Stefan's Fit of the Dulong and Petit Data for the Velocity of Cooling in a Vacuum

  In March 1879 Stephan pointed out that the velocity of cooling data obtained by Dulong and Petit could be calculated with an radiation function involving the 4th power of the absolute temperatures. How might he have discovered this? We start as he did with the velocities of cooling for an ambient temperature of 0 °C. In a manner similar to that previously used we can assume the difference of the two rates is Δv=A+Be^λT where T is the absolute temperature of thermometer in the vacuum given in Kelvins. If we assume a value for λ we can use linear least squares to find the remaining coefficients and search for a value of λ which will minimize the error.

Next we subtract A from Δv to get an estimate of the rate for emission of heat radiation. The data appears to be fairly linear in a log-log plot so we look for another set of coefficients for a second fit. A little math allows us to convert the constant term into a factor.

This fit is fairly good but noticing the value of B we are tempted to replace it with an integer, n, to see what we get for a third fit. Assuming n=4 the only unknown is the coefficient a and a search can be used to obtain the best fit. The search works better than averaging ratios of Δv/(T⁴-T₀⁴) to find a.

One gets calculated values and errors very close to those of Stefan.

Supplemental (Apr 1): The averaging of the ratios procedure shows that using some other integer n to compute the coefficient a verifies that n=4 gives the least error.

Edit (Apr 1): Caught an error in the first fit so had to redo it. The fit assumed the formula shown above rather than the original formula shown, Δv=Ae^λT-B. This affected the second fit also since the coefficients were confused and B was added to Δv while A should have been subtracted. Using a log-log fit can can bias the errors somewhat so it is best to work with the original data. The second fit above suggests a 3rd power law and one finds mention of it in some publications. The mistake didn't affect the last fit or the comparison of power laws.

Supplemental (Apr 1): Curvature in a semi-log plot might have suggested the quadratic emission power law mentioned by Dulong and Petit. The slight curvature in the plot of the data in the second suggests the presence of a systematic error in the log of the velocity and even greater fit errors in the velocities themselves.

Absolute Zero of Temperature

  To arrive at the Dulong and Petit empirical formula for the velocity of cooling one does not need to know the absolute temperatures, just changes in temperature relative to the zero of the temperature scale one is using. They did however assume that the rates of emission and absorption of heat were the same function of the absolute temperature. Their use appears to have been influenced by Dalton's discussion of the existence of an absolute temperature in 1808. Dalton states the difference from absolute zero may be approximately 1500 °F below the zero of the Fahrenheit scale but one can interpret this as the heat content of a body expressed in terms of the standard unit of heat based on heat capacity which is not constant for all materials. In the Dulong and Petit empirical formula with the equivalence of emission and absorption the absolute temperature of the thermometer scale used is a common factor that can be removed from the exponential factor and included in the common coefficient.

In 1848 Thomson (Kelvin) proposed an absolute temperature scale based on Carnot's work on steam engines. Joule and Thomson worked on this and a few years later arrived at a value of -273.7 °C for the temperature of absolute zero.

Supplemental (Apr 1): In 1854 Thomson and Joule used the coefficient of expansion of air for an estimate of the absolute zero of temperature equal to 272.85 °C and also indicated where the 273.7 °C value came from. A value for the thermal coefficient of expansion for air, α, equal to 0.0036623/°C had been published prior to this by Regnault in 1842. The formula for the relative expansion is 1 + αΔT so for a temperature change of 100 °C a change of 100α is observed.