572 HEAT 52. Now if for any particular fluid at some one given pressure p, with infinitesimal excess 8p above this pressure for the higher pressure in the thermodynamic experiment, we find neither heating nor cooling effect in passing through the porous plug, the paddle has nothing to do ; that is, 8w = 0. If, with always the same pressure p, but with different values of v, that is to say, with the fluid given at different temperatures, but with pressures infinitely nearly the same, we always find the same result, 8w 0, it follows from (16) that for this particular fluid at the particular pressure of the experiment, and for all the temperatures of the experiment, we have - - = 1 (22). t dv Hence by integration t = Cv (23). Hence we infer that with this fluid for thermometric substance, with the particular pressure of the experiment, and throughout the range of temperatures for which experi ment has given us 8iv = 0, absolute temperature is shown on a scale graduated and numbered in simple proportion to the whole volume of the fluid. 53. If the thermodynamic test repeated for the same fluid at different pressures gives still the same result, we have, for all pressures and temperatures within the range for which the supposed result 8w = has been found by the experiment, t=f(p]v (24); where f(p) denotes a quantity which depends only on the pressure of the fluid and is independent of its density. 54. Joule and Thomson s experiments on the thermal effects of fluids in motion 1 showed that for pressures of from one to five or six atmos hydrogen gas, common air, nitrogen, oxygen, and carbonic acid, all somewhat approximately fulfil the condition of passing through the porous plug without change of temperature, hydrogen much more approximately, carbonic acid much less approxi mately, than any of the others. Hence we infer that absolute temperature is somewhat approximately propor tional to the volume of the fluid, if any one of these gases ba used as the thermometric fluid in a constant-pressure thermometer. We shall presently see that the requisite correction of this statement for the case of hydrogen is so small as to be almost within the limits of accuracy of the most accurate thermometric usage. 55. In the case of common air, nitrogen, oxygen, and carbonic acid, the experiments showed a slight cooling effect upon the fluid in passing through the porous plug ; in the case of hydrogen, a much smaller heating effect. According to the rigorous dynamical form of our statement of 51, we have no right to measure these heating and cooling effects on any scale of temperature, as we have not yet formed a thermometric scale. And it is interesting to remark that in point of fact the thermodynamic experiment described in that section involves the use of a differential thermoscope ( 13) and not of any intrinsic thermo- scope at all ; and in respect to this requisite it maybe contrasted with the thermodynamic investigation of 49 previously, which involved the use, not of any continuous thermoscope, but only of a single-temperature intrinsic ther moscope ( 14). Now, instead of reckoning on any thermo metric scale the cooling effect or the henting effect of passage through the plug, we have to measure the quantity of work (8w) required to annul it, in the case of the majority of gases ; and in the case of hydrogen, instead of reckoning on any thermometric scale the heating effect, we 1 Transactions Royal Society, June 1853, June 1854, June 1860, and June 1862. have to measure - 8w as explained in 51 . The experiments as actually made by Joule and Thomson simply gave the cooling effects and heating effects shown by mercury thermo meters in the tranquil stream towards and from the plug ; but the very thermometers that were used had been used by Joule in his original experiments determining the dynamical equivalent of heat, and again in his later experi ments by which for the first time the specific heat of air at constant pressure was measured with sufficient accuracy for our present purpose. Hence by putting together different experiments which had actually been made with those thermometers of Joule s, the operation of measuring 8w, at all events for the case of air, was virtually completed. Thus according to our present view the mercury thermo meters are merely used as a step in aid of the measurement of 8ic, and their scales may be utterly arbitrary, provided we know the quantity of work required to raise unit mass of any of the fluids concerned through the particular differences of temperature actually shown by the thermo meters in the Joule and Thomson experiment. The best way of doing this of course is to take advantage of the best measurements, that is to say Regnault s, of the thermal capacity of air at constant pressure, and then to calculate according to Joule s own measurement the dynamical equivalent of the heat required to warm water through one degree of his own thermometers. 56. Let K be the thermal capacity, pressure constant, of the fluid experimented on, J the dynamical equivalent of the thermal unit, and St the cooling effect (reckoned negative when the effect is rise of temperature), as measured by Joule s thermometers. AYe have 8w; = JK5i! (25). Hence (16) becomes
- dt *: (26).
The experiment showed St to be simply proportional to 8^ not merely for an infinitesimal difference of pressures but for pressures up to 5 or 6 atmos. For the case of hydrogen 2 the heating effect observed amounted, per 100 inches of mercury, to 100 of a degree centigrade at temperatures of 4 or 5 centigrade, and to - 155 of a degree centigrade at temperatures of from 89 to 93 centigrade. The investigation was not carried out in sufficient detail to give any law of variation of this effect with temperature, and it was not even absolutely proved to be greater for the higher than for the lower temperature. In the circumstances we may take the mean of the results for the higher and lower temperatures, say 13 per 100 inches of mercury, or - 039 per atmo. Hence if n denotes tha force per uiit of area in the pressure called " one atmo," we have St -039 Hence which gives by integration dv The arbitrary constant C depends on the unit adopted for tempera ture. Let tliis be such that the difference of temperature between freezing and boiling is 100 (which will make our arbitrary scale agree with the ordinary centigrade scale in respect to the difference between these two temperatures). Denote now by t the absolute temperature corresponding to C. The absolute temperature corresponding to 100 C. will be < + 100. Denote also by r and v lon , for the same two temperatures, the bulks of unit mass of hydrogen at any constant pressure within the limits of Joule and Thomson s experiments, say, from one to five or six atmos. Then by dividing the value of each member of (28) for C. by the difference of its values for and 100, we find _t n _v - -039JK/IT , 9 gv 100 v v Hence
2 Joule and Thomson, Transactions Royal Society, June I860.
