Question

Learning Goal: To use the Arrhenius equation to calculate the activation energy. As temperature rises, the average kinetic energy of molecules increases. In a chemical reaction, this means that a higher percentage of the molecules possess the required activation energy, and the reaction goes faster. This relationship is shown by the Arrhenius equation k=Ae−Ea/RT where k is the rate constant, A is the frequency factor, Ea is the activation energy, R = 8.3145 J/(K⋅mol) is the gas constant, and T is the Kelvin temperature. The following rearranged version of the equation is also useful: ln(k1/k2)=(Ea/R)(1/T2−1/T1) where k1 is the rate constant at temperature T1, and k2 is the rate constant at temperature T2.

The rate constant of a chemical reaction increased from 0.100 s−1 to 2.70 s−1 upon raising the temperature from 25.0 ∘C to 55.0 ∘C .

Part A

Calculate the value of (1/T2−1/T1) where T1 is the initial temperature and T2 is the final temperature.

Express your answer numerically.

Part B

Calculate the value of ln(k1/k2) where k1 and k2 correspond to the rate constants at the initial and the final temperatures as defined in part A.

Express your answer numerically.

Part C

What is the activation energy of the reaction?

Express your answer numerically in kilojoules per mole.

Answer 1

Answer 2

Part A:

To calculate the value of (1/T2 - 1/T1), we need to convert the temperatures from Celsius to Kelvin and then substitute the values into the equation.

Given: T1 = 25.0 °C = 25.0 + 273.15 K = 298.15 K T2 = 55.0 °C = 55.0 + 273.15 K = 328.15 K

Now we can calculate (1/T2 - 1/T1): (1/T2 - 1/T1) = (1/328.15 K - 1/298.15 K) = 0.003191 K⁻¹

Part B:

To calculate the value of ln(k1/k2), we need to use the given rate constants and plug them into the equation.

Given: k1 = 0.100 s⁻¹ k2 = 2.70 s⁻¹

Now we can calculate ln(k1/k2): ln(k1/k2) = ln(0.100 s⁻¹ / 2.70 s⁻¹) = ln(0.037037) ≈ -3.295

Part C:

To calculate the activation energy (Ea), we can rearrange the equation ln(k1/k2) = (Ea/R)(1/T2 - 1/T1) and solve for Ea.

Given: R = 8.3145 J/(K⋅mol)

Now we can rearrange the equation and solve for Ea: Ea = (ln(k1/k2) * R) / (1/T2 - 1/T1) Ea = (-3.295 * 8.3145 J/(K⋅mol)) / (0.003191 K⁻¹) Ea ≈ -27.339 kJ/mol

Therefore, the activation energy of the reaction is approximately 27.339 kJ/mol.