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.

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

The Science of Heat at the Beginning of the 19th Century

  So far our understanding of heat hasn't advanced much beyond what was known in the 1st decade of the 19th Century. Dalton in 1801 was the first to note that the relative expansion of gases with temperature was approximately the same for all gases. A year later Gay-Lussac published experimental results for the expansion of air, hydrogen, oxygen and nitrogen. In 1807 Young published a lecture on heat which touched on the measurement of the expansion of solids with changes in temperature and noted that the mercury thermometer may be off by 2 or 3 degrees Fehrenheit mid scale. Since heat was produced by friction he concluded that it was not a substance but instead a quality being associated with both the motion of the constituent particles of a body and the motion of a medium in the case of radiation. At the end of the same year the scientific community got a preliminary look at Fourier's Theory of Heat.

To improve one's understanding of heat one needs to take a closer look at cooling and heating and the mechanisms of the conduction and radiation processes. The study of heat was focused on this during much of the remainder of the 19th Century.

Bibliography

Dalton - On the Expansion of Elastic Fluids by Heat (1801)

Gay-Lussac - Recherches sur la dilatation des gaz et des vapeurs (1802)

  Art. IV. Expériences et résultats.

Gay-Lussac - Researches upon the Rate of Expansion of Gases and Vapors

  Part IV. Experiments and Results

Young - On the measures and the nature of heat (1807)

Fourier - Mémoire sur la Propagation de la Chaleur dans les Corps Solides (1807)

A Solution of Fourier's Equation for a Sphere and Comparison with the Experimental Cooling Data

  We can compare out cooling data with the way a sphere of water of comparable size will cool according to Fourier's Equation. To simplify our calculations we need to convert our data to the cgs system of units so we can use the thermodynamic constants for water.

From the fit of the cooling data we can determine some of the constants that we need to solve Fourier's Eqn.

Since we are doing a rough we can arbitrarily set hR=2 which makes the Λ in Fourier's boundary condition equal to -1. The diameter of the sphere is taken to be 10 cm so R=5 cm. From the values of ε which satisfy the boundary condition we get the values of μ for the terms of the series and can expand the function ΔT(r,0)=ΔT₀, the initial temperature difference from T_a, to obtain the coefficients of the terms. Because the ε_k were not regularly spaced the coefficients were obtains by evaluating the table of functions sin(μr)/μr for each term and using a least squares method to minimize the deviation of the series from f(r,0)=1.

We can calculate the solutions for T(r,Δt) with steps of Δt=20 min to get the following plot. In the equation we have to use Δt given in seconds because we're working in cgs units. The solution at Δt=0 sec is a little wiggly because of the Gibbs phenomenon which prevents getting the series for ΔT(r,0)=ΔT₀ to match exactly at r=5 cm.

The best match of our cooling data with the Fourier solution is with the thermometer readings taken at a position that is about equal to 4.2 cm from the center of the sphere and the assumption of an ambient temperature, T_a, equal to about 23 °C.

In our cooling experiment the measuring cup holding the water would have offered some resistance to the heat trying to escape. The measuring cup was microwave safe so its composition was probably similar to that of Pyrex. In the sphere of water some of the water would be needed to model the resistance to the flow of heat by the measuring cup.

Edit (Mar 27): Corrected the misstatement T(r,0)=T₀ with ΔT(r,0)=ΔT₀ which was used in the calculation of the set of cooling curves.

Cooling with the Assumption of an Internal Resistance to Heat Flow in the Body

  I tried some electrical analogies for the cooling data posted a few days ago and a parallel process involving two cooling mechanisms resulted in the same equation for the temperature as a function of time but with a combined effective rate constant. A series circuit process could be considered analogous to a battery having an internal resistance with a voltage drop across it. This would be the Thevenin equivalent for a heated body. The differential equation for the temperature at the surface, T_s, is show below along with its solution where T_s=T₀ at t=0 and T_a is the ambient temperature.

The fit to the data collected confirms that this is a plausible explanation for the cooling mechanism.

The ambient temperature appears to be too high but the data doesn't track the cooling down all the way down to the ambient temperature. There was a warm coffee maker nearby which may have raised the effective ambient temperature. More data and more care is setting up the experiment are likely to produce better results.

The Solution of Fourier's Equation for a Sphere

  Fourier's presentation of the solution of his equation for the changes in the temperature a spherical body is difficult to follow and could benefit from a little cleanup. One can arrive at the boundary equation Fourier used by combining Newton's law of cooling with his law relating the change in heat content with the radial change in temperature.

Fourier reduces his temperature equation to a simpler one involving a factor and an auxiliary function. The factor can be found using a procedure similar to that for finding an integrating factor.

One next looks for the form of the elementary functions which are solutions of the auxiliary function U which one can do by arbitrarily setting U=e^φ. Substituting into the differential equation for U gives one for φ. The trick is to find something as simple as possible so we set φ equal to a linear function of r and t. The differential equation for φ then gives a relation connecting two of the coefficients and we can write one in terms of the other. Requiring that U → 0 as t → ∞ tells us that the coefficient of r is complex. So we get a class of functions that fit a simple form.

This gives us the general form of the elementary functions of the differential equation for T and like Fourier we can replace the complex exponential function with cosine and sine functions. Requiring that the functions to be bounded forces us to eliminate the cosine term. Next we consider the effect of the boundary equation on the form for T. We can eliminate the constant term involving the ambient temperature T_a which Fourier ignores by noting that T can be expressed as the sum of a specific solution involving just T_a and a general solution T_g which satisfies the equation dT_g/dr+hT_g=0. Finally, substituting the general form of T for T_g we get Fourier's constraint on the permissible functions for T, μR/tan(μR)=1-hR or, with ε=μR and λ=1-hR, ε/tanε= λ.

Note we can write T as a sum of the permissible functions ψ, that is, T_g=Σ_kT_kψ_k. The values for the coefficients T_k are determined by the expansion of the initial distribution of T=T(r,0) in terms of the permitted functions. One can use the constraint condition to find a table of permitted values for ε=μR as a function of λ.

Supplemental (Mar 20): I noticed a minor inconsistency in the argument above. It was assumed that U → 0 as t → ∞ but that apparently like Fourier ignores the ambient temperature T_a. There are to ways of correcting this, first one could assume that T is measured relative to T_a or alternatively one could assume the condition only applies to the general elementary functions for T_g.