \(\mathrm{NO}\) pressure in Run 2 by the \(\mathrm{NO}\) pressure in Run 1: \[\frac{P_{o}\mathrm{NO}_2}{P_{o}\mathrm{NO}_1} \approx 0.5\] To find the order of the reaction with respect to NO, we can set up the following equation: \[\left(\frac{P_{o}\mathrm{NO}_2}{P_{o}\mathrm{NO}_1}\right)^{x} = \frac{\text{Rate}_2}{\text{Rate}_1}\] Solving for x: \[0.5^{x} = 0.24\] \[x \approx 2\] So the order of the reaction with respect to NO is approximately 2.

Examine the change on one reactant's pressure while again keeping the other constant

Let's now examine Runs 3 and 4. In these runs, the initial pressure of NO gas is kept constant while the initial pressure of H₂ gas changes. From Run 3 to Run 4: \(P_{o} \mathrm{NO}\) is constant. \(P_{o} \mathrm{H}_{2}\) changes from 38.5 kPa to 19.6 kPa (approximately a factor of 0.51) Rate changes from 0.213 kPa s⁻¹ to 0.105 kPa s⁻¹ (approximately a factor of 0.49)

Determine the effect of the change in H₂ pressure on the reaction rate

Divide the rate from Run 4 by the rate from Run 3 and compare it to the change in H₂ pressure: \[\frac{\text{Rate}_4}{\text{Rate}_3} = \frac{0.105}{0.213} \approx 0.49 \] Now divide the \(\mathrm{H}_{2}\) pressure in Run 4 by the \(\mathrm{H}_{2}\) pressure in Run 3: \[\frac{P_{o}\mathrm{H}_{2}_4}{P_{o}\mathrm{H}_{2}_3} \approx 0.51\] To find the order of the reaction with respect to H₂, set up the following equation: \[\left(\frac{P_{o}\mathrm{H}_{2}_4}{P_{o}\mathrm{H}_{2}_3}\right)^{y} = \frac{\text{Rate}_4}{\text{Rate}_3}\] Solving for y: \[0.51^{y} = 0.49\] \[y \approx 1\] So the order of the reaction with respect to H₂ is approximately 1.

Write the rate law expression

Based on our analysis on steps 2 and 4, the order of the reaction with respect to NO = 2, and the order of the reaction with respect to H₂ = 1. Thus, the rate law expression is given by: Rate = k * [NO]² * [H₂] where k is the rate constant and [NO] and [H₂] represent the concentrations of NO and H₂, respectively.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of the rates at which chemical reactions proceed and the factors that affect these rates. It's a crucial area of chemistry because it helps us understand reaction mechanisms and the behavior of reactants over time. For example, when examining the reaction where nitrogen monoxide (NO) and hydrogen (H₂) gas react to form nitrogen (N₂) and water (H₂O), kinetics tells us not just that the products will form, but how quickly they'll form under certain conditions.

The rate of a reaction can be affected by a variety of factors, such as the concentration of reactants, temperature, and the presence of a catalyst. In the provided exercise, we see the initial pressures of the reactants used to study the reaction rate, which is common practice in gaseous reactions where pressure is directly proportional to concentration.

Rate Constant

The rate constant, symbolized by the letter 'k', is a proportionality factor in the rate law of a chemical reaction. It is unique for each reaction at a given temperature, and provides a quantitative measure of how fast a reaction occurs. In mathematical terms, the rate constant translates the relationship between reactant concentrations and the rate at which products are formed.

To find the rate constant, one usually needs to know the rate law expression and the concentrations of the reactants along with the actual rate of the reaction. Although we don't compute the actual value of 'k' in the provided exercise, understanding its conceptual role is important for analyzing reaction rates. It represents the intrinsic reaction speed and is influenced predominantly by the nature of the reactants and the temperature at which the reaction is taking place.

Reaction Order

The reaction order indicates the dependency of the reaction rate on the concentration of each reactant. It tells us how the rate of the reaction will change when the concentration of one of the reactants changes. In mathematical terms, each reactant's concentration is raised to a power called the 'order' with respect to that reactant. The overall reaction order is the sum of these individual orders.

In the exercise, we established the order of the reaction with respect to NO is 2 and with respect to H₂ is 1 — implying a second-order dependency on NO and first-order on H₂. This information is critical because it allows us to predict how the reaction rate will change under different conditions, making it a cornerstone concept in chemical kinetics.

Initial Reactant Pressures

For reactions involving gases, the role of pressure is vital. The initial pressures of reactants can be directly related to their molar concentrations through the ideal gas law. Thus, by varying the initial pressures of reactants and observing the changes in reaction rate, one can determine the order of the reaction with respect to each gas. This approach is evident in the exercise, where the pressure variation of NO and H₂ influences the reaction rate and subsequently reveals the reaction orders.

In practical terms, knowing how the initial pressures (or concentrations) impact the rate can help control a reaction—whether to hasten the process in industrial applications or to slow it down for safety reasons. Understanding this relationship is fundamental for scientists and engineers when scaling reactions from laboratory to industrial scales.