# Newton's law of cooling for corpse

1. Jul 28, 2007

### PsiPhi

1. The problem statement, all variables and given/known data
The time it takes for a deceased human body to reach room temperature.
Room temperature = 25 degrees C
Initial temperature of corpse = 37 degrees C

2. Relevant equations
I used Newton's law of cooling: $$\frac{dT}{dt} = -k(T - T_{room})$$
where T is a function of t(time in seconds), T(room) is the room temperature and k is the experimental constant.
3. The attempt at a solution
Well the solution to the D.E was simple, it is $$T(t) = 12e^{-kt} + 25$$
Ok, now I don't know enough information to determine k(the experimental constant). I generated various values of T(t) until it reach 25degreesC. For k = 1, it took 100secs to get to 25.(this seemed unreasonable). For k = 0.001, it took approximately 6 hrs(this seemed reasonable). Now for my problem is there way I can determine k, without resorting to plugging in random values of k. I was thinking of using the heat capacity of water to approximate the human, but then how is that related to k?

Thanks

Last edited: Jul 28, 2007
2. Jul 28, 2007

### Irid

If you want reliable data, you must kill a number of persons and measure the time, and deduce k from that.

If you want an estimate, I could suggest to look at a similar problem. It's Problem 3 - Hard Boiled Egg, from IPhO 2006 (www.ipho2006.org). In the web-site there's the problem as well as a solution.

3. Jul 28, 2007

### Gokul43201

Staff Emeritus
In the limit of a healthy airflow outside the corpse, the time constant, T = 1/k, can be approximated by T ~ RC (possibly with some factors of l/A and density - check dimensions), where R is the thermal resistance (1/conductance) of the body and C is its heat capacity.