Every one knows that heat can produce motion. That it possesses vast
motive power no one can doubt, in these days when the steam-engine is
everywhere so well known.
To heat also are due the vast movements which take place on the earth.
It causes the agitations of the atmosphere, the ascension of clouds, the
fall of rain and of meteors, the currents of water which channel the
surface of the globe, and of which man has thus far employed but a small
portion. Even earthquakes and volcanic eruptions are the result of heat.
From this immense reservoir we may draw the moving force necessary for
our purposes. Nature, in providing us with combustibles on all sides,
has given us the power to produce, at all times and in all places, heat
and the impelling power which is the result of it. To develop this
power, to appropriate it to our uses, is the object of heat-engines.
The study of these engines is of the greatest interest, their importance
is enormous, their use is continually increasing, and they seem destined
to produce a great revolution in the civilized world.
Already the steam-engine works our mines, impels our ships, excavates
our ports and our rivers, forges iron, fashions wood, grinds grains,
spins and weaves our cloths, transports the heaviest burdens, etc. It
appears that it must some day serve as a universal motor, and be
substituted for animal power, waterfalls, and air currents.
Over the first of these motors it has the advantage of economy, over the
two others the inestimable advantage that it can be used at all times
and places without interruption.
If, some day, the steam-engine shall be so perfected that it can be set
up and supplied with fuel at small cost, it will combine all desirable
qualities, and will afford to the industrial arts a range the extent of
which can scarcely be predicted. It is not merely that a powerful and
convenient motor that can be procured and carried anywhere is
substituted for the motors already in use, but that it causes rapid
extension in the arts in which it is applied, and can even create
entirely new arts.
The most signal service that the steam-engine has rendered to England is
undoubtedly the revival of the working of the coal-mines, which had
declined, and threatened to cease entirely, in consequence of the
continually increasing difficulty of drainage, and of raising the
coal.[5] We should rank second the benefit to iron manufacture, both by
the abundant supply of coal substituted for wood just when the latter
had begun to grow scarce, and by the powerful machines of all kinds, the
use of which the introduction of the steam-engine has permitted or
facilitated.
Iron and heat are, as we know, the supporters, the bases, of the
mechanic arts. It is doubtful if there be in England a single industrial
establishment of which the existence does not depend on the use of these
agents, and which does not freely employ them. To take away to-day from
England her steam-engines would be to take away at the same time her
coal and iron. It would be to dry up all her sources of wealth, to ruin
all on which her prosperity depends, in short, to annihilate that
colossal power. The destruction of her navy, which she considers her
strongest defence, would perhaps be less fatal.
The safe and rapid navigation by steamships may be regarded as an
entirely new art due to the steam-engine. Already this art has permitted
the establishment of prompt and regular communications across the arms
of the sea, and on the great rivers of the old and new continents. It
has made it possible to traverse savage regions where before we could
scarcely penetrate. It has enabled us to carry the fruits of
civilization over portions of the globe where they would else have been
wanting for years. Steam navigation brings nearer together the most
distant nations. It tends to unite the nations of the earth as
inhabitants of one country. In fact, to lessen the time, the fatigues,
the uncertainties, and the dangers of travel—is not this the same as
greatly to shorten distances?[6]
The discovery of the steam-engine owed its birth, like most human
inventions, to rude attempts which have been attributed to different
persons, while the real author is not certainly known. It is, however,
less in the first attempts that the principal discovery consists, than
in the successive improvements which have brought steam-engines to the
condition in which we find them to-day. There is almost as great a
distance between the first apparatus in which the expansive force of
steam was displayed and the existing machine, as between the first raft
that man ever made and the modern vessel.
If the honor of a discovery belongs to the nation in which it has
acquired its growth and all its developments, this honor cannot be here
refused to England. Savery, Newcomen, Smeaton, the famous Watt, Woolf,
Trevithick, and some other English engineers, are the veritable creators
of the steam-engine. It has acquired at their hands all its successive
degrees of improvement. Finally, it is natural that an invention should
have its birth and especially be developed, be perfected, in that place
where its want is most strongly felt.
Notwithstanding the work of all kinds done by steam-engines,
notwithstanding the satisfactory condition to which they have been
brought to-day, their theory is very little understood, and the attempts
to improve them are still directed almost by chance.
The question has often been raised whether the motive power of heat[7]
is unbounded, whether the possible improvements in steam-engines have an
assignable limit,—a limit which the nature of things will not allow to
be passed by any means whatever; or whether, on the contrary, these
improvements may be carried on indefinitely. We have long sought, and
are seeking to-day, to ascertain whether there are in existence agents
preferable to the vapor of water for developing the motive power of
heat; whether atmospheric air, for example, would not present in this
respect great advantages. We propose now to submit these questions to a
deliberate examination.
The phenomenon of the production of motion by heat has not been
considered from a sufficiently general point of view. We have considered
it only in machines the nature and mode of action of which have not
allowed us to take in the whole extent of application of which it is
susceptible. In such machines the phenomenon is, in a way, incomplete.
It becomes difficult to recognize its principles and study its laws.
In order to consider in the most general way the principle of the
production of motion by heat, it must be considered independently of any
mechanism or any particular agent. It is necessary to establish
principles applicable not only to steam-engines[8] but to all imaginable
heat-engines, whatever the working substance and whatever the method by
which it is operated.
Machines which do not receive their motion from heat, those which have
for a motor the force of men or of animals, a waterfall, an air-current,
etc., can be studied even to their smallest details by the mechanical
theory. All cases are foreseen, all imaginable movements are referred to
these general principles, firmly established, and applicable under all
circumstances. This is the character of a complete theory. A similar
theory is evidently needed for heat-engines. We shall have it only when
the laws of Physics shall be extended enough, generalized enough, to
make known beforehand all the effects of heat acting in a determined
manner on any body.
We will suppose in what follows at least a superficial knowledge of the
different parts which compose an ordinary steam-engine; and we consider
it unnecessary to explain what are the furnace, boiler, steam-cylinder,
piston, condenser, etc.
The production of motion in steam-engines is always accompanied by a
circumstance on which we should fix our attention. This circumstance is
the re-establishing of equilibrium in the caloric; that is, its passage
from a body in which the temperature is more or less elevated, to
another in which it is lower. What happens in fact in a steam-engine
actually in motion? The caloric developed in the furnace by the effect
of the combustion traverses the walls of the boiler, produces steam, and
in some way incorporates itself with it. The latter carrying it away,
takes it first into the cylinder, where it performs some function, and
from thence into the condenser, where it is liquefied by contact with
the cold water which it encounters there. Then, as a final result, the
cold water of the condenser takes possession of the caloric developed by
the combustion. It is heated by the intervention of the steam as if it
had been placed directly over the furnace. The steam is here only a
means of transporting the caloric. It fills the same office as in the
heating of baths by steam, except that in this case its motion is
rendered useful.
