46. Let a mass of air, occupying initially a given volume _V_, under a
pressure _P_, at a temperature _t_, be compressed to a less volume _V′_,
and allowed to part with heat until it sinks to its primitive
temperature _t_. The quantity of heat which is evolved may be
determined, according to Carnot’s theory, when the particular value of
μ, corresponding to the temperature _t_, is known. For, by § 30,
equation (6), we have
_v_(_dq_)/(_dv_) = (_Ep_{0}v_{0}_)/(μ),
where _dq_ is the quantity of heat absorbed, when the volume is allowed
to increase from _v_ to _v_ + _dv_; or the quantity evolved by the
reverse operation. Hence we deduce
_dq_ = (_Ep_{0}v_{0}_)/(μ) (_dv_)/(_v_). (8)
Now, (_Ep_{0}v_{0}_)/(μ) is constant, since the temperature remains
unchanged; and therefore we may at once integrate the second number. By
taking it between the limits _V′_ and _V_, we thus find
_Q_ = (_Ep_{0}v_{0}_)/(μ) log (_V_)/(_V′_)[60], (9)
where _Q_ denotes the required amount of heat evolved by the compression
from _V_ to _P′_. This expression may be modified by employing the
equations _PV_ = _P′V′_ = _p_{0}v_{0}_(1 + _Et_); and we thus obtain
_Q_ = (_EPV_)/(μ(1 + _Et_)) log (_V_)/(_V′_) = (_EP′V′_)/(μ(1 + _Et_)) log (_V_)/(_V′_). (10)
From this result we draw the following conclusion:
47. _Equal volumes of all elastic fluids, taken at the same temperature
and pressure, when compressed to smaller equal volumes, disengage equal
quantities of heat._
This extremely remarkable theorem of Carnot’s was independently laid
down as a probable experimental law by Dulong, in his “_Recherches sur
la Chaleur Spécifique des Fluides Élastiques_,” and it therefore affords
a most powerful confirmation of the theory.[61]
48. In some very remarkable researches made by Mr. Joule upon the heat
developed by the compression of air, the quantity of heat produced in
different experiments has been ascertained with reference to the amount
of work spent in the operation. To compare the results which he has
obtained with the indications of theory, let us determine the amount of
work necessary actually to produce the compression considered above.
49. In the first place, to compress the gas from the volume _v_ + _dv_
to _v_, the work required is _pdv_, or, since
_pv_ = _p_{0}v_{0}_(1 + _Et_),
_p_{0}v_{0}_(1 + _Et_)(_dv_)/(_v_).
Hence, if we denote by _W_ the total amount of work necessary to produce
the compression from _V_ to _V′_, we obtain, by integration,
_W_ = _p_{0}v_{0}_(1 + _Et_) log (_V_)/(_V′_).