Question

For the decomposition of a peroxide, the activation energy is \(17.4 \mathrm{~kJ} / \mathrm{mol}\). The rate constant at \(25^{\circ} \mathrm{C}\) is \(0.027 \mathrm{~s}^{-1}\). (a) What is the rate constant at \(65^{\circ} \mathrm{C}\) ? (b) At what temperature will the rate constant be \(25 \%\) greater than the rate constant at \(25^{\circ} \mathrm{C}\) ?

Short answer

Answer: The rate constant at \(65^{\circ} \mathrm{C}\) is approximately \(0.178\, s^{-1}\). The rate constant will be \(25 \%\) greater than the rate constant at \(25^{\circ} \mathrm{C}\) when the temperature is approximately \(39.47^{\circ} \mathrm{C}\).

Step-by-step solution

Use the Arrhenius equation

The Arrhenius equation relates the rate constant of a reaction, activation energy, and temperature. It is given as: \(k = Ae^{\frac{-E_a}{RT}}\) Here, \(k\) is the rate constant, \(A\) is the pre-exponential factor (frequency factor), \(E_a\) is the activation energy (\(17.4 \mathrm{~kJ/mol}\)), \(R\) is the gas constant (\(8.314 \mathrm{J/(mol \cdot K)}\)), \(T\) is the temperature in Kelvin.

Convert temperatures to Kelvin

To work with the temperatures in the Arrhenius equation, we need them in Kelvin. For part (a): \(25^{\circ} \mathrm{C} = 25 + 273.15 = 298.15\,K\) And for part (b): \(65^{\circ} \mathrm{C} = 65 + 273.15 = 338.15\,K\)

Part (a) - Step 3: Find the rate constant at 65°C

First, since we have \(k\) at \(25^{\circ} \mathrm{C}\), we will find the pre-exponential factor \(A\) using the given rate constant. \(\begin{aligned} A & = \frac{k}{e^{\frac{-E_a}{RT}}} \\ A & = \frac{0.027}{e^{\frac{-17,400}{8.314 \cdot 298.15}}} \end{aligned}\) Now, we can use \(A\) to find the rate constant \(k\) at \(65^{\circ} \mathrm{C}\) or \(338.15\,K\). \(k_{65} = Ae^{\frac{-E_a}{R \cdot 338.15}}\)

Part (a) - Step 4: Calculate the rate constant at 65°C

Solve the equation to find \(k_{65}\): \(k_{65} = \frac{0.027}{e^{\frac{-17,400}{8.314 \cdot 298.15}}} \cdot e^{\frac{-17,400}{8.314 \cdot 338.15}}\) \(k_{65} \approx 0.178\, s^{-1}\) The rate constant at \(65^{\circ} \mathrm{C}\) is approximately \(0.178\, s^{-1}\).

Part (b) - Step 5: Set up the equation for 25% greater rate constant

The rate constant at \(25^{\circ} \mathrm{C}\) is \(0.027\, s^{-1}\). We want to find the temperature at which the rate constant is \(25\%\) greater, so we need to find the temperature at which the rate constant is \(0.027(1+0.25) = 0.03375\, s^{-1}\). Using the Arrhenius equation and the value of \(A\) we found earlier: \(k = Ae^{\frac{-E_a}{RT}}\) \(0.03375 = \frac{0.027}{e^{\frac{-17,400}{8.314 \cdot 298.15}}} \cdot e^{\frac{-17,400}{8.314 \cdot T}}\)

Part (b) - Step 6: Solve for the temperature T

To solve for the temperature, we can rearrange the formula and use logarithms to isolate T: \(T = \frac{-17,400}{8.314\, ln\Big(\frac{0.03375}{0.027} \cdot e^{\frac{-17,400}{8.314 \cdot 298.15}}\Big)}\)

Part (b) - Step 7: Calculate the temperature T

Solve for the temperature T: \(T \approx 312.62 \,K\) We need to convert the result back to Celsius: \(312.62\,K - 273.15 \approx 39.47^{\circ} \mathrm{C}\) The rate constant will be \(25 \%\) greater than the rate constant at \(25^{\circ} \mathrm{C}\) when the temperature is approximately \(39.47^{\circ} \mathrm{C}\).

Key concepts

Activation Energy

Activation energy, symbolized as \(E_a\), is the minimum amount of energy required for a chemical reaction to occur. It's akin to climbing a hill before rolling down the other side; molecules need a certain energy boost to reach the top, called the transition state, before they can transform into products.

The importance of understanding activation energy lies in its influence on the reaction rate. A higher \(E_a\) means that fewer molecules have the necessary energy to react at a given temperature, leading to a slower reaction. Conversely, a lower \(E_a\) means more molecules can overcome the energy barrier, making the reaction faster. This is critical when considering factors such as catalysts, which lower the \(E_a\) to speed up a reaction, or temperature changes, as seen in the textbook exercise where the decomposition of a peroxide becomes more rapid at higher temperatures due to the decreased relative effect of the activation energy barrier.

Rate Constant

The rate constant, denoted by \(k\), is a proportionality constant in the rate equation of a chemical reaction that links the reaction rate to the concentrations of reactants. It provides insight into the speed of a reaction under a set of conditions.

Changes in temperature can significantly affect the rate constant, which is illustrated in the exercise where calculating \(k\) at different temperatures using the Arrhenius equation shows how a reaction quickens with rising temperatures. Higher temperatures give reactant molecules more kinetic energy, increasing the frequency of collisions and the likelihood of achieving the activation energy required for reaction. This temperature dependence is why the rate constant changes from \(0.027 \text{ s}^{-1}\) at \(25^{\text{o}} C\) to a much higher value at \(65^{\text{o}} C\). Understanding the rate constant is crucial for anyone studying chemical kinetics, as it supports predictions about how fast a reaction will proceed and how conditions can be adjusted to control the reaction rate.

Chemical Kinetics

Chemical kinetics is the study of reaction rates and the factors that affect them. It's a vital area of study because it helps predict the speed at which reagents are consumed and products are formed, allowing for control over industrial processes, biological systems, and environmental changes.

In the context of the exercise, kinetics concepts explain how temperature affects reaction rates through the rate constant and activation energy. By using the Arrhenius equation, chemists can model how even slight temperature changes can have a substantial effect on the rate constant and thus the overall speed of a reaction. Chemical kinetics merges theoretical calculations, such as those in the exercise, with experimental data to help us understand and predict how various factors—like temperature, concentration, and catalyst presence—impact reaction rates, emphasizing the real-world relevance of these parameters in scientific and industrial applications.