Question

The rate constant for the reaction of hydrogen with iodine is \(2.45 \times 10^{-4} \mathrm{M}^{-1} \mathrm{s}^{-1}\) at \(302^{\circ} \mathrm{C}\) and \(0.950 \mathrm{M}^{-1} \mathrm{s}^{-1}\) at \(508^{\circ} \mathrm{C}\). a. Calculate the activation energy and Arrhenius preexponential factor for this reaction. b. What is the value of the rate constant at \(400 .^{\circ} \mathrm{C} ?\)

Short answer

The activation energy (Ea) for the reaction of hydrogen with iodine is approximately \(93,874\, J\, mol^{-1}\), and the Arrhenius preexponential factor (A) is approximately \(4.17 \times 10^{10} \, M^{-1} s^{-1}\). The rate constant at \(400^{\circ}\mathrm{C}\) is approximately \(9.55 \times 10^{-3}\, M^{-1}s^{-1}\).

Step-by-step solution

Convert temperatures to Kelvin

First, we need to convert the temperatures 302°C and 508°C to Kelvin. We can do this by adding 273.15 to the given temperatures. This will give us: \[T_1 = 302 + 273.15 = 575.15\,K\] \[T_2 = 508 + 273.15 = 781.15\,K\] Now, we can use these temperatures in the Arrhenius equation.

Write two Arrhenius equations

We are given k values at two different temperatures, so we can write two separate Arrhenius equations using the values of k, T1, and T2: \[k_1 = Ae^{\frac{-Ea}{R \cdot T_1}}\] for the rate constant at \(T_1\) \[k_2 = Ae^{\frac{-Ea}{R \cdot T_2}}\] for the rate constant at \(T_2\)

Divide the two equations to solve for Ea

Divide the first equation by the second equation to eliminate A: \[\frac{k_1}{k_2} = \frac{Ae^{\frac{-Ea}{R \cdot T_1}}}{Ae^{\frac{-Ea}{R \cdot T_2}}}\] Now, we can cancel A on both sides and rewrite the equation to solve for Ea as follows: \[Ea = \frac{R \cdot (T_1T_2)}{T_2 - T_1} \cdot \ln(\frac{k_1}{k_2})\] Enter the given values for k1, k2, T1, and T2, and solve for Ea: \[Ea = \frac{(8.314 \,J\, mol^{-1}K^{-1}) (575.15 \, K)(781.15 \, K)}{(781.15 \, K) - (575.15 \, K)} \cdot \ln(\frac{2.45 \times 10^{-4}\, M^{-1}s^{-1}}{0.950 \, M^{-1}s^{-1}})\] \[Ea \approx 93,874\, J\, mol^{-1}\]

Solve for A using one of the Arrhenius equations

Now that we have the value for Ea, we can find A using one of the Arrhenius equations, for example, the one with T1: \[k_1 = Ae^{\frac{-Ea}{R \cdot T_1}}\] Rearrange this equation to solve for A: \[A = \frac{k_1}{e^{\frac{-Ea}{R \cdot T_1}}}\] Plug in the values of k1, Ea, R, and T1 to calculate A: \[A = \frac{2.45 \times 10^{-4}\, M^{-1}s^{-1}}{e^{\frac{-93,874 \, J\, mol^{-1}}{(8.314 \, J\, mol^{-1} K^{-1})(575.15\,K)}}\] \[A \approx 4.17 \times 10^{10} \, M^{-1} s^{-1}\]

Calculate the rate constant at 400°C

Convert the temperature 400°C to Kelvin and use the Arrhenius equation to find the rate constant at this temperature: \[T_3 = 400 + 273.15 = 673.15\,K\] \[k_3 = Ae^{\frac{-Ea}{R \cdot T_3}}\] Use the values of A and Ea we calculated earlier and plug them into the equation: \[k_3 = (4.17 \times 10^{10} \, M^{-1} s^{-1})e^{\frac{-93,874 \, J\, mol^{-1}}{(8.314 \, J\, mol^{-1} K^{-1})(673.15\,K)}}\] \[k_3 \approx 9.55 \times 10^{-3}\, M^{-1}s^{-1}\] So, the rate constant at 400°C is approximately \(9.55 \times 10^{-3} \,M^{-1}s^{-1}\).

Key concepts

Rate Constant

The rate constant, denoted as 'k', is a crucial parameter in the study of chemical kinetics, which signifies the speed at which a chemical reaction occurs. In the context of the Arrhenius equation, it is temperature-dependent, reflecting how even small changes in temperature can significantly alter the rate of a reaction. For the reaction between hydrogen and iodine mentioned in our example, we see two different rate constants at two distinct temperatures, suggesting that the rate increases at higher temperatures. Understanding the rate constant allows chemists to predict how quickly a reaction will proceed under given conditions, making it an essential element in the design of chemical processes and the safe handling of reactive materials.

When calculating the rate constant, it's critical to ensure correct temperature conversion to Kelvin and the usage of consistent units. The proper application of the Arrhenius equation provides the rate constant at any desired temperature, as demonstrated in the textbook solution.

Activation Energy

Activation energy, represented by 'Ea', is the minimum amount of energy that reacting particles must possess for a reaction to occur. It is a threshold that must be overcome for reactants to transform into products. A higher activation energy implies that fewer molecules have the requisite energy to react at a given temperature, thus the reaction rate is slower. In our example, finding the activation energy involves rearranging and solving the Arrhenius equation, showing that this energy barrier is significant when it comes to understanding and predicting the behavior of chemical reactions.

Comprehension of activation energy is essential when exploring reaction mechanisms since it determines the sensitivity of the rate of reaction to temperature changes. For example, a reaction with low activation energy will show only a modest increase in rate with rising temperature, while a reaction with high activation energy may exhibit a dramatic rate increase under the same conditions.

Preexponential Factor

The preexponential factor or frequency factor, denoted by 'A' in the Arrhenius equation, is another key term in chemical kinetics. It encompasses factors such as the frequency of collisions and the orientation of reactant molecules. This factor is often presumed to remain constant over the temperature range of interest and is determined by the molecular properties and reaction specifics. It's a measure of how many particles have the proper orientation and the probability that particles will collide with sufficient energy to lead to a productive reaction.

In the given example, solving for the preexponential factor required using known values of temperature, rate constants, and activation energy. This numerical value, which often seems large, is indicative of the likelihood of a reaction occurring and provides insight into the nature of the chemical process under study.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes, examining how different experimental conditions influence the speed of a chemical reaction, including temperature, pressure, and concentration of reactants. It helps to develop an understanding of reaction mechanisms and the factors that can be manipulated to control the pace of a reaction. In the case of the example problem, chemical kinetics principles are applied to determine how temperature affects the rate of the reaction between hydrogen and iodine.

Knowledge of chemical kinetics is fundamental in various industries, from pharmaceuticals to manufacturing, since it allows scientists and engineers to optimize conditions for the desired reaction rate and yield. Through exercises like the one provided, students grasp the practical application of kinetics, unlocking the potential to innovate and improve upon existing processes in scientific research and industrial practice.