Rumford had noted the very great evolution of heat when gun-metal was rubbed by a blunt borer, and had come to the reasonable conclusion that what was evolved in apparently unlimited quantity by the abrasion or cutting down of a negligible quantity of materials could not be a material substance. He had also made a rough estimate of the relation between the work spent in driving the borer by horse-power and the heat generated. Joule's method of determining the work-equivalent of heat was a refinement of Rumford's, but differed in the all-important respect that accurate means were employed for measuring the expenditure of work and the gain of heat. He stirred a liquid, such as water or mercury, in a kind of churn driven by a falling weight. The range of descent of the weight enabled the work consumed to be exactly estimated, and a sensitive thermometer in the liquid measured the rise of temperature; thus the heat produced was accurately determined. The rise of temperature was very slight, and the change of state of the liquid, and therefore any possible change in its internal energy, was infinitesimal. The experiments were carried out with great care, and included very exact measurements of the various corrections—for example, the amount of work spent at pulleys and pivots without affecting the liquid, and the loss of heat by radiation. The experiments proved that the work spent on the liquid and the heat produced were in direct proportion to one another. He found, finally, in 1850, that 772 foot-pounds of work at Manchester generated one British thermal unit, that is, as much heat as sufficed to raise a pound of water from 60° F. to 61° F. An approximation to this conclusion was contained in the paper which he communicated to the British Association at Oxford in 1847.
The results of a later determination made with an improved apparatus, and completed in 1878, gave a very slightly higher result. When corrected to the corresponding Fahrenheit degree on the air thermometer it must be increased by somewhat less than one per cent. The exact relation has been the subject during the last twenty years of much refined experimental work, but without any serious alteration of the number indicated above.
It is probable that in consequence of the conference which he had with Joule at Oxford Thomson had his thoughts turned for some time almost exclusively to the dynamical theory of heat engines. He worked at the subject almost continuously for a long time, sending paper after paper to the Edinburgh Royal Society. As we have seen, he had given Joule a description of Carnot's essay on the Motive Power of Heat and the conclusions, or some of them, therein contained. Joule's result, and the thermodynamic law which it established, gave the key to the correction of Carnot's theory necessary to bring it into line with a complete doctrine of energy, which should take account of work done against frictional resistances.
Mayer of Heilbronn had endeavoured to determine the dynamical equivalent of heat in 1842, by calculating from the knowledge available at the time of the two specific heats of air—the specific heat at constant pressure and the specific heat at constant volume—the heat value of the work spent in compressing air from a given volume to a smaller one. The principle of this determination is easily understood, but it involves an assumption that is not always clearly perceived. Let the air be imagined confined in a cylinder closed by a frictionless piston, which is kept from moving out under the air pressure by force applied from without. Let heat be given to the air so as to raise its temperature, while the piston moves out so as to keep the pressure constant. If the pressure be p and the increase of volume be dv, the work done against the external force is pdv. Let the rise of temperature be one degree of the Centigrade scale, and the mass of air be one gramme, the heat given to the gas is the specific heat Cp of the gas at constant pressure, for there is only slight variation of specific heat with temperature. But if the piston had been fixed the heat required for the same rise of temperature would have been Cv, the specific heat at constant volume. Now Mayer assumed that the excess of the specific heat Cp above Cv was the thermal equivalent of the work pdv done in the former case. Thus he obtained the equation J (Cp − Cv) = pdv, where J denotes the dynamical equivalent of heat and Cp, Cv are taken in thermal units. But if a be the coefficient of expansion of the air under constant pressure (that is 1 ⁄ 273), and v0 be the volume of the air at 0° C., we have dv = av0, so that J (Cp − Cv) = apv0. Now if p be one atmosphere, say 1.014 × 106 dynes per square centimetre, and the temperature be the freezing point of water, the volume of a gramme of air is 1 ⁄ .001293 in cubic centimetres. Hence
from which, if Cp − Cv is known, the value of J can be found.
In Mayer's time the difference of the specific heats of air was imperfectly known, and so J could not be found with anything like accuracy. From Regnault's experiments on the specific heat at constant pressure, and from the known ratio of the specific heats as deduced from the velocity of sound combined with Regnault's result, the value of Cp − Cv may be taken as .0686. Thus J works out to 42.2 × 106, in ergs per calorie, which is not far from the true value. Mayer obtained a result equivalent to 36.5 × 106 ergs per calorie.
The assumption on which this calculation is founded is that there is no alteration of the internal energy of the gas in consequence of expansion. If the air when raised in temperature, and at the same time increased in volume, contained less internal energy than when simply heated without alteration of volume, the energy evolved would be available to aid the performance of the work done against external forces, and less heat would be required, or, in the contrary case, more heat would be required, than would be necessary if the internal energy remained unaltered. Thus putting dW for pdv, the work done, e for the internal energy before expansion, and dH for the heat given to the gas, we have obviously the equation
JdH = de + dW
where de is the change of internal energy due to the alteration of volume, together with the alteration of temperature. If now the temperature be altered without expansion, no external work is done and dW for that case is zero. Let ∂e and ∂H be the energy change and the heat supplied, then in this case