The Dulong and Petit Empirical Law of Cooling

  In 1817 Dulong and Petit published some research on the velocity of cooling in a vacuum. They used a corrected mercury thermometer for the temperatures and measured the velocity of cooling in an evacuated chamber for a number of values of the excess temperature θ and environmental temperature θ₀ within the range of the mercury thermometer. Since the rate of cooling is less for shiny surfaces the surface of the thermometer and the inner wall of the enclosing vacuum chamber were blackened to optimize the exchange of heat.

They noted that the ratios of the changes in the velocity of cooling was the same for the same change in the background temperature which led them to assume the rate of cooling was an exponential function of the temperature differences.

So one might expect v=w₀+we^λθ for this function where the constant λ can be found by averaging the ratios. The value of λ can be used to reduce the curves for the different background temperatures to a common curve. Here θ is the excess temperature.

If we assume a value for w₀ and take the ratio of its difference from the observed velocity of cooling with that of some reference temperature θ₀ we can estimate the values of the two remaining unknown quantities λ and w and average the results to get a fit for the common curve.

Multiplying the fit formula by e^λθ₀ we can obtain a more general formula for the velocity of cooling with different background temperatures. For a given value of λ we can compute the exponential factors and fit the set of curve using linear least squares to determine the constants w and w₀. Note that their values are approximately equal which is what one would expect if the rate of emission was equal to the rate of absorption of the radiation.

Bibliography

Dulong et Petit - Recherches Sur la Mesure des Températures et sur le Lois de la communication de la chaleur (1817)

1st partition  2nd partition  3rd partition

velocity of cooling data

Dulong and Petit - Researches on the Measure of Temperatures, and on the Laws of the Communication of Heat (1819)

1st section  2nd section  3rd section  4th section

velocity of cooling data