If the rate of a reaction increases by a factor of 2.5 when the temperature is raised from 20 °C to 30 °C, what is the value of the activation energy in kJ/mol? By what factor does the rate of this reaction increase when the temperature is raised from 120 °C to 130 °C?

Verified step by step guidance

1. Identify that the problem involves the Arrhenius equation, which relates the rate constant k to the temperature T and activation energy E_a: k = A * e^(-E_a/(RT)), where A is the pre-exponential factor and R is the gas constant.
2. Use the given information that the rate increases by a factor of 2.5 when the temperature increases from 20 °C to 30 °C. This implies that k_2/k_1 = 2.5, where k_1 and k_2 are the rate constants at 20 °C and 30 °C, respectively.
3. Convert the temperatures from Celsius to Kelvin by adding 273.15 to each temperature: T_1 = 293.15 K and T_2 = 303.15 K.
4. Apply the Arrhenius equation in the form of the ratio of rate constants: ln(k_2/k_1) = (E_a/R) * (1/T_1 - 1/T_2). Substitute the known values to solve for E_a.
5. To find the factor by which the rate increases from 120 °C to 130 °C, repeat the process using T_1 = 393.15 K and T_2 = 403.15 K, and the previously calculated E_a.

Key Concepts

Arrhenius Equation

The Arrhenius equation describes how the rate of a chemical reaction depends on temperature and activation energy. It is expressed as k = A * e^(-Ea/RT), where k is the rate constant, 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 highlights that an increase in temperature generally leads to an increase in reaction rate due to more molecules having sufficient energy to overcome the activation barrier.

Activation Energy (Ea)

Activation energy 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 means that fewer molecules will have enough energy to react at a given temperature, resulting in a slower reaction rate. Understanding Ea is crucial for predicting how changes in temperature affect reaction rates.

Temperature Dependence of Reaction Rates

The rate of a chemical reaction typically increases with temperature due to the increased kinetic energy of the molecules involved. As temperature rises, more molecules collide with sufficient energy to overcome the activation energy barrier, leading to a higher reaction rate. This relationship is often quantified using the Arrhenius equation, which allows for the calculation of how much the rate will change with temperature variations.