260 PRINCIPLES OF THE MECHANICAL THEORY OF HEAT. 



If we designate the free lieat of a body by T, the latent heat, Avhich it may 

 contain, by L, the whole quantity of heat in the body will be U=T-|-L. 



W. Thomson has proposed for this quantity, first introduced into the doctrine 

 of heat by Olausius, the name of energij of the body ; but Clausius has recently 

 designated the two components of U as store of heat, T, (Warmeinhalt,) and 

 store of work, L, [Werldnlmlt.) 



If now so mucli heat be added to a body of the temperature f and the volume 

 V, in which the collective quantity of heat, U, (energy,) is contained, that its 

 temperature be increased by f °, its volume by v' cubic metres, and the quantity of 

 heat contained within it by U', it will not answer to ascribe to it this quantity of 

 heat U', because through the simultaneous expansion of the body by v' an external 

 work has been done which consumes a corresponding quantity of heat. 



If p be the pressure under which the body stands, then the external work 

 which coiTesponds to the enlargement of volume v' will be p v', supposing the 

 pressure p to remain unaltered during the whole expansion ; but the quantity of 

 heat corresponding to that work is iv-=A.p v'. Thus the quantity of heat which 

 must be supplied to the body in question in order to increase its temperature from 

 tio t-i-t'j its volume from v to v+v', and the heat contained in it from U to 

 U + li' is, 



q=W-\-Apv' .... I 



an equation which corresponds to the differential equation 



d Q=d U-j-A^ d V . . . la 

 in W'hich d Q designates the very small quantity of heat which must be supplied 

 to the body, in order for the interior heat" of the body to undergo the small aug- 

 mentation d U and its volume to be increased by the small magnitude d v. 



The equation I or rather a differential equation corresponding to it, is the 

 mathematical expression of the first Jaw of the mechanical theory of heat. By 

 the help of this equation we can calculate the quantity of heat which disappears 

 through a given change of volume of a body submitted to a given pressure, pro- 

 vided we know the whole wor^i done thereby ; which is, however, only the 

 case when we have to do exclusively tvith external work without the accession of 

 any internal work proceeding from molecular forces, and which evades a direct 

 measurement. The first law, therefore, of the mechanical theory of lieat suffices 

 only for the solution of correspondent problems when in the magnitude U' of the 

 equation I or in d U of the equation I a an internal work is not comprehended, 

 a condition which is only satisfied by bodies of a completely gaseous form. 



The folhnving examples will illustrate the application of the first law of the 

 mechanical theory of heat to gaseous botlies. In a hollow cylinder, (Fig. 9,) hav- 

 ing a transverse section of one square metre, is situated 

 at the distance of one metre from the bottom an easily 

 movable, hut air-tight, piston K ; the space of one 

 cubic metre, shut in by this piston, is filled with atmo- 

 spheric air at 0°, Avhile the pressure of the atmosphere 

 weighs on the piston ; and hence the total pressure which 

 tends to sink the piston is 10333 kilograms. 



If this mass of air, the pressure unchanged, be heated 

 to 273° it will be expanded to double its original vol- 

 ume; the piston, during this expansion, will be thrust 

 one metre higher, and thus be brought into the position K^. 

 The work so done is 10333 metre-kilograms, and 

 the quantity of heat, A j) r', consumed in doing this 

 work is, in that case, 



— ^-^=24.37 units of heat. 

 424 



The total heat, q, which must T)0 supplied to a cubic 



metre of air at 0° and under ' atmospheric pressure 



