Question

When boron trifluoride reacts with ammonia, the following reaction occurs: $$ \mathrm{BF}_{3}(g)+\mathrm{NH}_{3}(g) \longrightarrow \mathrm{BF}_{3} \mathrm{NH}_{3}(g) $$ The following data are obtained at a particular temperature: .$$ \begin{array}{ccll} \hline \text { Expt. } & {\left[\mathrm{BF}_{3}\right]} & {\left[\mathrm{NH}_{3}\right]} & \text { Initial Rate }(\mathrm{mol} / \mathrm{L} \cdot \mathrm{s}) \\ \hline 1 & 0.100 & 0.100 & 0.0341 \\ 2 & 0.200 & 0.233 & 0.159 \\ 3 & 0.200 & 0.0750 & 0.0512 \\ 4 & 0.300 & 0.100 & 0.102 \\ \hline \end{array} $$ (a) What is the order of the reaction with respect to \(\mathrm{BF}_{3}\), \(\mathrm{NH}_{3},\) and overall? (b) Write the rate expression for the reaction. (c) Calculate \(k\) for the reaction. (d) When \(\left[\mathrm{BF}_{3}\right]=0.533 \mathrm{M}\) and \(\mathrm{NH}_{3}=0.300 \mathrm{M},\) what is the rate of the reaction at the temperature of the experiment?

Short answer

Answer: The order with respect to \(\mathrm{BF}_{3}\) is 1, the order with respect to \(\mathrm{NH}_{3}\) is 2, and the overall order of the reaction is 3. The rate of the reaction when \(\left[\mathrm{BF}_{3}\right]=0.533 \mathrm{M}\) and \([\mathrm{NH}_{3}]=0.300 \mathrm{M}\) at the temperature of the experiment is \(0.924 \: \mathrm{mol} \: \mathrm{L}^{-1} \cdot \mathrm{s}^{-1}\).

Step-by-step solution

Determine the order with respect to \(\mathrm{BF}_{3}\)

Compare Experiments 1 and 3, where the concentration of \(\mathrm{NH}_{3}\) is constant. The concentration of \(\mathrm{BF}_{3}\) doubles from 0.100 M to 0.200 M, while the initial rate increases by a factor of 1.5 (0.0512 / 0.0341). Since the rate increases by a factor smaller than the change in \(\mathrm{BF}_{3}\) concentration, the order with respect to \(\mathrm{BF}_{3}\) is 1 (first order).

Determine the order with respect to \(\mathrm{NH}_{3}\)

Compare Experiments 1 and 2, where the concentration of \(\mathrm{BF}_{3}\) is constant. The concentration of \(\mathrm{NH}_{3}\) increases by a factor of 2.33 (0.233 / 0.100), while the initial rate increases by a factor of 4.66 (0.159 / 0.0341). Since the rate increases by approximately the square of the change in \(\mathrm{NH}_{3}\) concentration, the order with respect to \(\mathrm{NH}_{3}\) is 2 (second order).

Determine the overall order of the reaction

The overall order of the reaction is the sum of the orders with respect to each reactant. In this case, it is 1 (for \(\mathrm{BF}_{3}\)) + 2 (for \(\mathrm{NH}_{3}\)) = 3.

Write the rate expression for the reaction

Now we can write the rate expression for the reaction using the orders we obtained in the previous steps: $$ \text{Rate} = k[\mathrm{BF}_{3}]^1[\mathrm{NH}_{3}]^2 $$

Calculate the rate constant \(k\)

We can use data from any of the experiments to calculate the rate constant. Let's use Experiment 1 data: $$ k = \frac{\text{Rate}}{[\mathrm{BF}_{3}][\mathrm{NH}_{3}]^2} = \frac{0.0341}{(0.100)(0.100)^2} = 3.41\: \mathrm{L}^2\mathrm{mol}^{-2}\mathrm{s}^{-1} $$

Calculate the rate of the reaction for given concentrations

When \(\left[\mathrm{BF}_{3}\right]=0.533 \mathrm{M}\) and \(\mathrm{NH}_{3}=0.300 \mathrm{M},\) we can calculate the rate using the rate expression and the value of \(k\): $$ \text{Rate} = 3.41 \times (0.533)(0.300)^2 = 0.924 \: \mathrm{mol} \: \mathrm{L}^{-1} \cdot \mathrm{s}^{-1} $$ The rate of the reaction when \(\left[\mathrm{BF}_{3}\right]=0.533 \mathrm{M}\) and \(\mathrm{NH}_{3}=0.300 \mathrm{M}\) at the temperature of the experiment is \(0.924 \: \mathrm{mol} \: \mathrm{L}^{-1} \cdot \mathrm{s}^{-1}\).

Key concepts

Reaction Order

Understanding the reaction order is crucial when studying chemical kinetics, as it determines the relationship between the concentrations of the reactants and the reaction rate. A reaction's order is defined as the sum of the powers to which all reactant concentrations are raised in the rate law expression. For instance, if increasing a reactant’s concentration by a factor of two leads to a doubling of the reaction rate, the reactant is first order with respect to that reaction. In contrast, if the rate increases by a factor of four, then it is second order.

A simple way to determine the reaction order experimentally is to hold one reactant constant while varying another and noting the effect on the reaction rate. This comparison gives insight into how the rate depends on each reactant's concentration, thereby revealing the reaction order with respect to each reactant and the overall reaction order as a sum.

Rate Law Expression

The rate law expression quantifies the rate of a chemical reaction in terms of the concentrations of the reactants and the reaction orders. It is a mathematical equation that connects the rate of a reaction to the concentrations of reactants raised to their respective orders. A rate law can look like this: \[ \text{Rate} = k[A]^m[B]^n \], where \( k \) is the rate constant, \( [A] \) and \( [B] \) are the concentrations of reactants A and B, and \( m \) and \( n \) represent their respective reaction orders.

It's important to remember that the rate law and reaction orders must be determined experimentally; they cannot be deduced from the stoichiometry of the balanced equation alone. The rate law provides critical information for understanding the kinetics of a reaction and predicting how the reaction rate will change with varying reactant concentrations.

Rate Constant Calculation

The rate constant, symbolized as \( k \), is a proportionality constant in the rate law equation. It is unique for every reaction and changes only with temperature. Calculating the rate constant involves rearranging the rate law to solve for \( k \). The formula is as follows: \[ k = \frac{\text{Rate}}{[A]^m[B]^n} \]

By using experimentally determined initial rates and known reactant concentrations, the rate constant can be computed, which in turn enables us to predict the speed of the reaction under various conditions. Although the units of the rate constant can appear complex, they are a direct consequence of the reaction orders and ensure that the rate expression has the correct units of concentration per unit time.

Initial Rates Method

The initial rates method is a commonly used technique to determine the reaction order by measuring the initial rates of a reaction at different reactant concentrations. This method relies on the assumption that the initial rate of reaction is a true reflection of the reactant concentrations at the start, before any significant change happens as the reaction proceeds.

Practically, this involves performing multiple experiments where the concentration of one reactant is varied while the others are held constant. The initial rates obtained from these experiments can then be compared to see how the rate changes as a function of concentration changes. This method is particularly valuable because it simplifies the mathematical analysis by eliminating products from the rate expression and focusing solely on reactants at the outset of the reaction.