Problem 63 - Chemistry: An Atoms-Focused Approach

Question

Hydrogen gas reduces NO to \(\mathrm{N}_{2}\) in the following reaction: $$2 \mathrm{H}_{2}(g)+2 \mathrm{NO}(g) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{N}_{2}(g)$$ The initial reaction rates of four mixtures of \(\mathrm{H}_{2}\) and \(\mathrm{NO}\) were measured at \(900^{\circ} \mathrm{C}\) with the following results: $$\begin{array}{cccc}\text { Experiment } & \left[\mathrm{H}_{2}\right]_{0}(\mathrm{M}) & [\mathrm{NO}]_{0}(\mathrm{M}) & \begin{array}{c}\text { Initial } \\\\\text { Rate }(M / \mathrm{s})\end{array} \\\\\hline 1 & 0.212 & 0.136 & 0.0248 \\\\\hline 2 & 0.212 & 0.272 & 0.0991 \\\\\hline 3 & 0.424 & 0.544 & 0.793 \\\\\hline 4 & 0.848 & 0.544 & 1.59 \\\\\hline\end{array}$$ Determine the rate law and the rate constant for the reaction at \(900^{\circ} \mathrm{C}\).

Step-by-Step Solution

Answer

Question: Determine the rate law and the rate constant for the given reaction at 900°C based on the following data: Experiment 1: [H₂] = 0.212 M, [NO] = 0.136 M, Rate = 0.0248 M/s Experiment 2: [H₂] = 0.212 M, [NO] = 0.272 M, Rate = 0.0991 M/s Experiment 3: [H₂] = 0.424 M, [NO] = 0.136 M, Rate = 0.793 M/s Experiment 4: [H₂] = 0.848 M, [NO] = 0.136 M, Rate = 1.59 M/s Answer: The rate law for the reaction at 900°C is Rate = 0.424 [H₂] [NO]^2, with a rate constant (k) of 0.424 M⁻¹s⁻¹.

Step 1: Determine the order of the reaction with respect to H₂ and NO

First, let's write down the general rate law equation for the given reaction. It will be of the form: Rate = k [H₂]^m [NO]^n Our task is to find the values of m, n, and the rate constant k. To find the reaction order (m, n) with respect to each reactant, we will compare the experiments where the concentration of only one reactant is changing while the other remains constant. Comparing Experiments 1 and 2, [H₂] is constant, whereas [NO] is doubled (0.272M/0.136M = 2) and the rate is increased by a factor of 3.98 (0.0991/0.0248 ≈ 3.98). Using this information, we can write the equation: 2^n ≈ 3.98 hence, n ≈ 2 (since 2^2 = 4) Now, comparing Experiments 3 and 4, [NO] is constant, whereas [H₂] is doubled (0.848M/0.424M = 2) and the rate is increased by a factor of 2 (1.59/0.793 ≈ 2). Using this information, we can write the equation: 2^m ≈ 2 which gives us m ≈ 1 (since 2^1 = 2) Thus, the reaction order with respect to H₂ is 1, and the reaction order with respect to NO is 2.

Step 2: Write down the rate law equation with the determined reaction orders

Now that we have the reaction orders, we can write the rate law equation as: Rate = k [H₂]^1 [NO]^2

Step 3: Calculate the rate constant (k) using any of the experiments

We can now use the data from any of the experiments to determine k. Take Experiment 1 as an example: 0.0248 M/s = k (0.212 M)^1 (0.136 M)^2 Solving for k, we find k ≈ 0.424 M⁻¹s⁻¹. (Note: The value of k may vary slightly depending on the rounding) So the rate law for the reaction at 900°C is: Rate = 0.424 [H₂] [NO]^2

Key Concepts

Understanding Reaction Order

The concept of reaction order in chemical kinetics tells us how the concentration of reactants affects the rate of a chemical reaction. It is an important part of determining the rate law for any reaction. In our example, the reaction order was determined experimentally by observing how changes in the reactants' concentrations affected the reaction rate.
In this case, we examined the effects of hydrogen \(H_2\) and nitric oxide \(NO\) concentrations. By comparing different experiments where these concentrations varied systematically, we found that the reaction order with respect to \(H_2\) is 1. This means that as the concentration of \(H_2\) doubles, the rate of the reaction doubles as well.
The reaction order with respect to \(NO\), however, is 2. This indicates a quadratic relationship; when the concentration of \(NO\) doubles, the reaction rate increases by a factor of four, approximately. The overall reaction order is the sum of the individual orders, which in this case is 3 (1 for \(H_2\) and 2 for \(NO\)). Understanding these orders allows chemists to construct the rate law formula, which is crucial for predicting how a reaction progresses.

The Role of Rate Constant

In the context of chemical reactions, the rate constant, denoted as \(k\), serves as a proportionality factor in the rate law equation. It quantifies the speed of a reaction under specific conditions, such as temperature. For the experiment in question, the rate law was established with the formula: Rate = \(k [H_2]^1 [NO]^2\).

• The value of \(k\) was calculated to be approximately 0.424 M⁻¹s⁻¹ at 900°C.
• This indicates how fast the reaction proceeds at this particular temperature.

The rate constant itself does not depend on the concentrations of the reactants, but rather on factors such as temperature and the presence of a catalyst. It is crucial for chemists to determine this value when exploring new reactions, as it provides essential insights into the kinetic behavior.
Whenever performing calculations involving rate laws, careful measurement and control of temperature and other environmental conditions are necessary to ensure the accuracy and reproducibility of the rate constant.

Delving into Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with understanding the rates of chemical reactions and determining the factors affecting these rates. By analyzing chemical kinetics, scientists can develop detailed insights into reaction mechanisms, which specify the steps that occur from reactants to products.
In the exercise discussed, we applied chemical kinetics principles to determine both the reaction order and the rate constant for the reaction between \(H_2\) and \(NO\). These kinetic analyses allow us not only to write the rate law but also to predict how the reaction would behave under different concentrations or conditions.
Using the rate law, we can:

• Predict how adjustments in concentrations impact the reaction speed.
• Estimate the time needed for a reaction to reach a certain extent.
• Design conditions for industrial processes to optimize reaction efficiency.

Understanding chemical kinetics also helps in controlling reactions more efficiently by minimizing unwanted side reactions, maximizing the yield, or ensuring safe operations, making it a vital area of study in both research and practical applications.