Page:Encyclopædia Britannica, Ninth Edition, v. 11.djvu/604

Rh 570 HEAT pyro meter. In fact, the series of steam thermometers for the whole range from the lowest temperatures can be reproduced with the greatest ease in any part of the world by a person com mencing with no other material than a piece of sulphur and air to burn it in, 1 some pure water, and some pure mercury, and with no other apparatus than can be made by a moderately skilled glass-blower, and with no other standard of physical measurement of any kind than an accurate linear measure. He may assume the force of gravity to be that calculated for his latitude, with the ordi nary rough allowance for his elevation above the sea. and his omission to measure with higher accuracy the actual force of gravity in his locality can lead him into no thermo- metric error which is not incomparably less than the inevit able errors in the reproduction and use of the air thermo meter, or of mercury or other liquid thermometers. In temperatures above the highest for which mercury-steam pressure is not too great to be practically available, nothing Deville s hitherto invented but Deville s air thermometer with accurate hard porcelain bulb suited to resist the high temperature is available for accurate thermometry. 46. We have given the steam thermometer as our first example of thermodynamic thermometry because intelli gence in thermodynamics has been hitherto much retarded, and the student unnecessarily perplexed, and a mere quick sand has been given as a foundation for thermometry, by building from the beginning on an ideal substance called perfect gas, with none of its properties realized rigorously by any real substance, and with some of them (see MATTER, LIQUID, STEAM) unknown, and utterly unassignable, even by guess. But after having been moved by this reason to give the steam-pressure thermometer as our first theoretical example, we have been led into the preceding carefully de tailed examination of its practical qualities, and we have thus become convinced that though liitherto used in scientific investigations only for fixing the &quot;boiling point,&quot; and (through an inevitable natural selection) by practical en gineers for knowing the temperatures of their boilers by the pressures indicated by the Bourdon gauge, it is destined to be of great service both in the strictest scientific ther mometry and as a practical thermometer for a great variety of useful applications. 47. Example 2 (including example 1). Any case in which the stress is uniform pressure in all directions. Let p and v denote the pressure and volume. The condition of the substance (single, double, or triple, as the case may be) is de terminate when p and v are given, and it will therefore be spoken of shortly as the condition (p, v). Let e be the energy which must be communicated to the substance to bring it from any con veniently defined zero condition (p, v ) to any condition whatever (p, v). Remark that e is a function of the two independent variables p, v to be found by experiment, and that the finding of it by experiment is a perfectly determinate practical problem, which can be carried out without the aid of any thermoscope, and without any consideration whatever relating to temperature. We shall see in fact that accurate practical solutions of it for many different sub stances have been obtained by experiment (see THERMODYNAMICS). The absolute temperature t is also a function of p and v to be also determined by experiment, according to the equivalent definitions of 35 and 37. Let heat be communicated to the substance so as to cause its volume to increase by dv, the pressure being kept constant. The energy of the body will be augmented by %...**. dv At the same time the body in expanding and pressing out the matter around it does work to the extent of p. dv (4). Hence the whole work required to generate the heat given to it amounts to 1 Practically, the best ordinary chemical means of preparing sul phurous acid, as from sulphuric acid by heating with copper, might be adopted in preference to burning sulphur. Hence the ratio of (4) to (5), or (6), p (7), de_ dv is the &quot; work-ratio &quot; of 37. Hence by the definition Dt dp p dp de P ~r + p dv where Dt denotes the change of temperature produced by aug- Ope menting the pressure by dp, and at the same time preventing the tion substance from either giving heat to or taking heat from the sur- call rounding matter. To express this last condition analytically, let adia dv be the augmentation of volume (negative, of course, if dp be com positive) which it implies. The work done on the substance by sion the pressure from without is pdv, and the energy of the substance Rai] is augmented by just this amount, because of the condition to be expressed. Hence dc ,. de , whence But and so we have dv= dc dp dp -j- + p dv dt, dt 7 -- dp + -j-dv , dp dv de tdt_ dt dv da I , i- dp dp de d- V + VJ Eliminating Dt/dp from this by (7) we find de dt dc dt -P + (10). (11). dvj dp dp dv 48. This is a linear partial differential equation of the first order for the determination of t, supposing, as we do for the present, that e is a known function of p and v. The following graphical illustra tion of the well-known analytical process for finding the complete solutions of such equations shows exactly how much towards deter mination of temperature can be done with no other data from experiment than the values of and as functions of and v, dv dp and what additional information is required to fully determine t. First remark that (11) is the condition that be a factor rendering | de j de dvj dp of two independent variables p and v. that is to say, let &amp;lt;j&amp;gt; be such that d&amp;lt;b 1 / de i dtb -T- = -r(p + -r and -r- dv t dvj dp de de P + T~ ~r or t= &amp;lt;to = J$_ d$_ dtp dv dp Then every solution of the differential equation de a complete differential of a function 2 Let &amp;lt;/&amp;gt; be this function, de dp (12). fun ] tar call cut] by and seqi dp dv P + T dv _ dp (13) renders tp constant ; and conversely, every series of values of p and v which renders &amp;lt;f&amp;gt; constant constitutes a solution of (13). Now this differential equation may be solved graphically by taking p and v as rectangular coordinates of a point in a plane, and drawing the whole series of curves which satisfy it as follows. Commence with any point and calculate for its values of p and v the value of the second member of (13). Draw through this point an infinitesimal line in the direction of the tangent to the curve given by the value so found for -^ . &quot;With the altered values of p and v corresponding dv 2 This function is of great importance in practical thermodynamics : multiplied by t, it is equal to the excess of the energy of the sub- stance above its motivity. Motivity (defined by Sir Wm. Thomson, Proceedings M.S.E., 1879) is the amount of work obtainable by letting the substance pass from the state (p, v) in which it is given to the zero condition (p, V Q ), without either taking in heat from or giving out heat to matter at any other temperature than t. See THERMODYNAMICS. 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