### Cooling Bodies as a Natural Clock

#### By Mike Janssen

One example of a natural clock is what physicists and mathematicians know as Newton's Law of Cooling, which states that a "body at uniform temperature cools at a rate proportional to the difference between its temperature and that of the surroundings" (Ledder 7). In layman's terms, this means that an equation can be derived to model the change in temperature of a body with respect to its surroundings and the time it has been in those surroundings. We begin with a construction of Newton's Law of Cooling.

Newton's Law of Cooling is constructed by first assuming that the rate of cooling is proportional to the temperature difference between the body in question and its surroundings, as stated previously. One can let T(t) be the temperature of the body at time t and let S be the temperature of its surroundings (one can assume, in many cases, that this is a constant). Using basic differential calculus, −dT/dt is the derivative (rate of change) of the temperature. Its sign indicates that the temperature is going down (cooling). Putting this together, one finds the following ordinary differential equation, which is known as Newton's Law of Cooling: dT/dt = −k(T − S) takes a positive value.

Applying the basic techniques of ordinary differential equations, one finds the solution, since e^{c} = T_{0} − S for Newton's Law of Cooling:

(1) |

If one knows the initial temperature of the body T_{0}, the proportionality constant k (which can be discovered by taking two separate temperature readings), and the surrounding temperature S, one may easily discover approximately what time the body was at any previous temperature. Since we are interested in the time t, we may solve for t to yield the following equation:

(2) |

One potential application of this technique deals with discovering the time of death of a human being. For instance, if one finds a body at a certain time and takes a temperature reading and then waits a few minutes and takes another one (noting the time between readings), one may discover the necessary constants, and use them to trace back to an approximate time of death (assuming that the body was at an initial temperature of about 98.6° F).

As an example, suppose a body is found at a temperature of 85° F in an environment at 73 ° F. An hour later, the body is measured to be at 80 ° F. We must first find the value of the proportionality constant k, which we can then use to extrapolate backward to find the time of death. Substituting our values into equation (1) above, we have:

(3) |

That yields a constant value of k = 0.5389965... hour ^{−1}. Now, substituting that value, together with an initial temperature of 98.6° F and a final temperature of 85° F into equation (2),

(4) |

This result (1.4 hours since the time of death) is reasonable given our data concerning the time it took the body to reach 80° F.

Continuing with the time-of-death example, at time t = 0, the body is at a temperature 98.6° F. Any two objects in contact with one another will eventually reach the same temperature, so the ultimate condition (over a time interval I) is the temperature of the body's surroundings, T(I) = S (Schroeder 1). This natural clock is valid over the entire interval I. However, once the body has reached the temperature of its surroundings, the clock is invalid. If one discovers a body at a temperature of 70° F in an environment at 70° F, one cannot determine for certain how long the body has been there (though one can determine a minimum time, i.e. it has been there at least that long), as its temperature will no longer continue to fall once it has reached thermal equilibrium with its surroundings. There is one caveat, however (especially for the time-of-death problem): the uniform temperature assumption. Warm-blooded creatures, like human beings, generate heat internally to maintain a body temperature higher than that of their environment. Thus, with an internal temperature at 98.6° F and an external temperature of 70° F, the skin temperature will be somewhere in between the two, as it is at the skin that the heat transfer to a cooler environment takes place. Another assumption is that the temperature of the surroundings is constant. In this way, Newton's Law of Cooling is not a perfect system, but it can be used to give a first approximation of the time of death of an individual. It is worth noting that this method can be used on a wide variety of systems, from fast-cooling systems (such as a cup of coffee) or systems that cool much more slowly (such as a body of lava or magma, which may take years to cool). In fact, Lord Kelvin attempted to use this method to date the age of the earth. However, because many of his assumptions were incorrect, he found an age of 24,000,000 years, which falls far short of the currently-accepted value.

##### Sources:

- Ledder, Glenn. 2005. Differential Equations: A Modeling Approach, McGraw-Hill, 704 p.

- Schroader, Daniel V. 2000. An Introduction to Thermal Physics, Addison-Wesley-Longman, 442 p.