We easily recognize in the operations that we have just described the
re-establishment of equilibrium in the caloric, its passage from a more
or less heated body to a cooler one. The first of these bodies, in this
case, is the heated air of the furnace; the second is the condensing
water. The re-establishment of equilibrium of the caloric takes place
between them, if not completely, at least partially, for on the one hand
the heated air, after having performed its function, having passed round
the boiler, goes out through the chimney with a temperature much below
that which it had acquired as the effect of combustion; and on the other
hand, the water of the condenser, after having liquefied the steam,
leaves the machine with a temperature higher than that with which it
entered.
The production of motive power is then due in steam-engines not to an
actual consumption of caloric, but _to its transportation from a warm
body to a cold body_, that is, to its re-establishment of equilibrium—an
equilibrium considered as destroyed by any cause whatever, by chemical
action such as combustion, or by any other. We shall see shortly that
this principle is applicable to any machine set in motion by heat.
According to this principle, the production of heat alone is not
sufficient to give birth to the impelling power: it is necessary that
there should also be cold; without it, the heat would be useless. And in
fact, if we should find about us only bodies as hot as our furnaces, how
can we condense steam? What should we do with it if once produced? We
should not presume that we might discharge it into the atmosphere, as is
done in some engines;[9] the atmosphere would not receive it. It does
receive it under the actual condition of things, only because it fulfils
the office of a vast condenser, because it is at a lower temperature;
otherwise it would soon become fully charged, or rather would be already
saturated.[10]
Wherever there exists a difference of temperature, wherever it has been
possible for the equilibrium of the caloric to be re-established, it is
possible to have also the production of impelling power. Steam is a
means of realizing this power, but it is not the only one. All
substances in nature can be employed for this purpose, all are
susceptible of changes of volume, of successive contractions and
dilatations, through the alternation of heat and cold. All are capable
of overcoming in their changes of volume certain resistances, and of
thus developing the impelling power. A solid body—a metallic bar for
example—alternately heated and cooled increases and diminishes in
length, and can move bodies fastened to its ends. A liquid alternately
heated and cooled increases and diminishes in volume, and can overcome
obstacles of greater or less size, opposed to its dilatation. An
aeriform fluid is susceptible of considerable change of volume by
variations of temperature. If it is enclosed in an expansible space,
such as a cylinder provided with a piston, it will produce movements of
great extent. Vapors of all substances capable of passing into a gaseous
condition, as of alcohol, of mercury, of sulphur, etc., may fulfil the
same office as vapor of water. The latter, alternately heated and
cooled, would produce motive power in the shape of permanent gases, that
is, without ever returning to a liquid state. Most of these substances
have been proposed, many even have been tried, although up to this time
perhaps without remarkable success.
We have shown that in steam-engines the motive power is due to a
re-establishment of equilibrium in the caloric; this takes place not
only for steam-engines, but also for every heat-engine—that is, for
every machine of which caloric is the motor. Heat can evidently be a
cause of motion only by virtue of the changes of volume or of form which
it produces in bodies.
These changes are not caused by uniform temperature, but rather by
alternations of heat and cold. Now to heat any substance whatever
requires a body warmer than the one to be heated; to cool it requires a
cooler body. We supply caloric to the first of these bodies that we may
transmit it to the second by means of the intermediary substance. This
is to re-establish, or at least to endeavor to re-establish, the
equilibrium of the caloric.
It is natural to ask here this curious and important question: Is the
motive power of heat invariable in quantity, or does it vary with the
agent employed to realize it as the intermediary substance, selected as
the subject of action of the heat?
It is clear that this question can be asked only in regard to a given
quantity of caloric,[11] the difference of the temperatures also being
given. We take, for example, one body _A_ kept at a temperature of 100°
and another body _B_ kept at a temperature of 0°, and ask what quantity
of motive power can be produced by the passage of a given portion of
caloric (for example, as much as is necessary to melt a kilogram of ice)
from the first of these bodies to the second. We inquire whether this
quantity of motive power is necessarily limited, whether it varies with
the substance employed to realize it, whether the vapor of water offers
in this respect more or less advantage than the vapor of alcohol, of
mercury, a permanent gas, or any other substance. We will try to answer
these questions, availing ourselves of ideas already established.
We have already remarked upon this self-evident fact, or fact which at
least appears evident as soon as we reflect on the changes of volume
occasioned by heat: _wherever there exists a difference of temperature,
motive power can be produced_. Reciprocally, wherever we can consume
this power, it is possible to produce a difference of temperature, it is
possible to occasion destruction of equilibrium in the caloric. Are not
percussion and the friction of bodies actually means of raising their
temperature, of making it reach spontaneously a higher degree than that
of the surrounding bodies, and consequently of producing a destruction
of equilibrium in the caloric, where equilibrium previously existed? It
is a fact proved by experience, that the temperature of gaseous fluids
is raised by compression and lowered by rarefaction. This is a sure
method of changing the temperature of bodies, and destroying the
equilibrium of the caloric as many times as may be desired with the same
substance. The vapor of water employed in an inverse manner to that in
which it is used in steam-engines can also be regarded as a means of
destroying the equilibrium of the caloric. To be convinced of this we
need but to observe closely the manner in which motive power is
developed by the action of heat on vapor of water. Imagine two bodies
_A_ and _B_, kept each at a constant temperature, that of _A_ being
higher than that of _B_. These two bodies, to which we can give or from
which we can remove the heat without causing their temperatures to vary,
exercise the functions of two unlimited reservoirs of caloric. We will
call the first the furnace and the second the refrigerator.
If we wish to produce motive power by carrying a certain quantity of
heat from the body _A_ to the body _B_ we shall proceed as follows:
(1) To borrow caloric from the body _A_ to make steam with it—that is,
to make this body fulfil the function of a furnace, or rather of the
metal composing the boiler in ordinary engines—we here assume that the
steam is produced at the same temperature as the body _A_.
(2) The steam having been received in a space capable of expansion, such
as a cylinder furnished with a piston, to increase the volume of this
space, and consequently also that of the steam. Thus rarefied, the
temperature will fall spontaneously, as occurs with all elastic fluids;
admit that the rarefaction may be continued to the point where the
temperature becomes precisely that of the body _B_.
(3) To condense the steam by putting it in contact with the body _B_,
and at the same time exerting on it a constant pressure until it is
entirely liquefied. The body _B_ fills here the place of the
injection-water in ordinary engines, with this difference, that it
condenses the vapor without mingling with it, and without changing its
own temperature.[12]
The operations which we have just described might have been performed in
an inverse direction and order. There is nothing to prevent forming
vapor with the caloric of the body _B_, and at the temperature of that
body, compressing it in such a way as to make it acquire the temperature
of the body _A_, finally condensing it by contact with this latter body,
and continuing the compression to complete liquefaction.
