15. Let _CDF_{2}E_{2}_ be a cylinder, of which the curved surface is
perfectly impermeable to heat, with a piston also impermeable to heat,
fitted in it; while the fixed bottom _CD_, itself with no capacity for
heat, is possessed of perfect conducting power. Let _K_ be an
impermeable stand, such that when the cylinder is placed upon it the
contents below the piston can neither gain nor lose heat. Let _A_ and
_B_ be two bodies permanently retained at constant temperatures, _S°_
and _T°_, respectively, of which the former is higher than the latter.
Let the cylinder, placed on the impermeable stand, _K_, be partially
filled with water, at the temperature _S_, of the body _A_, and (there
being no air below it) let the piston be placed in a position _EF_, near
the surface of the water. The pressure of the vapor above the water will
tend to push up the piston, and must be resisted by a force applied to
the piston,[43] till the commencement of the operations, which are
conducted in the following manner:
[Illustration]
(1) The cylinder being placed on the body _A_, so that the water and
vapor may be retained at the temperature _S_, _let the piston rise any
convenient height EE_{1}, to a position E_{1}F_{1}, performing work by
the pressure of the vapor below it during its ascent_.
[During this operation a certain quantity, _H_, of heat, the amount of
latent heat in the fresh vapor which is formed, is abstracted from the
body _A_.]
(2) The cylinder being removed, and placed on the impermeable stand _K,
let the piston rise gradually, till, when it reaches a position
E_{2}F_{2}, the temperature of the water and vapor is T, the same as
that of the body B_.
[During this operation the fresh vapor continually formed requires
heat to become latent; and, therefore, as the contents of the cylinder
are protected from any accession of heat, their temperature sinks.]
(3) The cylinder being removed from _K_, and placed on _B, let the
piston be pushed down, till, when it reaches the position E_{3}F_{3},
the quantity of heat evolved and abstracted by B amounts to that which,
during the first operation, was taken from A_.
[Note of Nov. 5, 1881. The specification of this operation, with a view
to the return to the primitive condition, intended as the conclusion to
the four operations, is the only item in which Carnot’s temporary and
provisional assumption of the materiality of heat has effect. To exclude
this hypothesis, Prof. James Thomson has suggested the following
corrected specification for the third operation: _Let the piston be
pushed down, till it reaches a position E_{3}F_{3}, determined so as to
fulfil the condition, that at the end of the fourth operation the
primitive temperature S shall be reached_:[44]]
[During this operation the temperature of the contents of the cylinder
is retained constantly at _T°_, and all the latent heat of the vapor
which is condensed into water at the same temperature is given out to
_B_.]
(4) The cylinder being removed from _B_, and placed on the impermeable
stand, _let the piston be pushed down from E_{3}F_{3} to its original
position EF_.
[During this operation, the impermeable stand preventing any loss of
heat, the temperature of the water and air must rise continually, till
(since the quantity of heat evolved during the third operation was
precisely equal to that which was previously absorbed) at the
conclusion it reaches its primitive value, _S_, in virtue of Carnot’s
fundamental axiom.]
[Note of Nov. 5, 1881. With Prof. James Thomson’s correction of
operation (3), the words in virtue of “Carnot’s Fundamental Axiom”
must be replaced by “the condition fulfilled by operation (3),” in the
description of the results of operation (4).]
16. At the conclusion of this cycle of operations[45] the total thermal
agency has been the _letting down_ of _H_ units of heat from the body
_A_, at the temperature _S_, to _B_, at the lower temperature _T_; and
the aggregate of the mechanical effect has been a certain amount of
_work produced_, since during the ascent of the piston in the first and
second operations, the temperature of the water and vapor, and therefore
the pressure of the vapor on the piston, was on the whole higher than
during the descent, in the third and fourth operations. It remains for
us actually to evaluate this aggregate amount of work performed; and for
this purpose the following graphical method of representing the
mechanical effect developed in the several operations, taken from Mons.
Clapeyron’s paper, is extremely convenient.
17. Let _OX_ and _OY_ be two lines at right angles to one another. Along
_OX_ measure off distances _ON_{1}_, _N_{1}N_{2}_, _N_{2}N_{3}_,
_N_{3}O_, respectively proportional to the spaces described by the
piston during the four successive operations described above; and, with
reference to these four operations respectively, let the following
constructions be made:
(1) Along _OY_ measure a length _OA_, to represent the pressure of the
saturated vapor at the temperature _S_; and draw _AA_{1}_ parallel to
_OX_, and let it meet an ordinate through _N_{1}_, in _A_{1}_.
