Problem 64 - Chemistry: An Atoms-Focused Approach

Question

The rate of the reaction $$\mathrm{NO}_{2}(g)+\mathrm{CO}(g) \rightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g)$$ was determined in three experiments at \(225^{\circ} \mathrm{C} .\) The results are given in the following table: $$\begin{array}{cccc}\text { Experiment } & \left[\mathrm{NO}_{2}\right]_{0}(\mathrm{M}) & [\mathrm{CO}]_{0}(\mathrm{M}) & \begin{array}{c}\text { Initial Rate } \\\\(M / \mathrm{s})\end{array} \\\\\hline 1 & 0.263 & 0.826 & 1.44 \times 10^{-5} \\\\\hline 2 & 0.263 & 0.413 & 1.44 \times 10^{-5} \\\\\hline 3 & 0.526 & 0.413 & 5.76 \times 10^{-5} \\\\\hline\end{array}$$ a. Determine the rate law for the reaction. b. Calculate the value of the rate constant at \(225^{\circ} \mathrm{C}\) c. Calculate the rate of appearance of \(\mathrm{CO}_{2}\) when \(\left[\mathrm{NO}_{2}\right]=[\mathrm{CO}]=0.500 \mathrm{M}\)

Step-by-Step Solution

Answer

Answer: a. The rate law for the reaction is Rate = k[NO₂]². b. The rate constant at 225°C is 2.08 × 10⁻⁴ M⁻¹ s⁻¹. c. The rate of appearance of CO₂ when [NO₂]=[CO]=0.500 M is 5.20 × 10⁻⁵ M/s.

Step 1: Understand the initial rates method

The initial rates method is a method used to determine the rate law for a chemical reaction by analyzing the data of initial concentrations and initial rates of the reaction. We will be analyzing the given data to find the order of the reaction with respect to each reactant and the rate constant.

Step 2: Determine the order of reaction with respect to reactants

Compare experiments 1 and 2. The concentration of \(\mathrm{NO}_{2}\) is the same, but the concentration of \(\mathrm{CO}\) is halved. Since the initial rate stays the same in these two experiments, we can conclude that the reaction is zero-order with respect to \(\mathrm{CO}\). Now, compare experiments 2 and 3. The concentration of \(\mathrm{CO}\) is the same, but the concentration of \(\mathrm{NO}_{2}\) is doubled. As a result, the initial rate increased by a factor of 4. Therefore, we can say that the reaction is second-order with respect to \(\mathrm{NO}_{2}\).

Step 3: Find the rate law for the reaction

Based on the order of reaction for each reactant, we can write the rate law as follows: $$\text{Rate} = k[\mathrm{NO}_{2}]^2[\mathrm{CO}]^0$$ Since the reaction is zero-order with respect to \(\mathrm{CO}\), we can simplify the rate law as: $$\text{Rate} = k[\mathrm{NO}_{2}]^2$$

Step 4: Calculate the value of the rate constant

To find the rate constant \(k\), we can substitute the initial concentrations and initial rate from any of the three experiments into the rate law equation. We will use experiment 1 as an example: $$1.44 \times 10^{-5} M/s = k(0.263 M)^2$$ Solve for \(k\): $$k = \frac{1.44 \times 10^{-5} M/s}{(0.263 M)^2} = 2.08 \times 10^{-4} M^{-1} s^{-1}$$

Step 5: Calculate the rate of appearance of \(\mathrm{CO}_{2}\)

Now we can use the rate law and the rate constant to calculate the rate of appearance of \(\mathrm{CO}_{2}\) when \(\left[\mathrm{NO}_{2}\right]=\left[\mathrm{CO}\right]=0.500M\). According to the rate law: $$\text{Rate} = k[\mathrm{NO}_{2}]^2$$ Plugging in the known values, we get: $$\text{Rate} = (2.08 \times 10^{-4} M^{-1} s^{-1})(0.500 M)^2 = 5.20 \times 10^{-5} M/s$$ In conclusion, a. The rate law for the reaction is \(\text{Rate} = k[\mathrm{NO}_{2}]^2\). b. The value of the rate constant at \(225^{\circ} \mathrm{C}\) is \(2.08 \times 10^{-4} M^{-1} s^{-1}\). c. The rate of appearance of \(\mathrm{CO}_{2}\) when \(\left[\mathrm{NO}_{2}\right]=\left[\mathrm{CO}\right]=0.500 M\) is \(5.20 \times 10^{-5} M/s\).

Key Concepts

Reaction Order

In chemical kinetics, the reaction order is a crucial concept that helps in understanding how reactant concentrations influence the rate of a reaction. The reaction order is the power to which the concentration of a reactant is raised in the rate law equation. In our given chemical reaction, through comparison of the experiments, we determined the order of reaction by observing changes in concentration and the resulting effect on the rate.

When we look at reactions 1 and 2, the concentration of \([NO_2]\) remains constant, while that of \([CO]\) is halved, with no change in reaction rate. This indicates the reaction is zero-order with respect to \([CO]\).

• A zero-order reaction implies the rate is independent of the concentration of that particular reactant.

The analysis of experiments 2 and 3, where we doubled \([NO_2]\) while keeping \([CO]\) constant, shows a fourfold increase in the rate. This implies second-order dependence on \([NO_2]\).

• This means that the rate is proportional to the square of the concentration of \([NO_2]\).

Overall, knowing the reaction order is essential for constructing the correct rate law and predicting how the reaction progresses under different conditions.

Rate Constant

The rate constant, \(k\), is another vital aspect of the rate law equation, serving as a proportionality factor. It's a constant at a given temperature but can change with temperature variations. The rate constant provides insights into how quickly a reaction occurs, separate from concentration effects.

In our case, after determining the reaction orders, we derived the rate law: \(Rate = k[NO_2]^2[CO]^0\). With this knowledge, we can isolate \(k\) by using data from any of the experiments. For example, in Experiment 1, the initial rate and concentration of \([NO_2]\) allows us to solve for \(k\):

\[k = \frac{1.44 \times 10^{-5} \; M/s}{(0.263 \; M)^2} = 2.08 \times 10^{-4} \, M^{-1} s^{-1}\]

This value signifies the reaction's rate at \(225^\circ\mathrm{C}\) and is useful in predicting the reaction rate under similar experimental conditions or with similar changes in concentration.

Initial Rates Method

The initial rates method is a powerful analytical technique used in determining the rate law for chemical reactions. It involves measuring how fast a reaction proceeds when reactants are first mixed. By analyzing a series of experiments with different initial concentrations, we can deduce how changes in concentration affect the rate, thus revealing the reaction order for each reactant.

To utilize this method effectively, follow these key steps:

• Conduct a series of experiments altering the concentration of one reactant at a time while measuring the initial rate of reaction.
• Compare experiments to determine how variations in concentration impact the rate, helping in ascertaining the reaction order with respect to each reactant.

In our exercise, by maintaining one reactant constant and varying the other, we easily identified the reaction as zero-order with respect to \([CO]\) and second-order with respect to \([NO_2]\).

This method is particularly valuable because it isolates variables, allowing us to focus solely on the influence of concentration changes, without interference from other factors that might affect the reaction as it progresses.