# Newton's Law of Cooling

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You are a forensic doctor called to a murder scene. When the victim was discovered, the body temperature was measured and found to be 20°C. You arrive one hour later and find the body temperature at that time to be 18°C. Assuming that the ambient temperature remained constant in that intervening hour, give the police an estimate of the time of death. (Take 37°C as normal body temperature.)

• Newton's Law of Cooling -

Newton's law of convective cooling can be written in the form

dT/dt = -C(T - Ta)
where C is a constant that involves the heat capacity and cooling coefficient, and Ta is ambient temperature.

The solution to the differential equation is

• Newton's Law of Cooling -

Newton's law can be written in the form

dT/dt = -C(T - Ta)
where C i

• Newton's Law of Cooling -

• Newton's Law of Cooling -

• Newton's Law of Cooling -

The solution is
(T-Ta) = (To-Ta)e^-Ct
Where To = 37 C and Ta = ambient temperature
Your two data points give you two equations in three unknowns: t (when reported), Ta and C. The time of second measurement is t + 1 hr. More information is needed to make an estimate. You would need the value of ambient temperature or the Newton heat loss coefficient C

• Newton's Law of Cooling -

Oh, I see. So to make an estimate you'd have to make an assumption of ambient temperature, right? Assuming constant ambient temperature of 10°C, the time of death could be estimated to be at around 4.45 (i.e 4 to 4 1/2) hours before discovery. Is that correct?

• Newton's Law of Cooling -

I have not done the numbers but that sounds reasonable. Did they tell you to assume ambient was 10 C?