(2) Draw a curve _A_{1}PA_ such that, if _ON_ represent, at any instant
during the second operation, the distance of the piston from its
primitive position, _NP_ shall represent the pressure of the vapor at
the same instant.
(3) Through _A__{2} draw _A_{2}A_{3}_ parallel to _OX_, and let it meet
an ordinate through _N_{3}_ in _A_{3}_.
(4) Draw the curve _A_{3}A_ such that the abscissa and ordinate of any
point in it may represent respectively the distances of the piston from
its primitive position, and the pressure of the vapor, at each instant
during the fourth operation. The last point of this curve must,
according to Carnot’s fundamental principle, coincide with _A_, since
the piston is, at the end of the cycle of operations, again in its
primitive position, and the pressure of the vapor is the same as it was
at the beginning.
[Illustration]
18. Let us now suppose that the lengths, _ON_{1}_, _N_{1}N_{2}_,
_N_{2}N_{3}_, and _N_{3}O_, _represent numerically_ the volumes of the
spaces moved through by the piston during the successive operations.
It follows that the mechanical effect obtained during the first
operation will be _numerically represented_ by the area
_AA_{1}N_{1}O_; that is, the number of superficial units in this area
will be equal to the number of “foot-pounds” of work performed by the
ascending piston during the first operation. The work performed by the
piston during the second operation will be similarly represented by
the area _A_{1}A_{2}N_{2}N_{1}_. Again, during the third operation a
certain amount of work is spent on the piston, which will be
represented by the area _A_{2}A_{3}N_{3}N_{2}_; and lastly, during the
fourth operation, work is spent in pushing the piston to an amount
represented by the area _A_{3}AON_{3}_.
19. Hence the mechanical effect (represented by the area
_OAA_{1}A_{2}N_{2}_) which was obtained during the first and second
operations, exceeds the work (represented by _N_{2}A_{2}A_{3}AO_) spent
during the third and fourth, by an amount represented by the area of the
quadrilateral figure _AA_{1}A_{2}A_{3}_; and, consequently, it only
remains for us to evaluate this area, that we may determine the total
mechanical effect gained in a complete cycle of operations. Now, from
experimental data, at present nearly complete, as will be explained
below, we may determine the length of the line _AA_{1}_ for the given
temperature _S_, and a given absorption _H_, of heat, during the first
operation; and the length of _A_{2}A_{3}_ for the given lower
temperature _T_, and the evolution of the same quantity of heat during
the fourth operation: and the curves _A_{1}PA_{2}_, _A_{3}P′A_ may be
drawn as graphical representations of actual observations. The figure
being thus constructed, its area may be measured, and we are, therefore,
in possession of a graphical method of determining the amount of
mechanical effect to be obtained from any given thermal agency. As,
however, it is merely the area of the figure which it is required to
determine, it will not be necessary to be able to describe each of the
curves _A_{1}PA_{2}_, _A_{3}P′A_, but it will be sufficient to know the
difference of the abscissas corresponding to any equal ordinates in the
two; and the following analytical method of completing the problem is
the most convenient for leading to the actual numerical results.
20. Draw any line _PP′_ parallel to _OX_, meeting the curvilinear sides
of the quadrilateral in _P_ and _P′_. Let ξ denote the length of this
line, and _p_ its distance from _OX_. The area of the figure, according
to the integral calculus, will be denoted by the expression
∫_{_p_{3}_} ^{_p_{1}_} ξ_dp_,
where _p_{1}_ and _p_{3}_ (the limits of integration indicated according
to Fourier’s notation) denote the lines _OA_ and _N_{3}A_{3}_, which
represent respectively the pressures during the first and third
operations. Now, by referring to the construction described above, we
see that ξ is the difference of the volumes below the piston at
corresponding instants of the second and fourth operations, or instants
at which the saturated steam and the water in the cylinder have the same
pressure _p_, and consequently the same temperature, which we may denote
by _t_. Again, throughout the second operation the entire contents of
the cylinder possess a greater amount of heat by _H_ units than during
the fourth; and, therefore, at any instant of the second operation there
is as much more steam as contains _H_ units of latent heat than at the
corresponding instant of the fourth operation. Hence if _k_ denote the
latent heat in a unit of saturated steam at the temperature _t_, the
volume of the steam at the two corresponding instants must differ by
(_H_)/(_k_). Now, if σ denote the ratio of the density of the steam to
that of the water, the volume (_H_)/(_k_) of steam will be formed from
the volume σ (_H_)/(_k_) of water; and consequently we have, for the
difference of volumes of the entire contents at the corresponding
instants,
ξ = (1 - σ)(_H_)/(_k_).