Page:Scientific Memoirs, Vol. 1 (1837).djvu/381

M. Dulong has shown that the air, and all the other gases taken at the temperature of 0°, and under the pressure $0^{m}\centerdot 76$ of mercury, when compressed by ${\frac {1}{267}}$ of their volume, disengage a quantity of heat, capable of elevating the same volume of atmospheric air by $0.421$.

Suppose that we operate upon a kilogramme of air occupying a volume $v=0.770$ of a cubic metre, under the pressure of the atmosphere $p$, equivalent to $10230$ kilogrammes upon a square metre; we have

$pv=R(267+t),$

and

$Q=R(B-C\log p).$

If a variation be suddenly effected in $v$ by an infinitely small quantity $d\,v$, without there being any variation in the absolute quantity of heat $Q$, we shall have

$p\;d\,v+v\;d\,p=R\;d\,t,$

and

$0=R\left({\frac {d\,B}{d\,t}}-\log p{\frac {d\,C}{d\,t}}\right)-RC{\frac {d\,p}{p}},$

or preferably

${\frac {d\,t}{C}}R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p\right)=R{\frac {d\,p}{p}}={\frac {R}{p}}\left({\frac {R\;d\,t-p\;d\,v}{v}}\right)={\frac {R\;d\,t-p\;d\,v}{267+t}}.$

Now $R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p\right)$ being the partial differential of $Q$ in respect of $t$, $p$ remaining constant, is nothing else than the specific caloric of the air at a constant pressure; it is the number of unities of heat necessary to elevate a kilogramme of air under atmospheric pressure by one degree; we have therefore

$R\left({\frac {d\,B}{d\,t}}-{\frac {d\,C}{d\,t}}\log p\right)=0.267.$

Then substituting $-{\frac {v}{267}}$ for $d\,v$, and $0.421$ for $d\,t$, we arrive lastly at

${\frac {1}{C}}=1.41.$

{{{2}}}

This is the maximum effect producible by a quantity of heat, equal to that which would elevate by 1° a kilogramme of water taken at zero, passing from a body maintained at 1° to a body maintained at 0°. It is expressed in kilogrammes raised one metre high.

Having the value of $C$, which corresponds to $t=0$, it is interesting to know, setting out from this point, whether $C$ increases or decreases, and in what proportion. An experiment of MM. De Laroche and Bérard upon the variations experienced by the specific caloric of the air,