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.