A reaction has a rate constant of 0.000122 s⁻¹ at 27 °C and 0.228 s⁻¹ at 77 °C. a. Determine the activation barrier for the reaction.

Verified step by step guidance

1. Identify the given data: rate constants k1 = 0.000122 \text{ s}^{-1} at T1 = 27 \degree C and k2 = 0.228 \text{ s}^{-1} at T2 = 77 \degree C.
2. Convert the temperatures from Celsius to Kelvin: T1 = 27 + 273.15 \text{ K} and T2 = 77 + 273.15 \text{ K}.
3. Use the Arrhenius equation: k = A e^{-\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 temperature in Kelvin.
4. Apply the Arrhenius equation in its logarithmic form to find the activation energy: \ln\left(\frac{k2}{k1}\right) = \frac{E_a}{R}\left(\frac{1}{T1} - \frac{1}{T2}\right).
5. Solve for E_a (activation energy) by rearranging the equation: E_a = \frac{R \cdot \ln\left(\frac{k2}{k1}\right)}{\left(\frac{1}{T1} - \frac{1}{T2}\right)}.

Key Concepts

Rate Constant (k)

The rate constant (k) is a proportionality factor in the rate equation of a chemical reaction, indicating the speed of the reaction at a given temperature. It is influenced by factors such as temperature and the activation energy of the reaction. The units of k vary depending on the order of the reaction, and for first-order reactions, it is expressed in s⁻¹.

Arrhenius Equation

The Arrhenius equation relates the rate constant of a reaction to the temperature and activation energy, expressed as k = A * e^(-Ea/RT). Here, A is the pre-exponential factor, Ea is the activation energy, R is the universal gas constant, and T is the temperature in Kelvin. This equation helps in understanding how temperature affects reaction rates and is essential for calculating the activation barrier.

Activation Energy (Ea)

Activation energy (Ea) is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that reactants must overcome to form products. A higher activation energy indicates a slower reaction rate, while a lower activation energy suggests a faster reaction. Understanding Ea is crucial for predicting how changes in temperature will affect the rate of a reaction.