# Stefan and the T to the fourth power law

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The working of Stefan’s diathermometer to measure the thermal conductivity was explained earlier. Here we recount how the diathermometer helped also in the prediction of the T to the fourth power law of electromagnetic radiation purely by experimental means. It was an instance of scientific advancement, where the experimental outcome preceded the theoretical support. We use again content from the article [1] that recounts Stefan’s achievements with a modern perspective. For some direct translation we consult [2], a nice revisit of Stefan’s original paper. I claim no originality of the following content barring the exposition and opinions.

For the sake of starting somewhere let us begin in a time in the past when it was agreed that radiation heat transfer was not proportional to the temperature difference, particularly at high temperatures. But it was not clear what the proportionality is.

In 1817 Dulong and Petit – known for their chemical law for specific heat (example) – published experimental data of temperature evolution in a mercury container kept inside another chamber maintained at θ0. The gap between the mercury container and the outside chamber was evacuated to low pressure ‘p’ (lowest achieved was 2 torr), while the preheated mercury was cooled from a high temperature ? (maximum reached was 300deg C). For given θ,θ0, the time ‘w’ for cooling varied with pressure p. By extrapolating their data to zero pressures, they concluded that this effect was solely due to radiation heat transfer (as convection and conduction were removed). Based on their data and associated cooling rates, they proposed an empirical formula for radiation emittance R(?) as

where a = 1.0077 was a constant and μ depended on the density, form and surface area of the body that radiates. Since the outer chamber at θ0 also radiates, the net radiation emittance or exchange can be written – if we substitute ‘modern conventional’ nomenclature – as

Enter Stefan. He praised Dulong and Petit’s experiment for its simplicity but suspected the data. Because his previous experiment to measure thermal conductivity of air using his ingenious diathermometer has given him the insight that thermal conductivity doesn’t depend on pressure. This meant the extrapolated data of Dulong and Petit for air, contrary to what they believed, retains the effect of conduction heat transfer. Convection effects could be eliminated by lowering the air pressure in the gap, but not the conduction effect.

By adding a conduction term to the original energy balance equation (that only had radiation loss), Stefan was able to estimate the conduction contribution in Dulong and Petits’ data ranged between 10 and 50 percent. The 10 percent was with data obtained for un-silvered inner mercury chamber and 50 percent was for silvered inner mercury chamber. The silvering reduced the emissivity of the medium, thus revealing the conduction contribution more in the data. But this reasoning was not available to Stefan as emissivity, during his times, is yet to be delineated as a property and measured. Nevertheless, his prediction on conduction effect is objective and correct. Further, in a flight of insight (I don’t know what else to call it), by assuming both E and R to depend on T, Stefan was able to observe the data could be curve-fit using a relation the R = AT4. So Eq. (2) can be expressed as

where A depends on the material of the medium used (like silvered or un-silvered) and its size and shape. The notation T in Eq. (3) is to signify the temperature is in absolute scale (in Kelvins), whose significance Stefan was able to appreciate.

By dividing the temperature difference in Eq. (3) with a suitable constant X, Stefan was able to arrive at a cooling rate that agreed well with the experimental data and also with the cooling rate provided by Dulong and Petit (by using Eq. (2)). A sample is given in the table below, where ‘w’ represents cooling rates and subscripts DP and S are self-explanatory. Further details on the associated discussion by Stefan in his original paper can be read in reference [2], an excellent expository of this topic.

Although his formula gave a slightly better agreement with the experiment data, Stefan was not happy with his proposal, as Eq. (3) is yet to be checked for data at high temperatures. Only then its generality could be established.

Stefan then located in a nondescript book by Wullner, the experiments of Tyndall (1865) on the differences in radiation from a heated platinum wire to its surrounding. The platinum wire was heated to different temperatures between about 525 deg C (red heat) and 1200 deg C (white heat). Stefan found that the intensity of radiation increased from 10.4 to 122 correspondingly, the ratio of which is around 12. The T power four law gave a ratio of 11.6, lending more credence to its generality. Stefan went on to compare the predictions of his model with more experimental data from Draper (1878) and Ericsson (1872) with success. In the Ericsson data, he again used his knowledge of conduction effects and subtracted them from the data before finding his T4 agreed with it better than the Dulong and Petit model.

