Newton's Law of Cooling

From experimental observations it is known that (up to a ``satisfactory'' approximation) the surface temperature of an object changes at a rate proportional to its relative temperature. That is, the difference between its temperature and the temperature of the surrounding environment. This is what is known as Newton's law of cooling. Thus, if tex2html_wrap_inline27 is the temperature of the object at time t, then we have

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where S is the temperature of the surrounding environment. A qualitative study of this phenomena will show that k >0. This is a first order linear differential equation. The solution, under the initial condition tex2html_wrap_inline37 , is given by

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Hence,

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which implies

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This equation makes it possible to find k if the interval of time tex2html_wrap_inline47 is known and vice-versa.

Example: Time of Death Suppose that a corpse was discovered in a motel room at midnight and its temperature was tex2html_wrap_inline49 . The temperature of the room is kept constant at tex2html_wrap_inline51 . Two hours later the temperature of the corpse dropped to tex2html_wrap_inline53 . Find the time of death.

Solution: First we use the observed temperatures of the corpse to find the constant k. We have

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In order to find the time of death we need to remember that the temperature of a corpse at time of death is tex2html_wrap_inline59 (assuming the dead person was not sick!). Then we have

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which means that the death happened around 7:26 P.M.

One of our interested readers, E.P. Esterle, wrote a program that helps find the time of death based on the above notes. Click HERE to download it. Have fun with it.

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Author: Mohamed Amine Khamsi

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