Question

Given the same concentrations, the reaction $$ \mathrm{CO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow \mathrm{COCl}_{2}(g) $$ at \(250^{\circ} \mathrm{C}\) is \(1.50 \times 10^{3}\) times as fast as the same reaction at \(150^{\circ} \mathrm{C}\). Calculate the energy of activation for this reaction. Assume that the frequency factor is constant.

Short answer

The activation energy for the reaction is approximately \( 146.8 \) kJ/mol.

Step-by-step solution

Write down the Arrhenius equation

The Arrhenius equation in two-temperature form is given as follows : \( k_2 / k_1 = e^{(E_a / R) [(1 / T_1) - (1 / T_2)]} \) where \( k_2, k_1, T_1, T_2, E_a, R \) are rate constants at temperatures \( T_2, T_1 \) (in Kelvin), the respective temperatures, the activation energy and the gas constant respectively.

Substitute the given rate constant ratio and temperatures into the equation

From the problem, the rate of reaction at \( 250^{\circ} C \) (523K) is given to be \( 1.50 \times 10^{3} \) times as fast as the rate of reaction at \( 150^{\circ} C \) (423K). This gives us the ratio \( k_2 / k_1 = 1.50 \times 10^{3} \). Substituting these into the Arrhenius equation gives \( 1.50 \times 10^{3} = e^{(E_a / R) [(1 / 423) - (1 / 523)]} \)

Solve for the activation energy

Rearrange the equation from step 2 to solve for the activation energy, \( E_a \). This gives us \( E_a = R * \ln{(1.50 \times 10^{3})} / [(1 / 423) - (1 / 523)] \) which, using \( R = 8.314 \) J/mol.K (gas constant), simplifies to \( E_a = 146.8 \) kJ/mol.

Key concepts

Activation Energy

Activation energy, symbolized as \( E_a \), is a crucial concept in chemical kinetics, representing the minimum energy required for a reaction to occur. It's the energy barrier that reactant molecules must overcome to transform into products. In the context of our exercise, the reaction involving carbon monoxide and chlorine gas to produce phosgene (\( \mathrm{COCl}_{2} \) gas) requires a certain amount of energy to progress. This energy ensures that the reaction attains the necessary transition state where bonds can break and new ones form. The Arrhenius equation helps us in calculating this activation energy by relating the reaction rate constants at different temperatures, which is particularly helpful when the frequency factor or the pre-exponential factor remains constant, as assumed in our problem. Understanding \( E_a \) is essential because it dictates not only the speed of a reaction but also its feasibility under certain conditions.

Rate Constants

The rate constant, denoted as \( k \), is a fundamental parameter in expressing the speed of a chemical reaction at a specific temperature. It shows the proportionality between the reaction rate and the concentration of reactants according to the rate law for a reaction. In our exercise, we compare the rate constants of a reaction at two different temperatures, \( k_1 \) at \( 150^{\text{\( \circ \)}}C \) and \( k_2 \) at \( 250^{\text{\( \circ \)}}C \). The dramatic difference in rate constants, with the reaction being \( 1.50 \times 10^{3} \) times faster at the higher temperature, implies that temperature plays a significant role in reaction kinetics. By using the Arrhenius equation, we can quantify how \( k \) varies with temperature, which provides insight into the reaction mechanism and the sensitivity of the reaction rate to temperature changes.

Knowing the rate constant is useful in both theoretical studies of reaction mechanisms and practical applications such as chemical manufacturing, where manipulating reaction rates is often necessary.

Reaction Rate Temperature Dependence

The temperature dependence of reaction rates is a phenomenon where an increase in temperature typically results in a faster reaction. This can be explained by the Arrhenius equation, which describes how the rate constant \( k \) changes with temperature. As temperature rises, molecules have more kinetic energy, which increases the frequency of effective collisions and the likelihood that these collisions will have enough energy to surpass the activation energy barrier, leading to a higher reaction rate.

In our exercise, we observe that a temperature increase from \( 150^{\text{\( \circ \)}}C \) to \( 250^{\text{\( \circ \)}}C \) significantly accelerates the reaction, indicating a sensitive temperature dependence. Understanding this dependence allows chemists and engineers to control reaction rates by adjusting temperatures, which is essential in industries like pharmaceuticals where precise control over reaction conditions is critical for product consistency.

Reflecting on the exercise, one might enhance their grasp of concepts by visualizing the molecular dynamics at different temperatures, utilizing graphing tools to depict how the Arrhenius equation relates \( k \) and \( E_a \) or using simulation software to observe the effect of temperature changes on reaction rates in a virtual setting. All these approaches can help deepen the comprehension of how the reaction rate varies with temperature—an integral part of chemical kinetics.