A Statement of the Law

• T Decays Asymptotically to S as t Increases
  
  Isaac Newton found that the temperature of a hot object decreases at a rate proportional to the difference between itself and the surrounding temperature. Obversely, an object colder than its surroundings warms at a rate proportional to the same difference.
  
  The formula governing the law is "T/"t = c (T - S), where T is the object's temperature, S is the surrounding temperature, t is time and c is a constant of proportionality. "t is small, since the law is for an instantaneous rate of change.
  
  The solution to this differential equation is exponential in form, and is written in terms of the base e (= 2.71828...). Because c is negative, T goes to S as time, t, gets large.
  
  

Two Experiments

• One experiment to demonstrate this law is to heat a thermometer some 20 or 30 degrees above the ambient temperature. Then when the heat source is taken away, start a stopwatch to take readings every minute.
  
  A similar experiment is to warm some water in a pot, then record the temperature from a thermometer in the water at regular intervals as the water cools.
  
  A common variation is to ask the following question: If you want to cool your coffee as fast as possible, when do you add the milk--at the start or at the end of the cooling period? Students can use a larger container than a coffee cup, of course, and see what adding a fixed volume of cool fluid to a cooling mixture does to the rate of cooling. In one trial, the volume would be added at the start. In the second, the volume would be added at the end. Of course, between trials, the volumes and starting temperatures should all be equal. Again, the temperature is measured with a thermometer at regular intervals.
  
  

Graph

• Data points from the experiment can be graphed such that the natural log of the excess temperature (T-S) is graphed against time. The slope will equal the proportionality constant c.
  
  Students should probably also graph temperature T against time t, just to see the asymptotic (flattening) nature of the curve as time passes.
  
  This experiment is an opportunity to expose students to log paper. The excess temperature T-S can be graphed against time, t, and should come out a straight line.
  
  Graphing also affords an opportunity to expose students to algorithm properties, e.g., that ln exp [x] = x. The importance of taking the logarithm of T-S instead of just T can be pointed out as well, since taking the logarithm of S + (T(initial) - S) --- exp[ct] does not allow one to convert ct from an exponent into a coefficient. In other words, ln (A + exp[x]) isn't reduced with the same simplicity as ln exp [x].
  
  

Complications

• Note that there are three ways in which heat is lost in these experiments: radiation, convection and evaporation. Therefore, if the mix of these three happens to change as the temperature changes, the graph of the logarithm of the excess temperature could end up crooked or curved, instead of straight.
  
  For example, heat loss by evaporation would play a greater role at higher temperatures, when the water is near boiling. If the path of convection above a pot is blocked, c would vary. If the path is blocked during part of the experiment, the slope of the graph of ln(T-S) wouldn't be straight.
  
  Futhermore, it turns out that c is not truly a constant but increases with both S and T-S. The range of temperatures will probably not be wide enough to notice in a student lab project, however. See experimental results by Dulong and Petit on page 246 of Poynting's "A Text Book of Physics" for a graph of this variation.
  
  

Separating the Rate of Radiation and Convection

• Newton's original cooling experiment (1701) involved primarily heat loss by convection from hot iron, with some loss by conduction and radiation. An experiment to measure just the rate of radiation would require a vacuum surrounding the cooling object, to prevent heat loss by convection. The question is interesting because it gets into the properties of light (radiation), as well as the question of how to estimate the temperature of far away objects such as the sun. In astronomical cases, the radiation curve must be determined for S equal to the temperature of outer space, which Dulong and Petit tried to extrapolate to, by performing experiments at successively lower container temperatures (1817). Subsequent theoretical analysis led Stefan and Boltzmann to more exact formulations.