Question

When nitrogen dioxide reacts with carbon monoxide, the following reaction occurs. $$ \mathrm{NO}_{2}(g)+\mathrm{CO}(g) \longrightarrow \mathrm{NO}(g)+\mathrm{CO}_{2}(g) $$ The following data are obtained at a certain temperature: $$ \begin{array}{cccc} \hline \text { Expt. } & {\left[\mathrm{NO}_{2}\right]} & {[\mathrm{CO}]} & \text { Initial } \text { Rate }(\mathrm{mol} / \mathrm{L} \cdot \mathrm{s}) \\\ \hline 1 & 0.138 & 0.100 & 0.00565 \\ 2 & 0.189 & 0.200 & 0.0106 \\ 3 & 0.276 & 0.100 & 0.0226 \\ 4 & 0.276 & 0.300 & 0.0226 \\ \hline \end{array} $$ (a) What is the order of the reaction with respect to \(\mathrm{NO}_{2}, \mathrm{CO},\) and overall? (b) Write the rate expression of the reaction. (c) Calculate \(k\) for the reaction.

Short answer

(a) The order of the reaction with respect to NO2 is 2, with respect to CO is 0, and the overall order of the reaction is 2. (b) The rate expression of the reaction is: Rate = \(k[\mathrm{NO}_{2}]^2\) (c) The rate constant, k, for the overall reaction is approximately 0.0297 L/mol·s.

Step-by-step solution

(a) Determine the order of the reaction with respect to NO2, CO, and overall

First, analyze the experiments that have the same initial concentration of CO (Experiments 1 and 3), then compare the initial rates: Exp. 3 / Exp. 1: \(\frac{0.0226}{0.00565} = x^{2}\) Since the [CO] is the same, we can compare the initial rates directly to determine the order of the reaction with respect to NO2. To determine x, take the square root of both sides: \(x = 2\), which means the reaction is second-order with respect to NO2. Next, analyze experiments with the same initial concentration of NO2 (Experiments 3 and 4), then compare the initial rates: Exp. 4 / Exp. 3: \(\frac{0.0226}{0.0226} = y^{3}\) Since the [NO2] is the same, we can compare the initial rates directly to determine the order of the reaction with respect to CO. To determine y, note that since both rates are equal, any exponent would give a value of 1: \(y = 0\), which means the reaction is zero-order with respect to CO. The overall order of the reaction is the sum of exponents: Overall order = 2 + 0 = 2

(b) Write the rate expression of the reaction

Now that we have the order of the reaction with respect to each reactant, we can write the rate expression: Rate = \(k[\mathrm{NO}_{2}]^2[\mathrm{CO}]^0\) Since anything raised to the power of 0 equals 1: Rate = \(k[\mathrm{NO}_{2}]^2\)

(c) Calculate k for the reaction

Choose any of the experiments to calculate the rate constant, k (for example, Exp. 1): \(0.00565 = k[0.138]^2\) Solve for k: \(k \approx 0.0297 \mathrm{L/mol}\cdot \mathrm{s}\) So the rate constant, \(k\), for the overall reaction is approximately 0.0297 L/mol·s.

Key concepts

Reaction Order

Reaction order is a key concept in understanding the dynamics of a chemical reaction. It indicates how the rate of a reaction is affected by the concentration of the reactants. To grasp this, imagine that each reactant's concentration can influence the rate of the reaction in its unique way, depending on how many molecules or atoms of that reactant participate in the crucial reaction step.

For instance, in our example involving nitrogen dioxide (NO2) and carbon monoxide (CO), experiments were conducted under different concentrations to determine the reaction order. The data indicated a second-order dependence with respect to NO2, meaning the rate of reaction increases as the square of the concentration of NO2. Conversely, we found a zero-order dependence with respect to CO, signifying that changes in its concentration do not affect the reaction rate. This gives a deeper insight into how the reactants interact on a molecular level during the reaction.

Rate Expression

A rate expression, often called the rate law, communicates this relationship between the reaction rate and the concentrations of reactants in mathematical terms. Once you've determined the order of the reaction for each reactant through experimentation, you can construct this expression. It is a critical equation in chemical kinetics, marrying the concepts of reaction order and rate.

In our nitrogen dioxide and carbon monoxide reaction, we constructed the rate expression as: Rate = \(k[NO_{2}]^{2}[CO]^{0}\). Since any concentration to the zero power equals one, CO drops out of the expression, placing the spotlight on NO2 for dictating the reaction pace. This expression serves as a formula that enables us to predict the speed of the reaction under various concentrations of the reactants.

Rate Constant

The rate constant, symbolized as \(k\), is a proportionality factor that ties the rate of a reaction to the concentrations of the reactants raised to their respective orders. It's unique to each chemical reaction and is affected by factors like temperature and the presence of a catalyst. As a fundamental constant in the rate expression, it tells us how quickly a reaction proceeds under specified conditions.

For the reaction concerning NO2 and CO, once we had our rate expression in hand, we were able to solve for the rate constant using experimental data. By plugging in the concentration and initial rate values from any one experiment, we derived the rate constant \(k\) to be approximately 0.0297 L/mol·s. This value is more than just a number; it allows us to make quantitative predictions about the reaction rate and can even provide insight into the mechanism of the reaction itself.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed at which chemical reactions occur and the factors that affect this speed. At its core, it seeks to answer questions about how quickly reactants are transformed into products and what pathways they take during this transformation.

Our examination of the reaction between NO2 and CO is a practical foray into chemical kinetics, applying its principles to calculate the reaction order, establish the rate expression, and determine the rate constant. These elements combine to provide a comprehensive view of the reaction's behavior, enabling chemists and students alike to predict and control reaction outcomes effectively. In educational settings, mastering chemical kinetics is about recognizing patterns, understanding how to conduct experiments methodically, and becoming proficient in interpreting data to elucidate how different variables influence the progress of chemical reactions.