We shall obtain the relation in another way, which will illustrate another mode of dealing with a cycle of operations which Thomson employed. Any small step of change of a substance may be regarded as made up of a step of volume, say, followed by a step of temperature, that is, by an isothermal step followed by an adiabatic step. In this way any cycle of operations whatever may be regarded as made up of a series of Carnot cycles. But without regarding any cycle of a more general kind than Carnot's as thus compounded, we can draw conclusions from it by the dynamical theory provided only it is reversible. Suppose a gramme, say, of the substance to be taken at a specified temperature T in the lower phase, and to be changed to the other phase at that temperature. The heat taken in will be L and the expansion will be v2 − v1. Next, keeping the substance in the second phase, and in equilibrium with the first phase (that is, for example, if the second phase is saturated vapour, the saturation is to continue in the further change), let the substance be lowered in temperature by dT. The heat given out by the substance will be hdT, where h is the specific heat of the substance in the second phase. Now at the new temperature T − dT let the substance be wholly brought back to the second phase; the heat given out will be L − ∂L ⁄ ∂T . dT. Finally, let the substance, now again all in the first phase, be brought to the original temperature: the heat taken in will be cdt, where c is the specific heat in the first phase. Thus the net excess of heat taken in over heat given out in the cycle is (∂L ⁄ ∂T + c − h) dT. This must, in the indicator diagram for the changes specified, be the area of the cycle or (v2 − v1) ∂p ⁄ ∂T . dT. But by equation (B) L ⁄ T (v2 − v1) = ∂p ⁄ ∂T, and the area of the cycle is (L ⁄ T) dT. Equating the two expressions thus found for the area we get equation (C).
This relation was arrived at by Clausius in his paper referred to above, and the priority of publication is his: it is here given in the form which it takes when Thomson's scale of absolute temperature is used.
Regnault's experimental results for the heat required to raise unit mass of water from the temperature of melting ice to any higher temperature and evaporate it at that temperature enable the values of L ⁄ T and ∂L ⁄ ∂T to be calculated, and therefore that of h to be found. It appears that h is negative for all the temperatures to which Regnault's experimental results can be held to apply. This, as was pointed out by Thomson, means that if a mass of saturated vapour is made to expand so as at the same time to fall in temperature, it must have heat given to it, otherwise it will be partly condensed into liquid; and, on the other hand, if the vapour be compressed and made to rise in temperature while at the same time it is kept saturated, heat must be taken from it, otherwise the vapour will become superheated and so cease to be saturated.
It is convenient to notice here the article on Heat which Thomson wrote for the ninth edition of the Encyclopædia Britannica. In that article he gave a valuable discussion of ordinary thermometry, of thermometry by means of the pressures of saturated vapour of different substances—steam-pressure thermometers, he called them—of absolute thermodynamic thermometry, all enriched with new experimental and theoretical investigations, and appended to the whole a valuable synopsis, with additions of his own, of the Fourier mathematics of heat conduction.
First dealing with temperature as measured by the expansion of a liquid in a less expansible vessel, he showed how it is in reality numerically reckoned. This amounted to a discussion of the scale of an ordinary mercury-in-glass thermometer, a subject concerning which erroneous statements are not infrequently made in text-books. A sketch of Thomson's treatment of it is given here.
Considering this thermometer as a vessel consisting of a glass bulb and a long glass stem of fine and uniform bore, hermetically sealed and containing only mercury and mercury vapour, he explained the numerical relation between the temperature as shown by the instrument and the volumes of the mercury and vessel. The scale is really defined by the method of graduation adopted. Two points of reference are marked on the stem at which the top of the mercury stands when the vessel is immersed (1) in melting ice, (2) in saturated steam under standard atmospheric pressure. The stem is divided into parts of equal volume of bore between these two points and beyond each of them. For a centigrade thermometer the bore-space between the two points is divided into 100 equal parts, and the lower point of reference is marked 0 and the upper 100, and the other dividing marks are numbered in accordance with this along the stem. Each of these parts of the bore may be called a degree-space.
Now let the instrument contain in its bulb and stem, up to the mark 0, N degree-spaces, and let v be the volume of a degree-space at that temperature. The volume up to the mark 0 will be Nv, at that temperature; and if the substance of the vessel be quite uniform in quality and free from stress, N will be the same for all temperatures. If v0 be the volume of a degree-space at the temperature of melting ice the volume of the mercury at that temperature will be Nv0. If G be the expansion of the glass when the volume of a degree-space is increased from v0 to v by the rise of temperature, then v = v0 (1 + G). The volume of the mercury has been increased therefore to (N + n) v0 (1 + G) by the same rise of temperature, if the top of the column is thereby made to rise from the mark 0 so as to occupy n degree-spaces more than before. But if E be the expansion of the mercury between the temperature of melting ice and that which has now been attained, the volume of the mercury is also Nv0 (1 + E). Hence N (1 + E) = (N + n) (1 + G). This gives n = N (E − G) ⁄ (1 + G).
If we take, as is usual, n as measuring the temperature, and substitute for it the symbol t, we have, since N = 100 (1 + G100) ⁄ (E100 − G100),
In this reckoning the definition of any temperature, let us say 37° C., is the temperature of the vessel and its contents when the top of the mercury column stands at the mark 37 above 0, on the scale defined by the graduation of the instrument; but the numerical signification with relation to the volumes is given by equation (D). This shows that the numerical measure of any temperature involves both the expansion of the vessel and that of the glass vessel between the temperature of melting ice and the temperature in question. This result may be contrasted with the erroneous statement frequently made that equal increments of temperature correspond to equal increments of the volume of the thermometric substance. It also shows that different mercury-in-glass thermometers, however accurately made and graduated, need not agree when placed in a bath at any other temperature than 0° C. or 100° C. This fact, and the results of the comparison of thermometers made with different kinds of glass with the normal air thermometer, which was carried out by Regnault, were always insisted on by Thomson in his teaching when he dealt with the subject of heat. The scale of a mercury-in-glass thermometer is too often in text-books, and even in Acts of Parliament regarded as a perfectly definite thing, and the expansion of a gas is not infrequently defined by this indefinite scale, instead of being used as it ought to be, as the basis of definition of the scale of the gas thermometer. The whole treatment of the so-called gaseous laws is too often, from a logical point of view, a mass of confusion.