Problem 55 - Chemistry: The Central Science

Question

A certain first-order reaction has a rate constant of \(2.75 \times 10^{-2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\). What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if (a) \(E_{a}=75.5 \mathrm{~kJ} / \mathrm{mol} ;\) (b) \(E_{a}=125 \mathrm{~kJ} / \mathrm{mol} ?\)

Step-by-Step Solution

Answer

The value of the rate constant \(k\) at \(60^{\circ}C\) is: (a) \(1.22 \times 10^{-1} ~s^{-1}\) when \(E_{a} = 75.5 \mathrm{~kJ} / \mathrm{mol}\) (b) \(3.28 \times 10^{-1} ~s^{-1}\) when \(E_{a} = 125 \mathrm{~kJ} / \mathrm{mol}\)

Step 1: Write down the given information

We are given the following data: Initial rate constant, \(k_1 = 2.75 \times 10^{-2} ~s^{-1}\) Initial temperature, \(T_1 = 20^{\circ}C = 293K\) (always convert temperatures to Kelvin) Final temperature, \(T_2 = 60^{\circ}C = 333K\) Activation energy for (a), \(E_{a1} = 75.5 kJ/mol = 75500J/mol\) Activation energy for (b), \(E_{a2} = 125 kJ/mol = 125000J/mol\) R (gas constant) = \(8.314 J/molK\) We will use the Arrhenius equation to find the rate constant at \(60^{\circ}C\).

Step 2: Write down the Arrhenius equation

The Arrhenius equation relates the rate constant, activation energy, and temperature as follows: \(k_2 = k_1 \times e^{(\frac{-E_a}{R})}(\frac{1}{T_1} - \frac{1}{T_2})\) We will use this equation to solve for the rate constant (\(k_2\)) at the final temperature.

Step 3: Calculate the value of k at 60°C for (a)

Using the given activation energy for (a), which is \(E_{a1} = 75500J/mol\), we can find the value of \(k_2\): \(k_2 = (2.75 \times 10^{-2}) \times e^{(\frac{-75500}{8.314})}(\frac{1}{293} - \frac{1}{333})\) Calculating the rate constant: \(k_2 ≈ 1.22 \times 10^{-1} ~s^{-1}\) So, the value of \(k_2\) for (a) is \(1.22 \times 10^{-1} ~s^{-1}\).

Step 4: Calculate the value of k at 60°C for (b)

Using the given activation energy for (b), which is \(E_{a2} = 125000 J/mol\), we can find the value of \(k_2\): \(k_2 = (2.75 \times 10^{-2}) \times e^{(\frac{-125000}{8.314})}(\frac{1}{293} - \frac{1}{333})\) Calculating the rate constant: \(k_2 ≈ 3.28 \times 10^{-1} ~s^{-1}\) So, the value of \(k_2\) for (b) is \(3.28 \times 10^{-1} ~s^{-1}\). To summarize, the value of the rate constant \(k\) at \(60^{\circ}C\) is: (a) \(1.22 \times 10^{-1} ~s^{-1}\) when \(E_{a} = 75.5 \mathrm{~kJ} / \mathrm{mol}\) (b) \(3.28 \times 10^{-1} ~s^{-1}\) when \(E_{a} = 125 \mathrm{~kJ} / \mathrm{mol}\)

Key Concepts

Rate Constant Determination

In chemical kinetics, the rate constant is a crucial parameter that helps us understand how fast a reaction occurs. It's generally denoted as "k" and it varies with temperature and the nature of the reacting substances. For a first-order reaction, such as the one in the exercise, the rate constant can be determined using data from experiments or calculated using the Arrhenius equation.

The Arrhenius equation is:

• \[ k = A imes e^{-\frac{E_a}{RT}} \]

Here, "A" represents the frequency factor, "E_a" is the activation energy, "R" is the gas constant \(8.314 \, \text{J/molK}\), and "T" is the absolute temperature in Kelvin. When provided with or required to find activation energy and rate constants at two different temperatures, the equation can be manipulated to solve for the unknown "k".

This manipulation often involves taking the natural logarithm of both sides and applying temperature values to find "k" under different conditions. Adjusting the temperature, as we do in the exercise, allows us to see how the rate constant changes with temperature, emphasizing that rate constant determination is tightly linked to temperature and energy changes.

Activation Energy

Activation energy \(E_a\) is the minimum amount of energy needed for reactants to transform into products in a chemical reaction. It is a barrier that needs to be overcome for the reaction to proceed. Higher activation energies often mean that a reaction is slower, as fewer molecules have the necessary energy to surpass the barrier.

In the context of the problem, the activation energies given were \(75.5 \, \text{kJ/mol}\) and \(125 \, \text{kJ/mol}\). These values are crucial for determining how the rate of reaction changes with temperature. The Arrhenius equation, specifically the component \(e^{-\frac{E_a}{RT}}\), illustrates this relationship by showing that higher activation energies make the exponential factor smaller, reducing the rate constant at a given temperature.

Thus, activation energy is not only a measure of how much energy is required for a reaction to occur but also a critical factor in understanding and predicting temperatures' impacts on reaction rates, helping chemists control and manage reactions in industrial and laboratory settings.

Temperature Dependence of Reaction Rates

Temperature is one of the most significant factors influencing the speed of chemical reactions. According to the Arrhenius equation, as temperature increases, the rate constant \(k\) generally increases as well. This is because an increase in temperature raises the average kinetic energy of the molecules, providing more energy to surpass the activation energy barrier.

In the formula \(e^{-\frac{E_a}{RT}}\), an increase in temperature \(T\) leads to a decrease in the negative exponent, making the whole exponential term larger, thereby increasing \(k\). Consequently, the reaction speed increases.

In our specific exercise, when the temperature rises from \(293K\) to \(333K\), we see a noticeable difference in the calculated rate constants \(k_2\) for each given activation energy. This demonstrates that even small changes in temperature can significantly affect reaction rates. This concept is crucial for designing and controlling chemical processes, where reaction rates must be optimized for efficiency, safety, and yield.