By our first operations there would have been at the same time
production of motive power and transfer of caloric from the body _A_ to
the body _B_. By the inverse operations there is at the same time
expenditure of motive power and return of caloric from the body _B_ to
the body _A_. But if we have acted in each case on the same quantity of
vapor, if there is produced no loss either of motive power or caloric,
the quantity of motive power produced in the first place will be equal
to that which would have been expended in the second, and the quantity
of caloric passed in the first case from the body _A_ to the body _B_
would be equal to the quantity which passes back again in the second
from the body _B_ to the body _A_; so that an indefinite number of
alternative operations of this sort could be carried on without in the
end having either produced motive power or transferred caloric from one
body to the other.
Now if there existed any means of using heat preferable to those which
we have employed, that is, if it were possible by any method whatever to
make the caloric produce a quantity of motive power greater than we have
made it produce by our first series of operations, it would suffice to
divert a portion of this power in order by the method just indicated to
make the caloric of the body _B_ return to the body _A_ from the
refrigerator to the furnace, to restore the initial conditions, and thus
to be ready to commence again an operation precisely similar to the
former, and so on: this would be not only perpetual motion, but an
unlimited creation of motive power without consumption either of caloric
or of any other agent whatever. Such a creation is entirely contrary to
ideas now accepted, to the laws of mechanics and of sound physics. It is
inadmissible.[13] We should then conclude that _the maximum of motive
power resulting from the employment of steam is also the maximum of
motive power realizable by any means whatever_. We will soon give a
second more rigorous demonstration of this theory. This should be
considered only as an approximation. (See page 59.)
We have a right to ask, in regard to the proposition just enunciated,
the following questions: What is the sense of the word _maximum_ here?
By what sign can it be known that this maximum is attained? By what sign
can it be known whether the steam is employed to greatest possible
advantage in the production of motive power?
Since every re-establishment of equilibrium in the caloric may be the
cause of the production of motive power, every re-establishment of
equilibrium which shall be accomplished without production of this power
should be considered as an actual loss. Now, very little reflection
would show that all change of temperature which is not due to a change
of volume of the bodies can be only a useless re-establishment of
equilibrium in the caloric.[14] The necessary condition of the maximum
is, then, _that in the bodies employed to realize the motive power of
heat there should not occur any change of temperature which may not be
due to a change of volume_. Reciprocally, every time that this condition
is fulfilled the maximum will be attained. This principle should never
be lost sight of in the construction of heat-engines; it is its
fundamental basis. If it cannot be strictly observed, it should at least
be departed from as little as possible.
Every change of temperature which is not due to a change of volume or to
chemical action (an action that we provisionally suppose not to occur
here) is necessarily due to the direct passage of the caloric from a
more or less heated body to a colder body. This passage occurs mainly by
the contact of bodies of different temperatures; hence such contact
should be avoided as much as possible. It cannot probably be avoided
entirely, but it should at least be so managed that the bodies brought
in contact with each other differ as little as possible in temperature.
When we just now supposed, in our demonstration, the caloric of the body
_A_ employed to form steam, this steam was considered as generated at
the temperature of the body _A_; thus the contact took place only
between bodies of equal temperatures; the change of temperature
occurring afterwards in the steam was due to dilatation, consequently to
a change of volume. Finally, condensation took place also without
contact of bodies of different temperatures. It occurred while exerting
a constant pressure on the steam brought in contact with the body _B_ of
the same temperature as itself. The conditions for a maximum are thus
found to be fulfilled. In reality the operation cannot proceed exactly
as we have assumed. To determine the passage of caloric from one body to
another, it is necessary that there should be an excess of temperature
in the first, but this excess may be supposed as slight as we please. We
can regard it as insensible in theory, without thereby destroying the
exactness of the arguments.
A more substantial objection may be made to our demonstration, thus:
When we borrow caloric from the body _A_ to produce steam, and when this
steam is afterwards condensed by its contact with the body _B_, the
water used to form it, and which we considered at first as being of the
temperature of the body _A_, is found at the close of the operation at
the temperature of the body _B_. It has become cool. If we wish to begin
again an operation similar to the first, if we wish to develop a new
quantity of motive power with the same instrument, with the same steam,
it is necessary first to re-establish the original condition—to restore
the water to the original temperature. This can undoubtedly be done by
at once putting it again in contact with the body _A_; but there is then
contact between bodies of different temperatures, and loss of motive
power.[15] It would be impossible to execute the inverse operation, that
is, to return to the body _A_ the caloric employed to raise the
temperature of the liquid.
This difficulty may be removed by supposing the difference of
temperature between the body _A_ and the body _B_ indefinitely small.
The quantity of heat necessary to raise the liquid to its former
temperature will be also indefinitely small and unimportant relatively
to that which is necessary to produce steam—a quantity always limited.
The proposition found elsewhere demonstrated for the case in which the
difference between the temperatures of the two bodies is indefinitely
small, may be easily extended to the general case. In fact, if it
operated to produce motive power by the passage of caloric from the body
_A_ to the body _Z_, the temperature of this latter body being very
different from that of the former, we should imagine a series of bodies
_B_, _C_, _D_ ... of temperatures intermediate between those of the
bodies _A_, _Z_, and selected so that the differences from _A_ to _B_,
from _B_ to _C_, etc., may all be indefinitely small. The caloric coming
from _A_ would not arrive at _Z_ till after it had passed through the
bodies _B_, _C_, _D_, etc., and after having developed in each of these
stages maximum motive power. The inverse operations would here be
entirely possible, and the reasoning of page 52 would be strictly
applicable.
According to established principles at the present time, we can compare
with sufficient accuracy the motive power of heat to that of a
waterfall. Each has a maximum that we cannot exceed, whatever may be, on
the one hand, the machine which is acted upon by the water, and
whatever, on the other hand, the substance acted upon by the heat. The
motive power of a waterfall depends on its height and on the quantity of
the liquid; the motive power of heat depends also on the quantity of
caloric used, and on what may be termed, on what in fact we will call,
the _height of its fall_,[16] that is to say, the difference of
temperature of the bodies between which the exchange of caloric is made.
In the waterfall the motive power is exactly proportional to the
difference of level between the higher and lower reservoirs. In the fall
of caloric the motive power undoubtedly increases with the difference of
temperature between the warm and the cold bodies; but we do not know
whether it is proportional to this difference. We do not know, for
example, whether the fall of caloric from 100 to 50 degrees furnishes
more or less motive power than the fall of this same caloric from 50 to
zero. It is a question which we propose to examine hereafter.
We shall give here a second demonstration of the fundamental proposition
enunciated on page 56, and present this proposition under a more general
form than the one already given.
When a gaseous fluid is rapidly compressed its temperature rises. It
falls, on the contrary, when it is rapidly dilated. This is one of the
facts best demonstrated by experiment. We will take it for the basis of
our demonstration.[17]
If, when the temperature of a gas has been raised by compression, we
wish to reduce it to its former temperature without subjecting its
volume to new changes, some of its caloric must be removed. This caloric
might have been removed in proportion as pressure was applied, so that
the temperature of the gas would remain constant. Similarly, if the gas
is rarefied we can avoid lowering the temperature by supplying it with a
certain quantity of caloric. Let us call the caloric employed at such
times, when no change of temperature occurs, _caloric due to change of
volume_. This denomination does not indicate that the caloric appertains
to the volume: it does not appertain to it any more than to pressure,
and might as well be called _caloric due to the change of pressure_. We
do not know what laws it follows relative to the variations of volume:
it is possible that its quantity changes either with the nature of the
gas, its density, or its temperature. Experiment has taught us nothing
on this subject. It has only shown us that this caloric is developed in
greater or less quantity by the compression of the elastic fluids.
