22. In the ideal air-engine imagined by Carnot four operations performed
upon a mass of air or gas enclosed in a closed vessel of variable volume
constitute a complete cycle, at the end of which the medium is left in
its primitive physical condition; the construction being the same as
that which was described above for the steam-engine, a body _A_,
permanently retained at the temperature _S_, and _B_ at the temperature
_T_; an impermeable stand _K_; and a cylinder and piston, which in this
case contains a mass of air at the temperature _S_, instead of water in
the liquid state, at the beginning and end of a cycle of operations. The
four successive operations are conducted in the following manner:
(1) The cylinder is laid on the body _A_, so that the air in it is kept
at the temperature _S_; and the piston is allowed to rise, performing
work.
(2) The cylinder is placed on the impermeable stand _K_, so that its
contents can neither gain nor lose heat, and the piston is allowed to
rise farther, still performing work, till the temperature of the air
sinks to _T_.
(3) The cylinder is placed on _B_, so that the air is retained at the
temperature _T_, and the piston is pushed down till the air gives out to
the body _B_ as much heat as it had taken in from _A_, during the first
operation.
[Note of Nov. 5, 1881. To eliminate the assumption of the materiality
of heat, make Professor James Thomson’s correction here also; as above
in § 15; or take Maxwell’s rearrangement of the cycle described in the
foot-note to § 15, p. 144.]
(4) The cylinder is placed on _K_, so that no more heat can be taken in
or given out, and the piston is pushed down to its primitive position.
23. _At the end of the fourth operation the temperature must have
reached its primitive value S, in virtue of_ CARNOT’S _axiom_.
24. Here, again, as in the former case, we observe that work is
performed by the piston during the first two operations; and during the
third and fourth work is spent upon it, but to a less amount, since the
pressure is on the whole less during the third and fourth operations
than during the first and second, on account of the temperature being
lower. Thus, at the end of a complete cycle of operations, mechanical
effect has been obtained; and the thermal agency from which it is drawn
is the taking of a certain quantity of heat from _A_, and _letting it
down_, through the medium of the engine, to the body _B_ at a lower
temperature.
25. To estimate the actual amount of effect thus obtained, it will be
convenient to consider the alterations of volume of the mass of air in
the several operations as extremely small. We may afterwards pass by the
integral calculus, or, practically, by summation to determine the
mechanical effect whatever be the amplitudes of the different motions of
the piston.
26. Let _dq_ be the quantity of heat absorbed during the first
operation, which is evolved again during the third; and let _dv_ be the
corresponding augmentation of volume which takes place while the
temperature remains constant, as it does during the first operation.[46]
The diminution of volume in the third operation must be also equal to
_dv_, or only differ from it by an infinitely small quantity of the
second order. During the second operation we may suppose the volume to
be increased by an infinitely small quantity φ; which will occasion a
diminution of pressure and a diminution of temperature, denoted
respectively by ω and τ. During the fourth operation there will be a
diminution of volume and an increase of pressure and temperature, which
can only differ, by infinitely small quantities of the second order,
from the changes in the other direction, which took place in the second
operation, and they also may, therefore, be denoted by φ, ω, and τ,
respectively. The alteration of pressure during the first and third
operations may at once be determined by means of Mariotte’s law, since
in them the temperature remains constant. Thus, if, at the commencement
of the cycle, the volume and pressure be _v_ and _p_, they will have
become _v_ + _dv_ and _pv_/(_v_ + _dv_) at the end of the first
operation. Hence the diminution of pressure during the first operation
is _p_ − _pv_/(_v_ + _dv_) or _pdv_/(_v_ + _dv_) and therefore, if we
neglect infinitely small quantities of the second order, we have
_pdv_/_v_ for the diminution of pressure during the first operation;
which to the same degree of approximation, will be equal to the increase
of pressure during the third. If _t_ + τ and _t_ be taken to denote the
superior and inferior limits of temperature, we shall thus have for the
volume, the temperature, and the pressure at the commencements of the
four successive operations, and at the end of the cycle, the following
values respectively:
(1) _v_, _t_ + τ, _p_;
(2) _v_ + _dv_, _t_ + τ, _p_(1 − (_dv_)/(_v_));
(3) _v_ + _dv_ + φ, _t_, _p_(1 − (_dv_)/(_v_)) − ω;
(4) _v_ + φ, _t_, _p_ − ω;
(5) _v_, _t_ + τ, _p_.
Taking the mean of the pressures at the beginning and end of each
operation, we find
(1) _p_(1 − ½(_dv_)/(_v_)),
(2) _p_(1 − (_dv_)/(_v_)) − ½ω,
(3) _p_(1 − ½(_dv_)/(_v_))) − ω,
(4) _p_ − ½ω,
which, as we are neglecting infinitely small quantities of the second
order, will be the expressions for the mean pressures during the four
successive operations. Now, the mechanical effect gained or spent,
during any of the operations, will be found by multiplying the mean
pressure by the increase or diminution of volume which takes place; and
we thus find
(1) _p_(1 − ½(_dv_)/(_v_))_dv_,
(2) {_p_(1 − (_dv_)/(_v_)) − ½ω}φ,
(3) {_p_(1 − ½(_dv_)/(_v_)) − ω}_dv_,
(4) (_p_ − ½ω)φ.
for the amounts gained during the first and second, and spent during the
third and fourth operations; and hence, by addition and subtraction, we
find
ω_dv_ − _p_φ(_dv_)/(_v_), or (_v_ω − _p_φ)(_dv_)/(_v_),
for the aggregate amount of mechanical effect gained during the cycle of
operations. It only remains for us to express this result in terms of
_dq_ and τ, on which the given thermal agency depends. For this purpose
we remark that φ and ω are alterations of volume and pressure which take
place along with a change of temperature τ, and hence, by the laws of
compressibility and expansion, we may establish a relation[47] between
them in the following manner:
Let _p_{0}_ be the pressure of the mass of air when reduced to the
temperature zero, and confined in a volume _v_{0}_; then, whatever be
_v_{0}_, the product _p_{0}v_{0}_ will, by the law of compressibility,
remain constant; and, if the temperature be elevated from 0 to _t_ + τ,
and the gas be allowed to expand freely without any change of pressure,
its volume will be increased in the ratio of 1 to 1 + _E_(_t_ + τ),
where _E_ is very nearly equal to .00366 (the Centigrade scale of the
air-thermometer being referred to), whatever be the gas employed,
according to the researches of Regnault and of Magnus on the expansion
of gases by heat. If, now, the volume be altered arbitrarily with the
temperature continually at _t_ + τ, the product of the pressure and
volume will remain constant; and therefore we have
_pv_ = _p_{0}v_{0}_{1 + _E_(_t_ + τ)}.
Similarly,
(_p_ − ω)(_v_ + φ) = _p_{0}v_{0}_{1 + _Et_}.