Substituting time for k

Arrhenius Equation: $\ln k = - \frac{E_{a}}{RT} + \ln A$

Glenda Marshall DIS 3M
Posts: 68
Joined: Sat Sep 27, 2014 3:00 am

Substituting time for k

Hello!
Problem #14.93 in the textbook says

"Raw milk sours in about 4 hours at 28 degrees celsius but in about 48 hours in a fridge at 5 degrees celsius. What is the activation energy for the souring of milk?"

When I looked at this problem I knew I had to use the Arrhenius Equation comparing two reactions but I was baffled as far as how to get the values of the rate constants. The solutions manual said that

rate at 28 degrees/rate at 5 degrees = k1/k2 = T2/T1

My question is, how do you know when it is okay to substitute temperature for the rate constant and why they used the inverse? Also, why were they able to say that the rate is equal to the k value. Thanks!

Neil DSilva 1L
Posts: 70
Joined: Fri Sep 26, 2014 2:02 pm

Re: Substituting time for k

The units for the rate of milk spoiling are $\frac{[milk]}{hr}$.

I believe we assume a first order reaction, so $rate = k[milk]$.
Then, the units of k and k' are both $\frac{1}{hr}$.

This is why you can rewrite $\frac{k}{k'}$ as $\frac{t'}{t}$ since k and t (time in hours) are inversely related.

We're able to substitute k values for the rates using the following analysis:
$\frac{rate(28^{\circ })}{rate(5^{\circ })} = \frac{k[milk]}{k'[milk]} = \frac{k}{k'}$. (The concentration of milk is the same in both scenarios, so the concentration of milk cancels out.)
Last edited by Neil DSilva 1L on Fri Feb 20, 2015 9:47 pm, edited 1 time in total.

Glenda Marshall DIS 3M
Posts: 68
Joined: Sat Sep 27, 2014 3:00 am

Re: Substituting time for k

That was so unbelievably helpful! Thank you!