[Illustration:
FIG. 1.
]
This preliminary idea being established, let us imagine an elastic
fluid, atmospheric air for example, shut up in a cylindrical vessel,
_abcd_ (Fig. 1), provided with a movable diaphragm or piston, _cd_. Let
there be also two bodies, _A_ and _B_, kept each at a constant
temperature, that of _A_ being higher than that of _B_. Let us picture
to ourselves now the series of operations which are to be described:
(1) Contact of the body _A_ with the air enclosed in the space _abcd_ or
with the wall of this space—a wall that we will suppose to transmit the
caloric readily. The air becomes by such contact of the same temperature
as the body _A_; _cd_ is the actual position of the piston.
(2) The piston gradually rises and takes the position _ef_. The body _A_
is all the time in contact with the air, which is thus kept at a
constant temperature during the rarefaction. The body _A_ furnishes the
caloric necessary to keep the temperature constant.
(3) The body _A_ is removed, and the air is then no longer in contact
with any body capable of furnishing it with caloric. The piston
meanwhile continues to move, and passes from the position _ef_ to the
position _gh_. The air is rarefied without receiving caloric, and its
temperature falls. Let us imagine that it falls thus till it becomes
equal to that of the body _B_; at this instant the piston stops,
remaining at the position _gh_.
(4) The air is placed in contact with the body _B_; it is compressed by
the return of the piston as it is moved from the position _gh_ to the
position _cd_. This air remains, however, at a constant temperature
because of its contact with the body _B_, to which it yields its
caloric.
(5) The body _B_ is removed, and the compression of the air is
continued, which being then isolated, its temperature rises. The
compression is continued till the air acquires the temperature of the
body _A_. The piston passes during this time from the position _cd_ to
the position _ik_.
(6) The air is again placed in contact with the body _A_. The piston
returns from the position _ik_ to the position _ef_; the temperature
remains unchanged.
(7) The step described under number 3 is renewed, then successively the
steps 4, 5, 6, 3, 4, 5, 6, 3, 4, 5; and so on.
In these various operations the piston is subject to an effort of
greater or less magnitude, exerted by the air enclosed in the cylinder;
the elastic force of this air varies as much by reason of the changes in
volume as of changes of temperature. But it should be remarked that with
equal volumes, that is, for the similar positions of the piston, the
temperature is higher during the movements of dilatation than during the
movements of compression. During the former the elastic force of the air
is found to be greater, and consequently the quantity of motive power
produced by the movements of dilatation is more considerable than that
consumed to produce the movements of compression. Thus we should obtain
an excess of motive power—an excess which we could employ for any
purpose whatever. The air, then, has served as a heat-engine; we have,
in fact, employed it in the most advantageous manner possible, for no
useless re-establishment of equilibrium has been effected in the
caloric.
All the above-described operations may be executed in an inverse sense
and order. Let us imagine that, after the sixth period, that is to say
the piston having arrived at the position _ef_, we cause it to return to
the position _ik_, and that at the same time we keep the air in contact
with the body _A_. The caloric furnished by this body during the sixth
period would return to its source, that is, to the body _A_, and the
conditions would then become precisely the same as they were at the end
of the fifth period. If now we take away the body _A_, and if we cause
the piston to move from _ef_ to _cd_, the temperature of the air will
diminish as many degrees as it increased during the fifth period, and
will become that of the body _B_. We may evidently continue a series of
operations the inverse of those already described. It is only necessary
under the same circumstances to execute for each period a movement of
dilatation instead of a movement of compression, and reciprocally.
The result of these first operations has been the production of a
certain quantity of motive power and the removal of caloric from the
body _A_ to the body _B_. The result of the inverse operations is the
consumption of the motive power produced and the return of the caloric
from the body _B_ to the body _A_; so that these two series of
operations annul each other, after a fashion, one neutralizing the
other.
The impossibility of making the caloric produce a greater quantity of
motive power than that which we obtained from it by our first series of
operations, is now easily proved. It is demonstrated by reasoning very
similar to that employed at page 56; the reasoning will here be even
more exact. The air which we have used to develop the motive power is
restored at the end of each cycle of operations exactly to the state in
which it was at first found, while, as we have already remarked, this
would not be precisely the case with the vapor of water.[18]
We have chosen atmospheric air as the instrument which should develop
the motive power of heat, but it is evident that the reasoning would
have been the same for all other gaseous substances, and even for all
other bodies susceptible of change of temperature through successive
contractions and dilatations, which comprehends all natural substances,
or at least all those which are adapted to realize the motive power of
heat. Thus we are led to establish this general proposition:
_The motive power of heat is independent of the agents employed to
realize it; its quantity is fixed solely by the temperatures of the
bodies between which is effected, finally, the transfer of the caloric._
We must understand here that each of the methods of developing motive
power attains the perfection of which it is susceptible. This condition
is found to be fulfilled if, as we remarked above, there is produced in
the body no other change of temperature than that due to change of
volume, or, what is the same thing in other words, if there is no
contact between bodies of sensibly different temperatures.
Different methods of realizing motive power may be taken, as in the
employment of different substances, or in the use of the same substance
in two different states—for example, of a gas at two different
densities.
This leads us naturally to those interesting researches on the aeriform
fluids—researches which lead us also to new results in regard to the
motive power of heat, and give us the means of verifying, in some
particular cases, the fundamental proposition above stated.[19]
We readily see that our demonstration would have been simplified by
supposing the temperatures of the bodies _A_ and _B_ to differ very
little. Then the movements of the piston being slight during the periods
3 and 5, these periods might have been suppressed without influencing
sensibly the production of motive power. A very little change of volume
should suffice in fact to produce a very slight change of temperature,
and this slight change of volume may be neglected in presence of that of
the periods 4 and 6, of which the extent is unlimited.
If we suppress periods 3 and 5, in the series of operations above
described, it is reduced to the following:
(1) Contact of the gas confined in _abcd_ (Fig. 2) with the body _A_,
passage of the piston from _cd_ to _ef_.
[Illustration:
FIG. 2. FIG. 3.
]
(2) Removal of the body _A_, contact of the gas confined in _abef_ with
the body _B_, return of the piston from _ef_ to _cd_.
(3) Removal of the body _B_, contact of the gas with the body _A_,
passage of the piston from _cd_ to _ef_, that is, repetition of the
first period, and so on.
