Question

The rate constant of a reaction is \(1.5 \times 10^{7} \mathrm{~s}^{-1}\) at \(50^{\circ} \mathrm{C}\) and \(4.5 \times 10^{7} \mathrm{~s}^{-1}\) at \(100^{\circ} \mathrm{C}\). What is the value of activation energy? a. \(2.2 \times 10^{3} \mathrm{~J} \mathrm{~mol}^{-1}\) b. \(2300 \mathrm{~J} \mathrm{~mol}^{-1}\) c. \(2.2 \times 10^{4} \mathrm{~J} \mathrm{~mol}^{-1}\) d. \(220 \mathrm{~J} \mathrm{~mol}^{-1}\)

Short answer

The value of activation energy is \(2.2 \times 10^4 \ \mathrm{J} \ \mathrm{mol}^{-1}\), option c.

Step-by-step solution

Understand the problem

We need to find the activation energy ( abla E_a abla E_aEa​nabla ) of a chemical reaction at two different temperatures given the rate constants. We use the Arrhenius equation and its related calculations to find this value.

Use the Two Temperature Arrhenius Equation

The formula connecting two rate constants at different temperatures through activation energy is: \[\ln \left(\frac{k_2}{k_1}\right) = \frac{E_a}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right)\] Here, \(k_1 = 1.5 \times 10^7 \ \mathrm{s}^{-1}, k_2 = 4.5 \times 10^7 \ \mathrm{s}^{-1}\), and we convert temperatures from Celsius to Kelvin, thus \(T_1 = 323 \ \mathrm{K}, T_2 = 373 \ \mathrm{K}\). \(R\) is the gas constant, \(8.314 \ \mathrm{J} \ \mathrm{mol}^{-1} \ \mathrm{K}^{-1}\).

Calculate the temperature difference

Convert the temperatures to Kelvin: - \(T_1 = 50 + 273 = 323 \ \mathrm{K}\)- \(T_2 = 100 + 273 = 373 \ \mathrm{K}\)Now, find the inverse temperature difference: \[\frac{1}{T_1} - \frac{1}{T_2} = \frac{1}{323} - \frac{1}{373} = 7.996 \times 10^{-4} \ \mathrm{K}^{-1}\]

Calculate the natural logarithm of rate constant ratio

Find the ratio and its natural logarithm:\[\frac{k_2}{k_1} = \frac{4.5 \times 10^7}{1.5 \times 10^7} = 3\]\[\ln\left(\frac{k_2}{k_1}\right) = \ln(3) = 1.0986\]

Determine the activation energy \(E_a\)

Plug the known values into the Arrhenius equation:\[1.0986 = \frac{E_a}{8.314} \times 7.996 \times 10^{-4}\]Rearrange and solve for \(E_a\):\[E_a = \frac{1.0986 \times 8.314}{7.996 \times 10^{-4}} = 2.2 \times 10^4 \ \mathrm{J} \ \mathrm{mol}^{-1}\]

Identify the answer from the choices

The calculated activation energy \(E_a\) is \(2.2 \times 10^4 \ \mathrm{J} \ \mathrm{mol}^{-1}\), which corresponds to option c.

Key concepts

Arrhenius Equation

The Arrhenius equation is a fundamental formula used to describe how the rate constant of a chemical reaction varies with temperature. It establishes a quantitative relationship between these parameters, allowing chemists to better predict reaction behaviors under different conditions. The general form of the Arrhenius equation is given by:- \[k = A e^{-E_a/RT}\] - Where:

• \(k\) is the rate constant, which is a measure of how fast a reaction proceeds.
• \(A\) is the pre-exponential factor or frequency factor, a constant that depends on the specific reaction.
• \(E_a\) represents the activation energy, the minimum energy required to initiate a reaction.
• \(R\) is the universal gas constant, approximately \(8.314 \ \mathrm{J} \ \mathrm{mol}^{-1} \ \mathrm{K}^{-1}\).
• \(T\) is the temperature in Kelvin.

Understanding this equation is crucial because it allows you to see how slight changes in temperature can lead to significant changes in the rate and mechanism of a chemical reaction.

Rate Constant

The rate constant, denoted as \(k\), is a crucial component in chemical kinetics equations, specifically the Arrhenius equation. It quantifies the speed of a chemical reaction at a particular temperature.
The rate constant is influenced by several factors:

• Nature of reactants: Different substances react with varying speeds due to intrinsic properties such as molecular structure.
• Temperature: As temperature rises, collisions between molecules become more frequent and energetic, often increasing the rate constant.
• Presence of catalysts: Catalysts can lower the activation energy, facilitating faster reactions without being consumed in the process.

In our exercise, we saw different rate constants at two distinct temperatures, illustrating its temperature dependence. Understanding how to calculate and interpret the rate constant helps predict how quickly a reaction will proceed under certain conditions.

Temperature Dependence

Temperature is a key factor in determining the speed of chemical reactions. According to chemical kinetics, as the temperature increases, the kinetic energy of molecules also increases. This makes them move faster, increasing the likelihood of successful collisions that lead to a reaction.
In the context of the Arrhenius equation, temperature affects the exponential factor \(e^{-E_a/RT}\). When the temperature \(T\) is high, the fraction \(-E_a/RT\) becomes less negative, increasing the rate constant \(k\), hence speeding up the reaction.
Understanding temperature dependence is crucial for controlling reactions in industrial processes, laboratories, and natural environments, where precise temperature control can significantly alter reaction rates.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical processes and the factors affecting them. It seeks to understand how different variables like concentration, temperature, and catalysts influence the speed of a reaction.
Key components in chemical kinetics include:

• Reaction Rate: The speed at which reactants are converted into products.
• Rate Laws: Mathematical expressions that describe the relationship between the rate of a reaction and the concentration of reactants.
• Activation Energy: The minimum amount of energy required to initiate a chemical reaction, crucially explained by the Arrhenius equation.
• Mechanisms: The step-by-step sequence of elementary reactions by which overall chemical change occurs.

Chemical kinetics is not just about predicting reaction speeds, but also about controlling and optimizing them for desired outcomes, be it in a laboratory synthesis or large-scale industrial production.