If a temperature increase from 10.0 °C to 20.0 °C doubles the rate constant for a reaction, what is the value of the activation energy for the reaction?

Verified step by step guidance

1. Identify the given information: initial temperature (T1) = 10.0 °C, final temperature (T2) = 20.0 °C, and the rate constant doubles, so k2 = 2k1.
2. Convert the temperatures from Celsius to Kelvin: T1 = 10.0 + 273.15 K and T2 = 20.0 + 273.15 K.
3. Use the Arrhenius equation in the form: \( \ln \left( \frac{k_2}{k_1} \right) = \frac{-E_a}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \), where \( E_a \) is the activation energy and \( R \) is the gas constant (8.314 J/mol·K).
4. Substitute the known values into the equation: \( \ln(2) = \frac{-E_a}{8.314} \left( \frac{1}{293.15} - \frac{1}{283.15} \right) \).
5. Solve for the activation energy \( E_a \) by isolating it on one side of the equation.

Key Concepts

Rate Constant and Temperature Relationship

The rate constant (k) of a chemical reaction is influenced by temperature, typically increasing with rising temperature. This relationship is often described by the Arrhenius equation, which shows that k is exponentially related to temperature. Understanding this concept is crucial for analyzing how changes in temperature affect reaction rates.

Arrhenius Equation

The Arrhenius equation, k = A * e^(-Ea/RT), relates the rate constant (k) to the activation energy (Ea), the frequency factor (A), and the temperature (T) in Kelvin. This equation allows us to calculate the activation energy when we know how the rate constant changes with temperature. It is fundamental for understanding the energy barrier that must be overcome for a reaction to occur.

Activation Energy

Activation energy (Ea) is the minimum energy required for a chemical reaction to proceed. 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. This concept is essential for determining how temperature changes can influence reaction kinetics.