The motive power resulting from the _ensemble_ of operations 1 and 2
will evidently be the difference between that which is produced by the
expansion of the gas while it is at the temperature of the body _A_, and
that which is consumed to compress this gas while it is at the
temperature of the body _B_.
Let us suppose that operations 1 and 2 be performed on two gases of
different chemical natures but under the same pressure—under atmospheric
pressure, for example. These two gases will behave exactly alike under
the same circumstances, that is, their expansive forces, originally
equal, will remain always equal, whatever may be the variations of
volume and of temperature, provided these variations are the same in
both. This results obviously from the laws of Mariotte and MM.
Gay-Lussac and Dalton—laws common to all elastic fluids, and in virtue
of which the same relations exist for all these fluids between the
volume, the expansive force, and the temperature.
Since two different gases at the same temperature and under the same
pressure should behave alike under the same circumstances, if we
subjected them both to the operations above described, they should give
rise to equal quantities of motive power.
Now this implies, according to the fundamental proposition that we have
established, the employment of two equal quantities of caloric; that is,
it implies that the quantity of caloric transferred from the body _A_ to
the body _B_ is the same, whichever gas is used.
The quantity of caloric transferred from the body _A_ to the body _B_ is
evidently that which is absorbed by the gas in its expansion of volume,
or that which this gas relinquishes during compression. We are led,
then, to establish the following proposition:
_When a gas passes without change of temperature from one definite
volume and pressure to another volume and another pressure equally
definite, the quantity of caloric absorbed or relinquished is always the
same, whatever may be the nature of the gas chosen as the subject of the
experiment._
Take, for example, 1 litre of air at the temperature of 100° and under
the pressure of one atmosphere. If we double the volume of this air and
wish to maintain it at the temperature of 100°, a certain quantity of
heat must be supplied to it. Now this quantity will be precisely the
same if, instead of operating on the air, we operate upon carbonic-acid
gas, upon nitrogen, upon hydrogen, upon vapor of water or of alcohol,
that is, if we double the volume of 1 litre of these gases taken at the
temperature of 100° and under atmospheric pressure.
It will be the same thing in the inverse sense if, instead of doubling
the volume of gas, we reduce it one half by compression. The quantity of
heat that the elastic fluids set free or absorb in their changes of
volume has never been measured by any direct experiment, and doubtless
such an experiment would be very difficult, but there exists a datum
which is very nearly its equivalent. This has been furnished by the
theory of sound. It deserves much confidence because of the exactness of
the conditions which have led to its establishment. It consists in this:
Atmospheric air should rise one degree Centigrade when by sudden
compression it experiences a reduction of volume of ¹⁄₁₁₆.[20]
Experiments on the velocity of sound having been made in air under the
pressure of 760 millimetres of mercury and at the temperature of 6°, it
is only to these two circumstances that our datum has reference. We
will, however, for greater facility, refer it to the temperature 0°,
which is nearly the same.
Air compressed ¹⁄₁₁₆, and thus heated one degree, differs from air
heated directly one degree only in its density. The primitive volume
being supposed to be _V_, the compression of ¹⁄₁₁₆ reduces it to _V_ −
¹⁄₁₁₆ _V_.
Direct heating under constant pressure should, according to the rule of
M. Gay-Lussac, increase the volume of air ¹⁄₂₆₇ above what it would be
at 0°: so the air is, on the one hand, reduced to the volume _V_ − ¹⁄₁₁₆
_V_; on the other, it is increased to _V_ + ¹⁄₂₆₇ _V_.
The difference between the quantities of heat which the air possesses in
both cases is evidently the quantity employed to raise it directly one
degree; so then the quantity of heat that the air would absorb in
passing from the volume _V_ − ¹⁄₁₁₆ _V_ to the volume _V_ + ¹⁄₂₆₇ _V_ is
equal to that which is required to raise it one degree.
Let us suppose now that, instead of heating one degree the air subjected
to a constant pressure and able to dilate freely, we inclose it within
an invariable space, and that in this condition we cause it to rise one
degree in temperature. The air thus heated one degree will differ from
the air compressed ¹⁄₁₁₆ only by its ¹⁄₁₁₆ greater volume. So then the
quantity of heat that the air would set free by a reduction of volume of
¹⁄₁₁₆ is equal to that which would be required to raise it one degree
Centigrade under constant volume. As the differences between the volumes
_V_ − ¹⁄₁₁₆ _V_, _V_, and _V_ + ¹⁄₂₆₇ _V_ are small relatively to the
volumes themselves, we may regard the quantities of heat absorbed by the
air in passing from the first of these volumes to the second, and from
the first to the third, as sensibly proportional to the changes of
volume. We are then led to the establishment of the following relation:
The quantity of heat necessary to raise one degree air under constant
pressure is to the quantity of heat necessary to raise one degree the
same air under constant volume, in the ratio of the numbers
¹⁄₁₁₆ + ¹⁄₂₆₇ to ¹⁄₁₁₆;
or, multiplying both by 116 × 267, in the ratio of the numbers 267 + 116
to 267.
This, then, is the ratio which exists between the capacity of air for
heat under constant pressure and its capacity under constant volume. If
the first of these two capacities is expressed by unity, the other will
be expressed by the number (267)/(267 + 116), or very nearly 0.700;
their difference, 1 − 0.700 or 0.300, will evidently express the
quantity of heat which will produce the increase of volume in the air
when it is heated one degree under constant pressure.
According to the law of MM. Gay-Lussac and Dalton, this increase of
volume would be the same for all other gases; according to the theory
demonstrated on page 87, the heat absorbed by these equal increases of
volume is the same for all the elastic fluids, which leads to the
establishment of the following proposition:
_The difference between specific heat under constant pressure and
specific heat under constant volume is the same for all gases._
It should be remarked here that all the gases are considered as taken
under the same pressure, atmospheric pressure for example, and that the
specific heats are also measured with reference to the volumes.
It is a very easy matter now for us to prepare a table of the specific
heat of gases under constant volume, from the knowledge of their
specific heats under constant pressure. Here is the table:
TABLE OF THE SPECIFIC HEAT OF GASES.
───────────────────────┬───────────────────────┬───────────────────────
NAMES OF GASES. │ Specific Heat under │Specific Heat at Const.
│ Const. Press. │ Vol.
───────────────────────┼───────────────────────┼───────────────────────
Atmospheric Air, │ 1.000 │ 0.700
Hydrogen Gas, │ 0.903 │ 0.603
Carbonic Acid, │ 1.258 │ 0.958
Oxygen, │ 0.976 │ 0.676
Nitrogen, │ 1.000 │ 0.700
Protoxide of Nitrogen, │ 1.350 │ 1.050
Olefiant Gas, │ 1.553 │ 1.253
Oxide of Carbon, │ 1.034 │ 0.734
───────────────────────┴───────────────────────┴───────────────────────
The first column is the result of the direct experiments of MM.
Delaroche and Bérard on the specific heat of the gas under atmospheric
pressure, and the second column is composed of the numbers of the first
diminished by 0.300.
