Newton's law of cooking chicken

Newton's law of cooling can be used to predict the time of death from the temperature of a corpse and the ambient, which is a silly and boring example. On the other hand, when we are cooking chicken and our better half asks when it will be done (and demands that it be done in a given time), then it becomes an interesting and useful equation. Let us say that it has already cooked for almost an hour, it is around half past seven, and dinner is supposed to be at eight.

The important point is that Newton's law of cooling is equally applicable to cooking because physical law is symmetrical, which means that cooling and heating behaves the same way. The law states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient, or

dT(t)/dt = -k(T(t)-Tambient)

where T is the temperature of the object, t is time, k is a constant, and Tambient is the ambient temperature. It is a differential equation that solves to 

T(t) = Tambient + (T(0) - Tambient)e^-kt

which we can use to calculate the temperature we have to cook the chicken at to be able to serve dinner at eight(-ish). The only problem is that we have to know the constant k which is specific to the particular chicken and filling we have in the oven. Luckily, we used an oven-thermometer, and we kind of remember how long it has cooked already. So, if it took 50 minutes to go from 10°C to 53°C with the oven at 150°C then we can calculate the constant as

k = -1/t ln((T(t)-Tambient) / (T(0)-Tambient))

that is

k = -1/50 * ln((53-150)/(10-150)) = 0.0073 min^-1

In turn, we can use this to calculate what temperature we have to use for the chicken to be done in another 30 minutes as

Tambient = (T(t) - T(0)e^-kt) / (1 - e^-kt) 

 which gives

Tambient = (80 - 53*e^-0.0073*30) / (1 - e^-0.0073*30) = 200°C

Luckily, we didn't have to do the calculations by hand because a bigger nerd than us have created a web-app where we just plug in the known values and get the missing one for free.

Finally, here is the money-shot.

Science, because it works bitch.

(Although the higher temperature did burn the skin a little bit, and it would have been jucier if it had cooked at 150°C the whole time.)