270 M. Achille Cazin on Internal Work in Gases. 



plicable to changes of every kind, can be demonstrated*, 



d\J = KdT + Ad((r^--p\dv, ... (4) 



do 

 where ■— is the partial derivative of p with respect to T, as de- 

 duced from equation (2) . 

 In the present problem, 



JQ=0 and dE = 0. 

 Hence 



</U=0; 



and the integration between the conditions i>„ T x ; v\ T ; gives 

 K(T,-T') = a£J(t^-^. ... (5) 



Such is the formula from which T' may be known when the 

 relation (2) is known. Each member of that equation expresses 

 in calories the internal work effected in the operation. It may 

 be easily seen that, if formula (3) be taken for that relation, we 

 shall find 



T^-T^O, 



which is not justified by experiment. I am going to apply for- 

 mula (5) to carbonic acid, for which formula (2) may be cal- 

 culated empirically within certain limits by means of the expe- 

 rimental data of M. Regnault. 



Mr. Rankine has calculated, following M. Regnault's experi- 

 ments on the heating of carbonic acid under a constant volume, 

 an empirical formula which is very convenient for the calculation, 

 and of which Messrs. Joule and Thomson have made use in their 

 researches on the eiflux of gases through narrow orifices f. 



This is the formula : — ■ 



P ~ T v W ( ' 



p , v , T are relative to one kilogramme of gas at the tempera- 

 ture of melting ice and under the pressure of one atmosphere. 

 From this formula may be derived 



dp _ 2ap v\ 



dT F ~^ F hF > 



* P. de Saint-Robert, Principes de Thermodynamique, p. 82 (1865). 

 t Philosophical Transactions of the Royal Society of London, vol. cxliv. 

 part 2, p. 337 (1854). 