The numbers of the first column and those of the second are here
referred to the same unit, to the specific heat of atmospheric air under
constant pressure.
The difference between each number of the first column and the
corresponding number of the second being constant, the relation between
these numbers should be variable. Thus the relation between the specific
heat of gases under constant pressure and the specific heat at constant
volume, varies in different gases.
We have seen that air when it is subjected to a sudden compression of
¹⁄₁₁₆ of its volume rises one degree in temperature. The other gases
through a similar compression should also rise in temperature. They
should rise, but not equally, in inverse ratio with their specific heat
at constant volume. In fact, the reduction of volume being by hypothesis
always the same, the quantity of heat due to this reduction should
likewise be always the same, and consequently should produce an
elevation of temperature dependent only on the specific heat acquired by
the gas after its compression, and evidently in inverse ratio with this
specific heat. Thus we can easily form the table of the elevations of
temperature of the different gases for a compression of ¹⁄₁₁₆.
TABLE OF THE ELEVATION OF TEMPERATURE<BR>OF
_Gases through the Effect of Compression_.
──────────────────────┬────────────────────────────────────────────────
NAMES OF GASES. │ Elevation of Temperature for a Reduction of
│ Volume of ¹⁄₁₁₆.
──────────────────────┼────────────────────────────────────────────────
│ °
Atmospheric Air, │ 1.000
Hydrogen Gas, │ 1.160
Carbonic Acid, │ 0.730
Oxygen, │ 1.035
Nitrogen, │ 1.000
Protoxide of Nitrogen,│ 0.667
Olefiant Gas, │ 0.558
Carbonic Oxide, │ 0.955
──────────────────────┴────────────────────────────────────────────────
A second compression of ¹⁄₁₁₆ (of the altered volume), as we shall
presently see, would also raise the temperature of these gases nearly as
much as the first; but it would not be the same with a third, a fourth,
a hundredth such compression. The capacity of gases for heat changes
with their volume. It is not unlikely that it changes also with the
temperature.
We shall now deduce from the general proposition stated on page 68 a
second theory, which will serve as a corollary to that just
demonstrated.
Let us suppose that the gas enclosed in the cylindrical space _abcd_
(Fig. 2) be transported into the space _a′b′c′d′_ (Fig. 3) of equal
height, but of different base and wider. This gas would increase in
volume, would diminish in density and in elastic force, in the inverse
ratio of the two volumes _abcd_, _a′b′c′d′_. As to the total pressure
exerted in each piston _cd_, _c′d′_, it would be the same from all
quarters, for the surface of these pistons is in direct ratio to the
volumes.
Let us suppose that we perform on the gas inclosed in _a′b′c′d′_ the
operations described on page 70, and which were taken as having been
performed upon the gas inclosed in _abcd_; that is, let us suppose that
we have given to the piston _c′d′_ motions equal to those of the piston
_cd_, that we have made it occupy successively the positions _c′d′_
corresponding to _cd_, and _e′f′_ corresponding to _ef_, and that at the
same time we have subjected the gas by means of the two bodies _A_ and
_B_ to the same variations of temperature as when it was inclosed in
_abcd_. The total effort exercised on the piston would be found to be,
in the two cases, always the same at the corresponding instants. This
results solely from the law of Mariotte.[21] In fact, the densities of
the two gases maintaining always the same ratio for similar positions of
the pistons, and the temperatures being always equal in both, the total
pressures exercised on the pistons will always maintain the same ratio
to each other. If this ratio is, at any instant whatever, unity, the
pressures will always be equal.
As, furthermore, the movements of the two pistons have equal extent, the
motive power produced by each will evidently be the same; whence we
should conclude, according to the proposition on page 68, that the
quantities of heat consumed by each are the same, that is, that there
passes from the body _A_ to the body _B_ the same quantity of heat in
both cases.
The heat abstracted from the body _A_ and communicated to the body _B_,
is simply the heat absorbed during the rarefaction of the gas, and
afterwards liberated by its compression. We are therefore led to
establish the following theorem:
_When an elastic fluid passes without change of temperature from the
volume U to the volume V, and when a similar ponderable quantity of the
same gas passes at the same temperature from the volume U′ to the volume
V′, if the ratio of U′ to V′ is found to be the same as the ratio of U
to V, the quantities of heat absorbed or disengaged in the two cases
will be equal._
This theorem might also be expressed as follows:
_When a gas varies in volume without change of temperature, the
quantities of heat absorbed or liberated by this gas are in arithmetical
progression, if the increments or the decrements of volume are found to
be in geometrical progression._
When a litre of air maintained at a temperature of ten degrees is
compressed, and when it is reduced to one half a litre, a certain
quantity of heat is set free. This quantity will be found always the
same if the volume is further reduced from a half litre to a quarter
litre, from a quarter litre to an eighth, and so on.
If, instead of compressing the air, we carry it successively to two
litres, four litres, eight litres, etc., it will be necessary to supply
to it always equal quantities of heat in order to maintain a constant
temperature.
This readily accounts for the high temperature attained by air when
rapidly compressed. We know that this temperature inflames tinder and
even makes air luminous. If, for a moment, we suppose the specific heat
of air to be constant, in spite of the changes of volume and
temperature, the temperature will increase in arithmetical progression
for reduction of volume in geometrical progression.
Starting from this datum, and admitting that one degree of elevation in
the temperature corresponds to a compression of ¹⁄₁₁₆, we shall readily
come to the conclusion that air reduced to ¹⁄₁₄ of its primitive volume
should rise in temperature about 300 degrees, which is sufficient to
inflame tinder.[22]
The elevation of temperature ought, evidently, to be still more
considerable if the capacity of the air for heat becomes less as its
volume diminishes. Now this is probable, and it also seems to follow
from the experiments of MM. Delaroche and Bérard on the specific heat of
air taken at different densities. (See the Mémoire in the _Annales de
Chimie_, t. lxxxv. pp. 72, 224.)
The two theorems explained on pp. 72 and 81 suffice for the comparison
of the quantities of heat absorbed or set free in the changes of volume
of elastic fluids, whatever may be the density and the chemical nature
of these fluids, provided always that they be taken and maintained at a
certain invariable temperature. But these theories furnish no means of
comparing the quantities of heat liberated or absorbed by elastic fluids
which change in volume at different temperatures. Thus we are ignorant
what relation exists between the heat relinquished by a litre of air
reduced one half, the temperature being kept at zero, and the heat
relinquished by the same litre of air reduced one half, the temperature
being kept at 100°. The knowledge of this relation is closely connected
with that of the specific heat of gases at various temperatures, and to
some other data that Physics as yet does not supply.
The second of our theorems offers us a means of determining according to
what law the specific heat of gases varies with their density.
