Page:Encyclopædia Britannica, Ninth Edition, v. 11.djvu/605

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iffer- il tion with arbi- r tion. tion ie to the other end of this infinitesimal line, calculate a fresh value of c ll, and continue the curve in the slightly altered direction thus dv found, and so on. Take another point anywhere infinitesirnally near this curve but not in it, and draw by a similar process the curve through it satisfying the equation. Take a third point infinitely near this second curve, and draw through it a third curve satisfying (13), and so on till the whole area of values^, v, possible for the sub stance in question, is filled with a series of curves one of which passes through, or infinitely nearly through, every point of the area. Assign arbitrarily a particular value of &amp;lt;p to each of these curves ; then graphically find f -^ and _$ for any or every value of dv dp p and v. Then either of the two second forms of equation (12) gives us explicitly a value of t for any values whatever of p and v. 49. The solution for t thus obtained involves the arbitrary assumption of a particular value of for each one of the series of curves which we have determinately traced. Hence, to render t wholly determinate, something more must be given than e as a function of p and v. Now the only thing that can be given respect ing temperature for any particular substance before we have a ther- mometric scale is the relation subsisting between p and v when the temperature is constant. This relation can, with merely a single- temperature-thermoscope ( 15 above), in addition to dynamical instruments, be determined for some one particular temperature ; and this, if e be known for every value of p and v, is the only additional knowledge required for the determination of t for every value of p and v. For let p =f(v) be the relation between p and v for some one particular temperature, t. If by this we eliminate _p from (12) we find &quot;i where, when p=f(v), becomes a known function of v alone. dv x &quot; Hence by integration we find 05), where F denotes a known function and C an arbitrary constant. &amp;gt;d Now trace the curve pf(v) on our diagram. It must generally r- cut every one of the previously drawn determinate series of i- curves. Hence equation (15), with two arbitrarily assigned of constants &amp;lt; and C, gives determinately the value of &amp;lt;p for every lute one of the diagram of curves, and thus &amp;lt;p is determined for jera- every value of p and v. Either of equations (12) then gives t from determinately as a function of p and v, with only the value t arbi- Its of trary. The information from experiment, regarding the properties iri- of the thermometric substance, on which this determination is t. founded, consists of a knowledge of the relation between p and v for any one temperature, and of the value of e - e Q for all values of p, v, (e denoting the unknown value of e for some particular values P(,v ). Although, theoretically, this information is attainable by purely dynamical operations and measurements, with no other thermal guidance or test than that afforded by a single-temperature-intrinsic- thermoscope ( 15), the whole of it has not in fact been explicitly ob tained for any one substance. But less than the whole of it suffices to make a perfect absolute thermometer of any given substance. 50. For this purpose it is not necessary to find t for all values of p and v : it is enough to know it for all values mutually related in any manner convenient for thermo- metric practice. For example, if we could find t for every value of v with p constant at some one particular chosen value, this would give a &quot;constant pressure&quot; absolute thermometer. Or again, if we find t for every value of p with v kept constant, this would give us a &quot;constant volume&quot; absolute thermometer. Let us now examine into the restricted dynamic and thermoscopic investigations upon any particular substance, which will suffice to allow us to make of it a standard absolute thermometer of one or other of these species. 51. Dynamical and tliermoscopic investigation required to graduate, according to the absolute scale, a constant-pressure thermometer of any particular fluid, Let a large quantity of fluid be given, and let proper mechanical means be taken to cause it to flow slowly and uniformly through a pipe, in one short length of which there is a fixed porous plug. If, as is the case with common air, nitrogen, oxygen, carbonic acid, and no doubt many other gases, the fluid leaves the plug cooler than it enters it, let there be a paddle in the stream flowing from the plug, and let this paddle be turned so as to stir the fluid and cause the temperature, when the rapids are fairly past and the eddies due to the stirring subsided, to be the same as in the stream flowing towards the plug. When, as in the case of hydrogen and of all ordinary liquids, the fluid flows away from the plug warmer than it entered it, let a uniform stream of water be kept flowing in a separate canal outside the tube round a portion of it in which the internal flow is from the plug, and by this means let the temperature of the internal fluid be brought to equality with that which it had on entering the plug. By a separate thermodynamic experiment find how much work would have to be spent in stirring the external stream of water by a paddle to warm it as much as it is warmed by conduction from the internal fluid across the separating tube. Returning now to the internal fluid flow ing towards and from the plug, let p + 8p be the pressure in the steady stream approaching the disturbed region, and p the pressure in the steady stream flowing from the dis turbed region ; and let oiv be the quantity of work done by the paddle per unit of mass of the fluid passing by, reckoned positive in the first case, that namely in which the paddle compensates a cooling effect experienced in passing through the porous plug. In the second case ow (in this case a positive quantity) must denote the work done by the paddle upon the supposed external stream of water in the separate thermodynamic experiment. It is to be reckoned per unit mass of the internal fluid, irrespectively of the rate of flow of the external water. Let t denote the temperature of the fluid according to the thermodynamic scale, and let ot denote the infinitely small change of temperature which it must experience to produce an infinitesimal expansion from volume v to volume v + Sw under constant pressure. We have (16). 1 8zt v Sp Proof. Let v + Sv, v, and c + Sc, e, be respectively the amounts of the volume and of the energy of the fluid per unit mass, in the tranquil stream before and after passing the disturbed region. The work done by an ideal piston pressing the fluid in towards the dis turbed region is (y+tpfo + tv), and the work done by the emergent scream upon an ideal piston moving before it is^;i&amp;gt;, each reckoned per unit of mass, of the fluid. The whole work done on the fluid per unit mass by these ideal pistons is pSv + v8p ; add to this Sw done by the paddle, and we find that, on the whole, an amount of work equal to pSv + vSp + Sw is done on the fluid in passing through the disturbed region. Hence e exceeds e + Sc by this amount ; that is to say, Se = })Sv + vSp + Siv (17). Now the paddle and plug together act so as to render the tempera tures equal in the tranquil streams at pressures _p and p + $p. But if there were change of temperature its analytical expressior would be . (18)., dt . dt * K-i-8 + -T-qp .... dv dp Hence Sv and 87; are so proportioned as to make this vanish. That is to say, we have dt Sv=- tp (19); and we have de de* dv dp hence (17) divided by p becomes dt_ _dt L da . JlL _ de - d P + V +*OL (20). dv M dp * dt_ + Sp dv dv Using this in (11) we find t dv = v + 57 (21). Dividing (21) by v, and taking the reciprocal of both members, we have the equation (16) which was to be proved.