Question

Consider the hypothetical reaction \(\mathrm{A}_{2}(g)+\mathrm{B}_{2}(g) \longrightarrow\) \(2 \mathrm{AB}(g),\) where the rate law is: $$ -\frac{\Delta\left[\mathrm{A}_{2}\right]}{\Delta t}=k\left[\mathrm{A}_{2}\right]\left[\mathrm{B}_{2}\right] $$ The value of the rate constant at \(302^{\circ} \mathrm{C}\) is \(2.45 \times 10^{-4} \mathrm{L} / \mathrm{mol}\) \(\mathrm{s},\) and at \(508^{\circ} \mathrm{C}\) the rate constant is 0.891 \(\mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\) . What is the activation energy for this reaction? What is the value of the rate constant for this reaction at \(375^{\circ} \mathrm{C} ?\)

Short answer

The activation energy for the given reaction is approximately \(149.7\,\mathrm{kJ}\cdot\mathrm{mol}^{-1}\). The value of the rate constant for this reaction at \(375^{\circ}\,\mathrm{C}\) is approximately \(0.0075\, \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\).

Step-by-step solution

Write down the Arrhenius equation

The Arrhenius equation is given by: $$ k = Ae^{\frac{-E_a}{RT}} $$ where \(k\) is the rate constant, \(A\) is the pre-exponential factor, \(E_a\) is the activation energy, \(R\) is the gas constant (8.314 J/mol K), and \(T\) is the absolute temperature in Kelvin.

Convert the given temperatures to Kelvin

To relate the rate constants and temperatures, we have to convert them to Kelvin using the following formula: $$ T(K) = T(^\circ C) + 273.15 $$ For the given temperatures: $$ T_1 = 302^\circ C + 273.15 = 575.15 K \\ T_2 = 508^\circ C + 273.15 = 781.15 K \\ T_3 = 375^\circ C + 273.15 = 648.15 K $$ where \(T_1\), \(T_2\), and \(T_3\) are the absolute temperatures at the given Celsius temperatures.

Apply the Arrhenius equation for the two given rate constants

Using the rate constants provided at the two temperatures \(k_1\) at \(T_1\) and \(k_2\) at \(T_2\), we now have two equations: $$ k_1 = Ae^{\frac{-E_a}{R T_1}} \\ k_2 = Ae^{\frac{-E_a}{R T_2}} $$ where \(k_1=2.45 \times 10^{-4}\,\mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\) and \(k_2=0.891\, \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\).

Solve for activation energy, \(E_a\)

To solve for the activation energy, divide the second equation by the first equation: $$ \frac{k_2}{k_1} = \frac{Ae^{\frac{-E_a}{R T_2}}}{Ae^{\frac{-E_a}{R T_1}}} $$ Solve for the activation energy, \(E_a\): $$ E_a = R\left(T_2-T_1\right) \frac{\ln \frac{k_1}{k_2}}{\ln \frac{k_1}{k_2} - 1} $$ Substitute the values and compute the activation energy, \(E_a\): $$ E_a = 8.314\, J \cdot mol^{-1}K^{-1} \cdot(781.15\,K-575.15\,K) \cdot \frac{\ln \frac{2.45 \times 10^{-4}}{0.891}}{\ln \frac{2.45 \times 10^{-4}}{0.891} - 1} \approx 149655\,J/mol $$ So, the activation energy is \(E_a \approx 149.7\,\mathrm{kJ}\cdot\mathrm{mol}^{-1}\).

Calculate the rate constant for the reaction at \(T_3\)

Now, we need to find the pre-exponential factor, \(A\), using one of the given rate constants. We can use the equation for \(k_1\) and \(T_1\): $$ A = \frac{k_1}{e^{\frac{-E_a}{R T_1}}} $$ Substitute the values and compute \(A\): $$ A = \frac{2.45 \times 10^{-4}\,\mathrm{L} / \mathrm{mol} \cdot \mathrm{s}}{e^{\frac{-149655\,J\cdot mol^{-1}}{8.314\, J \cdot mol^{-1}K^{-1}\cdot 575.15\, K}}} \approx 4.88 \times 10^{-6}\, \mathrm{L} / \mathrm{mol} \cdot \mathrm{s} $$ Now that we have the pre-exponential factor, we can calculate the rate constant at \(T_3\): $$ k_3 = Ae^{\frac{-E_a}{RT_3}} $$ Substitute the values and calculate \(k_3\): $$ k_3 = 4.88 \times 10^{-6}\, \mathrm{L} / \mathrm{mol} \cdot \mathrm{s} \cdot e^{\frac{-149655\, J/mol}{8.314\, J / mol\cdot K\cdot 648.15\,K}} \approx 0.0075\, \mathrm{L} / \mathrm{mol} \cdot \mathrm{s} $$ Therefore, the value of the rate constant for this reaction at \(375^\circ \mathrm{C}\) is approximately \(0.0075\, \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\).

Key concepts

Arrhenius Equation

The Arrhenius equation is a fundamental formula in chemistry that relates the rate constant ( k) of a chemical reaction to its temperature and activation energy. This equation is expressed as \( k = Ae^{-E_a/(RT)} \), where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

The pre-exponential factor ( A) represents the frequency of collisions between reacting molecules, while the term involving \( E_a \) signifies the probability that a given collision will overcome the activation energy. Essentially, the equation shows how temperature affects reaction rates by increasing how often molecules collide and have enough energy to react.

Using the Arrhenius equation, you can calculate the activation energy by measuring the rate constants of a reaction at different temperatures. This involves rearranging and applying logarithms to the equation, which illustrates the sensitivity of reaction rates to temperature changes.

Rate Constant

The rate constant, denoted as \( k \), is an essential parameter in chemical kinetics that indicates the speed of a reaction. Its value is determined experimentally and varies with temperature. For the reaction \( \text{A}_2(g) + \text{B}_2(g) \rightarrow 2\text{AB}(g) \), the rate law is given by \( -\frac{\Delta [\text{A}_2]}{\Delta t} = k[\text{A}_2][\text{B}_2] \), suggesting that both \( \text{A}_2 \) and \( \text{B}_2 \) affect the reaction rate.

The units of \( k \) depend on the overall order of the reaction. For this second-order reaction, the rate constant has units of \( \text{L/mol} \cdot \text{s} \). As temperature increases, the rate constant typically increases, allowing reactions to proceed more quickly. The Arrhenius equation shows us this phenomenon: at higher temperatures, the exponential term in the equation becomes larger, thereby increasing \( k \).

The rate constant is critical because it enables the prediction of reaction rates and can help in the design of chemical processes, environmental sciences, and even pharmacokinetics.

Reaction Kinetics

Reaction kinetics is a branch of chemistry that studies the speed of chemical reactions and the factors affecting these rates. Understanding kinetics is crucial to controlling reactions for industrial applications as well as biological processes.

Several factors influence reaction rates:

• Concentration of Reactants: Higher concentrations generally lead to faster reactions.
• Temperature: An increase in temperature typically increases reaction rates, aligned with the Arrhenius equation.
• Presence of a Catalyst: Catalysts lower activation energy, thereby speeding up reactions without being consumed.

Analyzing reaction kinetics involves studying rate laws and determining parameters like the rate constant and activation energy. These analyses help in designing chemical reactors and predicting how reactions will progress over time under different conditions.

By mastering reaction kinetics, chemists can optimize conditions to maximize yield and efficiency, addressing both practical and theoretical challenges in chemical research.