Let us suppose that the operations described on p. 70, instead of being
performed with two bodies, _A_, _B_, of temperatures differing
indefinitely small, were carried on with two bodies whose temperatures
differ by a finite quantity—one degree, for example. In a complete
circle of operations the body _A_ furnishes to the elastic fluid a
certain quantity of heat, which may be divided into two portions: (1)
That which is necessary to maintain the temperature of the fluid
constant during dilatation; (2) that which is necessary to restore the
temperature of the fluid from that of the body _B_ to that of the body
_A_, when, after having brought back this fluid to its primitive volume,
we place it again in contact with the body _A_. Let us call the first of
these quantities _a_ and the second _b_. The total caloric furnished by
the body A will be expressed by _a_ + _b_.
The caloric transmitted by the fluid to the body _B_ may also be divided
into two parts: one, _b′_, due to the cooling of the gas by the body
_B_; the other, _a′_, which the gas abandons as a result of its
reduction of volume. The sum of these two quantities is _a′_ + _b′_; it
should be equal to _a_ + _b_, for, after a complete cycle of operations,
the gas is brought back exactly to its primitive state. It has been
obliged to give up all the caloric which has first been furnished to it.
We have then
_a_ + _b_ = _a′_ + _b′_;
or rather,
_a_ − _a′_ = _b′_ − _b_.
Now, according to the theorem given on page 81, the quantities _a_ and
_a′_ are independent of the density of the gas, provided always that the
ponderable quantity remains the same and that the variations of volume
be proportional to the original volume. The difference _a_ − _a′_ should
fulfil the same conditions, and consequently also the difference _b′_ −
_b_, which is equal to it. But _b′_ is the caloric necessary to raise
the gas enclosed in _abcd_ (Fig. 2) one degree; _b′_ is the caloric
surrendered by the gas when, enclosed in _abef_, it is cooled one
degree. These quantities may serve as a measure for specific heats. We
are then led to the establishment of the following proposition:
_The change in the specific heat of a gas caused by change of volume
depends entirely on the ratio between the original volume and the
altered volume._ That is, the difference of the specific heats does not
depend on the absolute magnitude of the volumes, but only on their
ratio.
This proposition might also be differently expressed, thus:
_When a gas increases in volume in geometrical progression, its specific
heat increases in arithmetical progression._
Thus, _a_ being the specific heat of air taken at a given density, and
_a_ + _h_ the specific heat for a density one half less, it will be, for
a density equal to one quarter, _a_ + 2_h_; for a density equal to one
eighth, _a_ + 3_h_; and so on.
The specific heats are here taken with reference to weight. They are
supposed to be taken at an invariable volume, but, as we shall see, they
would follow the same law if they were taken under constant pressure.
To what cause is the difference between specific heats at constant
volume and at constant pressure really due? To the caloric required to
produce in the second case increase of volume. Now, according to the law
of Mariotte, increase of volume of a gas should be, for a given change
of temperature, a determined fraction of the original volume, a fraction
independent of pressure. According to the theorem expressed on page 76,
if the ratio between the primitive volume and the altered volume is
given, that determines the heat necessary to produce increase of volume.
It depends solely on this ratio and on the weight of the gas. We must
then conclude that:
_The difference between specific heat at constant pressure and specific
heat at constant volume is always the same, whatever may be the density
of the gas, provided the weight remains the same._
These specific heats both increase accordingly as the density of the gas
diminishes, but their difference does not vary.[23]
Since the difference between the two capacities for heat is constant, if
one increases in arithmetical progression the other should follow a
similar progression: thus one law is applicable to specific heats at
constant pressure.
We have tacitly assumed the increase of specific heat with that of
volume. This increase is indicated by the experiments of MM. Delaroche
and Bérard: in fact these physicists have found 0.967 for the specific
heat of air under the pressure of 1 metre of mercury (see Mémoire
already cited), taking for the unit the specific heat of the same weight
of air under the pressure of 0^m.760.
According to the law that specific heats follow with relation to
pressures, it is only necessary to have observed them in two particular
cases to deduce them in all possible cases: it is thus that, making use
of the experimental result of MM. Delaroche and Bérard which has just
been given, we have prepared the following table of the specific heat of
air under different pressures:
SPECIFIC HEAT OF AIR.
────────────────────────┬────────────────────────────────────────────
Pressure in Atmospheres.│Specific Heat, that of Air under Atmospheric
│ Pressure being 1.
────────────────────────┼────────────────────────────────────────────
¹⁄₁₀₂₄ │ 1.840
¹⁄₅₁₂ │ 1.756
¹⁄₂₅₆ │ 1.672
¹⁄₁₂₈ │ 1.588
¹⁄₆₄ │ 1.504
¹⁄₃₂ │ 1.420
¹⁄₁₆ │ 1.336
⅛ │ 1.252
¼ │ 1.165
½ │ 1.084
1 │ 1.000
2 │ 0.916
4 │ 0.832
8 │ 0.748
16 │ 0.664
32 │ 0.580
64 │ 0.496
128 │ 0.412
256 │ 0.328
512 │ 0.244
1024 │ 0.160
────────────────────────┴────────────────────────────────────────────
The first column is, as we see, a geometrical progression, and the
second an arithmetical progression.
We have carried out the table to the extremes of compression and
rarefaction. It may be believed that air would be liquefied before
acquiring a density 1024 times its normal density, that is, before
becoming more dense than water. The specific heat would become zero and
even negative on extending the table beyond the last term. We think,
furthermore, that the figures of the second column here decrease too
rapidly. The experiments which serve as a basis for our calculation have
been made within too contracted limits for us to expect great exactness
in the figures which we have obtained, especially in the outside
numbers.
Since we know, on the one hand, the law according to which heat is
disengaged in the compression of gases, and on the other, the law
according to which specific heat varies with volume, it will be easy for
us to calculate the increase of temperature of a gas that has been
compressed without being allowed to lose heat. In fact, the compression
may be considered as composed of two successive operations: (1)
compression at a constant temperature; (2) restoration of the caloric
emitted. The temperature will rise through the second operation in
inverse ratio with the specific heat acquired by the gas after the
reduction of volume,—specific heat that we are able to calculate by
means of the law demonstrated above. The heat set free by compression,
according to the theorem of page 81, ought to be represented by an
expression of the form
_s_ = _A_ + _B_ log _v_,
_s_ being this heat, _v_ the volume of the gas after compression, _A_
and _B_ arbitrary constants dependent on the primitive volume of the
gas, on its pressure, and on the units chosen.
The specific heat varying with the volume according to the law just
demonstrated, should be represented by an expression of the form
_z_ = _A′_ + _B′_ log _v_,
_A′_ and _B′_ being the different arbitrary constants of _A_ and _B_.
The increase of temperature acquired by the gas, as the effect of
compression, is proportional to the ratio (_s_)/(_z_) or to the relation
(_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_). It can be represented by this
ratio itself; thus, calling it _t_, we shall have
_t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_).
If the original volume of the gas is 1, and the original temperature
zero, we shall have at the same time _t_ = 0, log _v_ = 0, whence _A_ =
0; _t_ will then express not only the increase of temperature, but the
temperature itself above the thermometric zero.
