Question

At room temperature (20 °C), milk turns sour in about 64 hours. In a refrigerator at 3 °C, milk can be stored three times as long before it sours.
(a) Estimate the activation energy of the reaction that causes the souring of milk.
(b) How long should it take milk to sour at 40 °C?

Answer 1

Answer: Since k2 corresponds to 64 hours, the time for the milk to sour at 40 C is 64 h / 9.38 = 6.8 hours.

Explanation:

At temperature T1, the Arrhenius Equation is:

k1 = Ae^(-Ea/RT1).

An equivalent equation can be written at T2:

k2 = Ae^(-Ea/RT2).

If these equations are divided, then A cancels:

k1/k2 = e^(-Ea/RT1)/e^(-Ea/RT2)

Taking the natural log:

ln(k1/k2) = (Ea/RT2)-(Ea/RT1);

or:

ln(k1/k2) = (Ea/R)(1/T2 - 1/T1)

We can infer from the question that the milk sours 3 times as fast at the higher temperature (let's call it T1), so we can arbitrarily call k2 = 1 and k1 = 3.

a) Substitute:

ln(3) =  (Ea/R)(1/276.15 K - 1/293.15 K).

We get Ea/R = 5231.6. Multiply this by whatever value of R you choose to get Ea in your favorite energy unit. Remember the sig figs.

b) Again, let's let the lower temperature = T2, since we have defined k2 = 1:

ln(k1) = (5231.6)(1/276.15 K - 1/313.15);

ln(k1) = 2.24, so k1 = 9.38.

Since k2 corresponds to 64 hours, the time for the milk to sour at 40 C is 64 h / 9.38 = 6.8 hours.