Newton's Law of Cooling (Time of death)

Question

A detective was asked to estimate the time of death of a man. At 8:20 p.m., it's body temperature was at 27 degrees celsius. The detective found out that the air condition in the room was set to 23 degrees celsius and has been running for about a week now. How long has the man been dead?

Answer 1

Hi Mitsuha!

 

Newton's law of cooling describes a link between the rate at which a body cools and the difference between its temperature and the ambient temperature (of its environment).  If that ambient temperature is held constant, the law simplifies to the form:

 

T(t) - T_a = (T_o-T_a)e^-kt

 

where

 

T_a = (constant) ambient temperature

T_o = original temperature of the object (at t=0)

 

 

We can take the original temperature of the body to be about 37 C (average human body temperature).  So, presumably, at the time of death (t = 0), we had T_o = 37 C.

 

The problem is, as written, the question does not give (I think) enough information to solve for k, if we do not know the time since the death.  We would have to know the temperature of the body at another known non-zero time.  For example, if they measured the temperature again ten minutes later, or something like that.  Perhaps they want an assumption based on the air conditioner being one for "about a week," but I am not sure what such an assumption would be.

 

If there is any more information about this, please let me know.  I will also think about it more, and perhaps I (or another tutor) can see something else in it later.

 

[NOTE:  It also occurs to me that, perhaps, there is a standard cooling constant (k) value for a known quantity like the average human body given somewhere in whatever class materials you have, or online]