Stefan and the Diathermometer
Once in a while, it is useful to revisit the history of inventions and discoveries that we now take for granted. Such revisits to the masters and their works provide a perspective on how to think about the unknown from whatever was known - right or wrong. Instances where a correct discovery results from wrong beliefs like the Carnot's ideal heat engine theory from Caloric theory of heat that modeled heat as a fluid substance. Or data from experiments providing conclusive shape to a model before it was analytically predicted, like the T power four law of radiation - as we will see later in a subsequent note.
We now revisit the ingenuity of an experiment performed by Josef Stefan, that measured something that was thought impossible. Much of what I write below is available in . I also claim no originality of the equations discussed.
Thermal conductivity of a substance gives a measure of energy that is conducted as heat across a length maintained at a temperature difference. Fourier Law, which relates heat flux across a locally stationary substance to be in direct proportion with the associated temperature gradient, is the constitutive relation that defines thermal conductivity (the essay in that link explains it in detail).
It was a time whether gases would conduct heat remained in debate. Joseph Priestly in 1780 thought he had measured "the power to conduct heat" for gases using experiments, but it was not to be. He had rather measured the specific heat of gases and not the thermal conductivity. Specific heat was a concept not well understood until the experiments of Joseph Black in 1840. In 1786 Count Rumsford attempted to measure the power to conduct heat by "artificial airs" or gases, but stumbled on to the discovery of a new mode of heat transfer, convection. So he thought gases cannot conduct heat. According to  (which borrows the insight, I presume, from ) , this thought remained unchallenged for several decades, largely due to the reputation of Rumsford. Until 1861, when Magnus showed conclusively that gases do conduct heat, by electrically heating platinum wires surrounded by different gases.
In 1860 Maxwell published his dynamical theory of gases. He calculated a theoretical value of the thermal conductivity of a gas and showed its dependence on the temperature and pressure. Further, he made an assertion (as mentioned in ) that it would be impossible to measure by direct experiments the thermal conductivity of a gas, as the radiation heat transfer from the gas to the surrounding would always remain orders higher than the conduction within the gas even when circulation (convection) was prevented.
Two years later, Rudolph Clausius thought Maxwell had treated thermal conductivity incompletely. He went on to show that the thermal conductivity was dependent on temperature but independent of pressure for ideal gases. He even measured the thermal conductivity of air (by treating it as an ideal gas) to be k = 0.0115 W/mK. Maxwell went on to revise his work and obtained for air a value of k = 0.0218 W/mK.
And this is where Josef Stefan entered. To defy Maxwell's words .
Wikipedia has a brief life history of Josef Stefan. We quote an inspiring paragraph from  on his academic stature before proceeding to his experiment.
[...] In addition to his scientific and administrative talents, Stefan was a warm and beloved teacher. He gave very energetic, animated lectures and was said to be exhausted upon their completion. His students not only felt comfortable around him but were motivated to do high-level scientific research. One of his students later remarked on the collegial atmosphere that Stefan maintained at the cramped, underfunded Institute on Erdbergstrasse, far from the central campus buildings of the University, "Nothing diminishes the excellence of his character, the magic [Stefan] worked on the young academics. That magic could only be experienced personally; Erdberg stayed with me my whole life as a symbol of serious, inspired experimental activity"
That student who felt that way was Ludwig Boltzmann, Stefan's first Ph. D. student.
Josef Stefan was aware of the works of Maxwell, Clausius and Magnus. He was also determined to measure thermal conductivity of gases through direct experiments. He was aware of the detrimental effect convection of the gas can have on such a measurement (across a set temperature difference, convection can force the gas to physically move due to buoyancy carrying enthalpy along with it, increasing the heat flux enormously). In one of his earlier attempts to construct a device, he heated air from the top and cooled it from the bottom to negate the convection effect - a thermally stratified air column. But he was unable to control the heat loss to the surrounding.
After such imperfect subsequent attempts (one more is given in ), he constructed the diathermometer. In order to prevent the convection effects on stationary air column, Stefan first struck on the idea of using transient measurements. He conceived a method where the heat conduction across a small gap filled with a gas can be equated with the enthalpy gained by the gas under transient conditions. In modern parlance this equality is First Law of Thermodynamics applied to a fixed volume of gas, with the heat transfer equated to local temperature gradient through Fourier Law (pdf) and the enthalpy gained by the gas calculated as a product of specific heat at constant volume and temperature raise. In equation form this unique energy balance can be written as
Here k is thermal conductivity of the gas (measured in W/mK), is the temperature difference across a gap of length, is the specific heat of the gas at constant volume.
The use of specific heat at constant volume is correct - when we look at the experiment performed - but perhaps fortuitous. The understanding that gases in principle have also and one must use it in an energy balance is a more recent understanding. Even some modern text books carry this error of using in the energy conservation statement while dealing with convection.
Rearranging and integrating Eq. (1) can be recast as
where '0' in the subscript denotes initial temperature difference across the gap filled by the gas. Stefan, at this stage, perhaps through the works of Clausius, realized that for an ideal gas at fixed volume, the relative change in temperature is equal to the relative change in pressure (via the equation of state). This leads Eq. (2) to be revised as
Based on these equations, Stefan constructed his diathermometer - part of the schematic is shown here.
The gas in question is sent through the valve opening marked I into the small gap between concentric cylinders ABCD and GHJK. The small gap ensures minimal convection (gravity acting downwards in the picture). The pressure in the GHJK chamber is measured by a manometer connected to the limb marked M. The apparatus, after reaching internal thermal equilibrium, is kept in a constant temperature bath. In a transient (time dependent) process, heat is conducted through the walls AB and DC (actually one cylindrical surface), through the gas in the gap and into the GHJK chamber. The measured pressure difference for each time instant provides the unknowns in Eq. (3), resulting in the direct measurement of thermal conductivity of the gas in the gap.
Stefan measured the thermal conductivity of air to be k = 0.0234 W/mK, which is 11% off todays accepted value of k = 0.0263 W/mK (at 300 K). It also compared well (about 7%) with Maxwell's earlier theoretical predictions. Stefan went on to measure the 'k' of several gases including hydrogen, nitrous oxide, methane, carbon monoxide and carbon dioxide.
The radiation effect that was thought to mire such an experiment was completely negated in Stefan's ingenuous diatermometer. Interestingly, he used it again to prove experimentally the T to the fourth power radiation law. In a subsequent note.
Narasimhan, A., (2013), "The Scientific Legacy of Josef Stefan," Chapter 11, pp. 200-220, in Jožef Stefan: His Scientific Legacy on the 175th Anniversary of His Birth, ed. John Crepeau, Bentham Press. [DOI: 10.2174/97816080547701130101 | Product Link]
 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
 A.C. Burr, Notes on the history of the thermal conductivity of gases, Isis 21 (1) (1934), pp. 169-186. Full Text via CrossRef
 Image of Stefan's Monument from http://www.carantha.net/science_and_literature.htm
 W.L. Reiter, The physical tourist Vienna: a random walk in science, Physics in Perspective 3 (2001), pp. 462-489. Full Text via CrossRef | View Record in Scopus | Cited By in Scopus (3) - An interesting read on the contributions of Vienna to Science