If a temperature increase from 20.0 °C to 35.0 °C triples 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) = 20.0 °C, final temperature (T2) = 35.0 °C, and the rate constant triples, so k2 = 3k1.
2. Convert the temperatures from Celsius to Kelvin by adding 273.15 to each temperature: T1 = 293.15 K and T2 = 308.15 K.
3. Use the Arrhenius equation in its ratio form: \( \frac{k_2}{k_1} = \frac{A e^{-E_a/(RT_2)}}{A e^{-E_a/(RT_1)}} = e^{-E_a/(RT_2) + E_a/(RT_1)} \), which simplifies to \( \frac{k_2}{k_1} = e^{E_a/R (1/T_1 - 1/T_2)} \).
4. Substitute the known values into the equation: \( 3 = e^{E_a/R (1/293.15 - 1/308.15)} \).
5. Solve for the activation energy \( E_a \) by taking the natural logarithm of both sides and rearranging the equation: \( E_a = R \cdot \ln(3) / (1/293.15 - 1/308.15) \), where R is the gas constant (8.314 J/mol·K).

Key Concepts

Rate Constant (k)

The rate constant (k) is a proportionality factor in the rate equation of a chemical reaction, indicating the speed at which the reaction occurs. It is temperature-dependent and reflects how changes in temperature can influence the frequency of collisions and the energy of the reacting molecules.

Arrhenius Equation

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

Activation Energy (Ea)

Activation energy (Ea) is the minimum energy required for a chemical reaction to occur. It represents the energy barrier that must be overcome for reactants to transform into products. A higher activation energy means a slower reaction rate, while a lower activation energy indicates a faster reaction.