This result combined with the equation A derived from the second law, gives an important expression for Carnot's function.
We shall not pursue this discussion further: so much is given to make clear how certain results as to the physical properties of substances were obtained, and to explain Thomson's scale of absolute thermodynamic temperature, which is by far the most important discovery within the range of theoretical thermodynamics.
There are several scales of temperature: in point of fact the scale of a mercury-in-glass thermometer is defined by the process of graduation, and therefore there are as many such scales as there are thermometers, since no two specimens of glass expand in precisely the same way. Equal differences of temperature do not correspond to equal increments of volume of the mercury: for the glass envelope expands also and in its own way. On the scale of a constant pressure gas thermometer changes of temperature are measured by variations of volume of the gas, while the pressure is maintained constant; on a constant volume gas thermometer changes of temperature are measured by alterations of pressure while the volume of the gas is kept constant. Each scale has its own independent definition, thus if the pressure of the gas be kept constant, and the volume at temperature 0° C. be v0 and that at any other temperature be v1 we define the numerical value t, this latter temperature, by the equation v = v0 (1 + Et), where E is 1 ⁄ 100 of the increase of volume sustained by the gas in being raised from 0° C. to 100° C. These are the temperatures of reference on an ordinary centigrade thermometer, that is, the temperature of melting ice and of saturated steam under standard atmospheric pressure, respectively. Thus t has the value (v ⁄ v0 − 1) ⁄ E, and is the temperature (on the constant pressure scale of the gas thermometer) corresponding to the volume v. Equal differences of temperature are such as correspond to equal increments of the volume at 0° C.
Similarly, on the constant volume scale we obtain a definition of temperature from the pressure p, by the equation t = (p ⁄ p0 − 1) ⁄ E', where p0 is the pressure at 0° C., and E' is 1 ⁄ 100 of the change of pressure produced by raising the temperature from 0° C. to 100° C.
For air E is approximately 1 ⁄ 273, and thus t = 273 (v − v0) ⁄ v0. If we take the case of v = 0, we get t = − 273. Now, although this temperature may be inaccessible, we may take it as zero, and the temperature denoted by t is, when reckoned from this zero, 273 + t. This zero is called the absolute zero on the constant pressure air thermometer. The value of E' is very nearly the same as that of E; and we get in a similar manner an absolute zero for the constant volume scale. If the gas obeyed Boyle's law exactly at all temperatures, E would not differ from E'.
It was suggested to Thomson by Joule, in a letter dated December 9, 1848, that the value of μ might be given by the equation μ = JE ⁄ (1 + Et). Here we take heat in dynamical units, and therefore the factor J is not required. With these units Joule's suggestion is that μ = E ⁄ (1 + Et), or with E = 1 ⁄ 273 μ = 1 ⁄ (273 + t), that is, μ = 1 ⁄ T where T is the temperature reckoned in centigrade degrees from the absolute zero of the constant pressure air thermometer.
The possibility of adopting this value of μ was shown by Thomson to depend on whether or not the heat absorbed by a given mass of gas in expanding without alteration of temperature is the equivalent of the work done by the expanding gas against external pressure. The heat H absorbed by the air in expanding from volume V to another volume V' at constant temperature is the integral of Mdv taken from the former volume to the latter. But by the value of M given on p. [121], if W be the integral of pdv, that is the work done by the air in the expansion, ∂W ⁄ ∂t = μH. The equation fulfilled by the gas at constant pressure (the defining equation for t), v = v0 (1 + Et), gives for the integral of pdv, that is W, the equation W = pv0 (1 + Et) log (V' ⁄ V), so that ∂W ⁄ ∂t = EW ⁄ (1 + Et). Thus μH = EW ⁄ (1 + Et).
Hence it follows that if μ = E ⁄ (1 + Et), the value of H will be simply W. Thus Joule's suggested value of μ is only admissible if the work done by the gas in expanding from a given volume to any other is the equivalent of the heat absorbed; or, which is the same thing, if the external work done in compressing the gas from one volume to another is the equivalent of the heat developed.
This result naturally suggests the formation of a new scale of thermometry by the adoption of the defining relation T = 1 ⁄ μ, where T denotes temperature. A scale of temperature thus defined is proposed in the paper by Joule and Thomson, "On the Thermal Effects of Fluids in Motion," Part II, which was published in the Philosophical Transactions for June 1854, and is what is now universally known as Thomson's scale of absolute thermodynamic temperature. It can, of course, be made to give 100 as the numerical value of the temperature difference between 0° C. and 100° C. by properly fixing the unit of T. This scale was the natural successor, in the dynamical theory, of one which Thomson had suggested in 1848, and which was founded, according to Carnot's idea, on the condition that a unit of heat should do the same amount of work in descending through each degree. This, as he pointed out, might justly be called an absolute scale, since it would be independent of the physical properties of any substance. In the same sense the scale defined by T = 1 ⁄ μ is truly an absolute scale.
The new scale gives a simple expression for the efficiency of a perfect engine working between two physically given temperatures, and assigns the numerical values of these temperatures; for the heat H taken in from the source in the isothermal expansion which forms the first operation of the cycle (p. [120]) is Mdv, and, as we have seen, the work done in the cycle is ∂p ⁄ ∂t . dtdv, or μHdt. If we adopt the expression 1 ⁄ T for μ, we may put dT for dt; and we obtain for the work done the expression HdT ⁄ T. The work done is thus the fraction dT ⁄ T of the heat taken in, and this is what is properly called the efficiency of the engine for the cycle.