Question

Calculate the ratio of rate constants for two thermal reactions that have the same Arrhenius preexponential term but have activation energies that differ by \(1.0,10 .,\) and \(30 . \mathrm{kJ} / \mathrm{mol}\) for \(T=298 \mathrm{K}\).

Short answer

The ratio of rate constants for the activation energy differences are: \(R_1 = 0.966\) for 1.0 kJ/mol, \(R_2 = 0.682\) for 10 kJ/mol, and \(R_3 = 0.223\) for 30 kJ/mol.

Step-by-step solution

Understanding the Arrhenius equation

The Arrhenius equation is used to describe the temperature dependence of the rate constants of a chemical reaction. It is given by the following formula: \(k = A \cdot e^{-\frac{E_a}{RT}}\) where, - \(k\) is the rate constant of the reaction, - \(A\) is the preexponential factor or the Arrhenius constant (which is same for both the given reactions), - \(E_a\) is the activation energy of the reaction, - \(R\) is the gas constant (8.314 J/mol·K), - \(T\) is the temperature in Kelvin (given as 298 K).

Calculate the rate constants for each reaction

Let's denote the activation energies of the two reactions as \(E_{a1}\) and \(E_{a2}\), where \(E_{a2} = E_{a1} + \Delta E_a\). The rate constants for each reaction at 298 K can be calculated using the Arrhenius equation: \(k_1 = A \cdot e^{-\frac{E_{a1}}{RT}}\) \(k_2 = A \cdot e^{-\frac{E_{a2}}{RT}}\)

Calculate the ratio of rate constants

Now, we need to find the ratio of these rate constants: \(R = \frac{k_1}{k_2} = \frac{A \cdot e^{-\frac{E_{a1}}{RT}}}{A \cdot e^{-\frac{E_{a2}}{RT}}}\) Since the preexponential factors cancel out in the ratio, we're left with: \(R = e^{\frac{E_{a1} - E_{a2}}{RT}} = e^{\frac{-\Delta E_a}{RT}}\) We will calculate this ratio for \(\Delta E_a\) equal to 1.0 kJ/mol, 10 kJ/mol, and 30 kJ/mol.

Calculate the ratio for each given activation energy difference

Let's calculate the rate constant ratios for each of the given values of activation energy difference: \(R_1 = e^{\frac{-1.0 \times 10^3 \, \text{J/mol}}{(8.314 \, \text{J/mol} \cdot \text{K})(298 \, \text{K})}}\) \(R_2 = e^{\frac{-10 \times 10^3 \, \text{J/mol}}{(8.314 \, \text{J/mol} \cdot \text{K})(298 \, \text{K})}}\) \(R_3 = e^{\frac{-30 \times 10^3 \, \text{J/mol}}{(8.314 \, \text{J/mol} \cdot \text{K})(298 \, \text{K})}}\) Calculating these ratios, we get: \(R_1 = 0.966\) \(R_2 = 0.682\) \(R_3 = 0.223\) Finally, we have the ratio of rate constants for each activation energy difference: - For a difference of 1.0 kJ/mol: \(R_1 = 0.966\) - For a difference of 10 kJ/mol: \(R_2 = 0.682\) - For a difference of 30 kJ/mol: \(R_3 = 0.223\)

Key concepts

Activation Energy

Activation energy, denoted as E_a, is the minimum amount of energy that reacting particles must possess in order to undergo a specific chemical reaction. Imagine it as a barrier that molecules must overcome to transform into products. When we increase the activation energy, fewer particles have the necessary energy to react, which leads to a slower reaction rate.

For example, in the provided exercise, reactions with different activation energies at the same temperature were considered. It showed that even a small increase in activation energy can significantly decrease the rate at which a reaction proceeds, emphasizing the sensitivity of chemical processes to this energy parameter.

Rate Constants

Rate constants, represented by the symbol k, are coefficients that quantify the speed of a chemical reaction at a given temperature. They are intrinsic to each reaction and depend on factors such as activation energy and temperature. The Arrhenius equation neatly expresses this relationship.

Understanding the concept of rate constants is essential for predicting how fast a reaction will proceed under various conditions. As illustrated in the step-by-step solution, a change in the activation energy alters the rate constant, hence, influencing the reaction rate.

Chemical Kinetics

Chemical kinetics is the study of reaction rates, how they change under different conditions, and what factors affect these rates. It encompasses various concepts, including the rate constants, activation energy, and the temperature dependence of reactions explained by the Arrhenius equation.

In the context of our problem, chemical kinetics helps us understand why different activation energies result in different reaction speeds. By applying kinetics principles, we can predict and control the outcome of chemical processes, which is crucial in fields such as pharmaceuticals, materials science, and environmental engineering.

Temperature Dependence

The rate at which chemical reactions occur is heavily influenced by temperature—a concept encapsulated in the Arrhenius equation. As temperature increases, molecular motion intensifies, resulting in more collisions with higher energies, thus overcoming the activation energy barrier more easily.

The temperature dependence of reaction rates is exponential, as shown by the negative exponent in the Arrhenius equation. Even small temperature changes can lead to significant alterations in reaction speed. This is why, in many industrial processes, precise temperature control is critical for achieving the desired reaction rates and yields.