Calculation Involving a Change in Temperature

Let's examine the general rule of thumb regarding the rate constant doubling for a 10 ° C increase in temperature. Assume a pre-exponential factor of 5.0 x 10¹⁶ s^-1. The ideal gas constant is 8.3145 J/mol K. Find the ratio of the two different rate constants for an activation energy of 147 kJ/mol and a change in temperature from 490 K to 500 K and 500 K to 510 K.

Solution:

A total of three temperatures are given at which to find the rate constants. We can insert these given values into Arrhenius' equation. We will calculate three rate constants, k₁, k₂, and k₃ at 490 K, 500 K, and 510 K respectively.

k = A*exp(-E/(RT))
k₁ = 5.0 x 10¹⁶*exp(-147,000/(8.3145*490)) = 10.689 s^-1
k₂ = 5.0 x 10¹⁶*exp(-147,000/(8.3145*500)) = 21.996 s^-1
k₃ = 5.0 x 10¹⁶*exp(-147,000/(8.3145*510)) = 44.001 s^-1

Now let's look at the ratios of these rate constants.

k₂/k₁ = 21.996/10.689 = 2.06

This is the increase in the rate constant by raising the temperature from 490 K to 500 K.

k₃/k₂ = 44.001/21.996 = 2.00

This is the increase in the rate constant by raising the temperature from 500 K to 510 K.

As you can see, the rate constant only doubles for a specific temperature change of 10 ° C.

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This project was funded in part by the National Science Foundation and is advised by Dr. Masel and Dr. Blowers at the University of Illinois.