Stefan also evaluated the temperature of the Sun using his T4 model along with the experimental data of Pouillet and Soret. Pouillet had earlier used Dulong and Petit’s model (Eq. (2) above and not the Dulong and Petit’s Law as the Wikipedia page claims) with his data to predict 1734 < Tsun < 2034K while Soret, measuring the radiant energy of the Sun, estimated 2446 < Tsun < 2546K. Stefan with his T4 model obtained 5859 < Tsun < 10420K for Pouillets’ data and 5580 < Tsun < 5838K. These are the first good estimates for the temperature of the Sun, the currently accepted value of which is Tsun = 5770K.

Stefan’s T4 model was not immediately accepted. Thankfully, his first Ph. D. student, Boltzmann was able to produce conclusive theoretical corroboration by deriving the T4 model using radiation pressure of light. This derivation can be read from the Wikipedia. Nevertheless, Stefan’s original paper was not viewed kindly by some even in the twentieth century. Here is an example cited in [2], a portion of commentary by Worthing and Halliday (1948) in a footnote in p.438 to their section on Boltzmanns fourth-power law (reproduced from [2]):

When, however, one considers the basis for Stefans deduction, it hardly seems fair to link his name with the law. The Irish physicist, John Tyndall (1820-1893) had reported the ratio of the radiation output of a platinum wire at 1200C to that at 525C as 11.7. Stefan noticed that the ratio of 1473 K to 798 K raised to the fourth power is 11.6, and stated the law empirically. There were at least two errors, however. Later work has shown that the ratio of the two radiances of platinum at these temperatures is more nearly 18.6 than 11.6, and that the radiation from platinum is far from black-body radiation and should not be assumed to follow the laws of blackbody radiation. Actually, Stefan applied his empirical law with some success to other data, but the same errors were always present. Stefan was an able physicist whose fame should rest on other accomplishments.

Emissivity (e) as a property that could be measured separately and that it also depended on T was not known by Stefan. Only for a black body the total radiant emittance R is proportional to T4, as in R/T4 = s, the Stefan-Boltzmann constant which is s = 5.67 × 10-8W/m2K4, 11 percent higher than what Stefan had estimated. For non-black bodies the ratio would be R/T4 = es.

Firstly, much of Stefan’s paper is on the data of Dulong and Petit and others (the ‘some success to other data’ in the above quote); Tyndall’s data receives a few paragraphs in a paper of 38 pages long – as pointed out in [2]. Secondly, Stefan’s contribution certainly shows he understood the relevance of his T4 suggestion, use of absolute temperatures (which Dulong and Petit missed) and was able to compare his model with more than one set of data at different temperature ranges before proposing its validity.

Being harsh on previous works in light of new knowledge that the previous researchers don’t have access to, is intellectual under-cutting. If there were two competing models in the past and one superseded the other, which was later found to be correct, then at least we have a case for criticism. But we cannot say Darwin’s evolution theory is wrong because he was unaware of genes or the Heliocentric theory of Copernicus in 1543 (or a Rutherford’s atom model around the start of the twentieth century) is simplistic because we now know of something better. Such myopic ‘critique’ doesn’t appreciate the perspective of ideas.

References

Narasimhan, A., (2013), "The Scientific Legacy of Josef Stefan," Chapter 11, pp. 200-220, in Joﾅｾef Stefan: His Scientific Legacy on the 175th Anniversary of His Birth, ed. John Crepeau, Bentham Press. [DOI: 10.2174/97816080547701130101 | Product Link]

1. Crepeau, J. (2007). Josef Stefan: His life and legacy in the thermal sciences Experimental Thermal and Fluid Science, 31(7), 795-803 DOI: 10.1016/j.expthermflusci.2006.08.005
2. Dougal, R. (1979). The centenary of the fourth-power law Physics Education, 14 (4), 234-238 DOI: 10.1088/0031-9120/14/4/312
3. Stefan-Boltzmann Law from Wikipedia