At 500K the rate of a bimolecular reaction is ten times the rate at 400K. Find the activation energy of this reaction: a. From Arrhenius’ law.

Sure, let's solve this step by step using the Arrhenius equation. The Arrhenius equation is given by:

k = Ae^(-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.

Given that the rate of the reaction at 500K is ten times the rate at 400K, we can write this as:

k2/k1 = 10

where k2 is the rate constant at 500K and k1 is the rate constant at 400K.

Substituting the Arrhenius equation into this gives:

Ae^(-Ea/500R) / Ae^(-Ea/400R) = 10

The pre-exponential factor A cancels out, leaving:

e^(-Ea/500R) / e^(-Ea/400R) = 10

This simplifies to:

e^(Ea/400R - Ea/500R) = 10

Taking the natural logarithm of both sides gives:

Ea/400R - Ea/500R = ln(10)

This simplifies to:

Ea(1/400 - 1/500) = ln(10)

Solving for Ea gives:

Ea = ln(10) / (1/400 - 1/500)

Substituting the value of ln(10) and simplifying gives the activation energy Ea. Remember to convert the gas constant R to the appropriate units to match the units of Ea.