a)
An object at temperatures T, placed in surroundings of a different temperature, will either gain or lose heat to come into equilibrium with its surrounding environment. This process can be modeled with a simple differential equation
dT/dt = -k[T-A] ; T(0)=T。
Here A is the ambient temperature, T。is the initial temperature of the object when introduced to the new environment, and k is a positive constant. Solve this differential equation for temperature as a function of time. Graph possible solution.
b)
Your friend, the rookie detective calls you in the middle of the night, saying something about a murder. He has forgotten, or never knew, the procedure to determine the time of death. He did remember to take the body temperature when he arrived at the crime scene and one hour later he took the temperature again. His first reading was 92° at 2:00 am., and the second was 88°. The body was found in a large basement with constant temperature 65°. Help your friend determine the time of death. Assume the normal living temperature is 98°.
Answers & Comments
Verified answer
a)
For dT/dt = -k[T-A] I get the general solution:
T=e^(-k*t)*(e^(k*t)*A+c)
With the initial condition T(0)=T[0] I get
the particular solution:
T=e^(-k*t)*((e^(k*t)-1)*A+T[0])
b) If convection is fully applicable then
Newton's law of cooling could be used, which
you might check:
(Change of temperature)/time is proportional to
[(initial temperature)-(ambient temperature)].
Hope it helps.