We need not consider the formula that we have just given as applicable
to very great changes in the volume of gases. We have regarded the
elevation of temperature as being in inverse ratio to the specific heat;
which tacitly supposes the specific heat to be constant at all
temperatures. Great changes of volume lead to great changes of
temperature in the gas, and nothing proves the constancy of specific
heat at different temperatures, especially at temperatures widely
separated. This constancy is only an hypothesis admitted for gases by
analogy, to a certain extent verified for solid bodies and liquids
throughout a part of the thermometric scale, but of which the
experiments of MM. Dulong and Petit have shown the inaccuracy when it is
desirable to extend it to temperatures far above 100°.[24]
According to a law of MM. Clement and Desormes, a law established by
direct experiment, the vapor of water, under whatever pressure it may be
formed, contains always, at equal weights, the same quantity of heat;
which leads to the assertion that steam, compressed or expanded
mechanically without loss of heat, will always be found in a saturated
state if it was so produced in the first place. The vapor of water so
made may then be regarded as a permanent gas, and should observe all the
laws of one. Consequently the formula
_t_ = (_A_ + _B_ log _v_)/(_A′_ + _B′_ log _v_)
should be applicable to it, and be found to accord with the table of
tensions derived from the direct experiments of M. Dalton.
We may be assured, in fact, that our formula, with a convenient
determination of arbitrary constants, represents very closely the
results of experiment. The slight irregularities which we find therein
do not exceed what we might reasonably attribute to errors of
observation.[25]
We will return, however, to our principal subject, from which we have
wandered too far—the motive power of heat.
We have shown that the quantity of motive power developed by the
transfer of caloric from one body to another depends essentially upon
the temperature of the two bodies, but we have not shown the relation
between these temperatures and the quantities of motive power produced.
It would at first seem natural enough to suppose that for equal
differences of temperature the quantities of motive power produced are
equal; that is, for example, the passage of a given quantity of caloric
from a body, _A_, maintained at 100°, to a body, _B_, maintained at 50°,
should give rise to a quantity of motive power equal to that which would
be developed by the transfer of the same caloric from a body, _B_, at
50°, to a body, _C_, at zero. Such a law would doubtless be very
remarkable, but we do not see sufficient reason for admitting it _à
priori_. We will investigate its reality by exact reasoning.
Let us imagine that the operations described on p. 70 be conducted
successively on two quantities of atmospheric air equal in weight and
volume, but taken at different temperatures. Let us suppose, further,
the differences of temperature between the bodies _A_ and _B_ equal, so
these bodies would have for example, in one of these cases, the
temperatures 100° and 100° − _h_ (_h_ being indefinitely small), and in
the other 1° and 1° − _h_. The quantity of motive power produced is, in
each case, the difference between that which the gas supplies by its
dilatation and that which must be expended to restore its primitive
volume. Now this difference is the same in both cases, as any one can
prove by simple reasoning, which it seems unnecessary to give here in
detail; hence the motive power produced is the same.
Let us now compare the quantities of heat employed in the two cases. In
the first, the quantity of heat employed is that which the body _A_
furnishes to the air to maintain it at the temperature of 100° during
its expansion. In the second, it is the quantity of heat which this same
body should furnish to it, to keep its temperature at one degree during
an exactly similar change of volume. If these two quantities of heat
were equal, there would evidently result the law that we have already
assumed. But nothing proves that it is so, and we shall find that these
quantities are not equal.
The air that we shall first consider as occupying the space _abcd_ (Fig.
2), and having 1 degree of temperature, can be made to occupy the space
_abef_, and to acquire the temperature of 100 degrees by two different
means:
(1) We may heat it without changing its volume, then expand it, keeping
its temperature constant.
(2) We may begin by expanding it, maintaining the temperature constant,
then heat it, when it has acquired its greater volume.
Let _a_ and _b_ be the quantities of heat employed successively in the
first of the two operations, and let _b′_ and _a′_ be the quantities of
heat employed successively in the second. As the final result of these
two operations is the same, the quantities of heat employed in both
should be equal. We have then
_a_ + _b_ = _a′_ + _b′_,
whence
_a′_ − _a_ = _b_ − _b′_.
_a′_ is the quantity of heat required to cause the gas to rise from 1°
to 100° when it occupies the space _abef_.
_a_ is the quantity of heat required to cause the gas to rise from 1° to
100° when it occupies the space _abcd_.
The density of the air is less in the first than in the second case, and
according to the experiments of MM. Delaroche and Bérard, already cited
on page 87, its capacity for heat should be a little greater.
The quantity _a′_ being found to be greater than the quantity _a_, _b_
should be greater than _b′_. Consequently, generalizing the proposition,
we should say:
_The quantity of heat due to the change of volume of a gas is greater as
the temperature is higher._
Thus, for example, more caloric is necessary to maintain at 100° the
temperature of a certain quantity of air the volume of which is doubled,
than to maintain at 1° the temperature of this same air during a
dilatation exactly equal.
These unequal quantities of heat would produce, however, as we have
seen, equal quantities of motive power for equal fall of caloric taken
at different heights on the thermometric scale; whence we draw the
following conclusion:
_The fall of caloric produces more motive power at inferior than at
superior temperatures._
Thus a given quantity of heat will develop more motive power in passing
from a body kept at 1 degree to another maintained at zero, than if
these two bodies were at the temperature of 101° and 100°.
The difference, however, should be very slight. It would be nothing if
the capacity of the air for heat remained constant, in spite of changes
of density. According to the experiments of MM. Delaroche and Bérard,
this capacity varies little—so little even, that the differences noticed
might strictly have been attributed to errors of observation or to some
circumstances of which we have failed to take account.
We are not prepared to determine precisely, with no more experimental
data than we now possess, the law according to which the motive power of
heat varies at different points on the thermometric scale. This law is
intimately connected with that of the variations of the specific heat of
gases at different temperatures—a law which experiment has not yet made
known to us with sufficient exactness.[26]
We will endeavor now to estimate exactly the motive power of heat, and
in order to verify our fundamental proposition, in order to determine
whether the agent used to realize the motive power is really unimportant
relatively to the quantity of this power, we will select several of them
successively: atmospheric air, vapor of water, vapor of alcohol.
Let us suppose that we take first atmospheric air. The operation will
proceed according to the method indicated on page 70. We will make the
following hypotheses: The air is taken under atmospheric pressure. The
temperature of the body _A_ is ¹⁄₁₀₀₀ of a degree above zero, that of
the body _B_ is zero. The difference is, as we see, very slight—a
necessary condition here.
The increase of volume given to the air in our operation will be ¹⁄₁₁₆ +
¹⁄₂₆₇ of the primitive volume; this is a very slight increase,
absolutely speaking, but great relatively to the difference of
temperature between the bodies _A_ and _B_.
The motive power developed by the whole of the two operations described
(page 70) will be very nearly proportional to the increase of volume and
to the difference between the two pressures exercised by the air, when
it is found at the temperatures 0°.001 and zero.
This difference is, according to the law of M. Gay-Lussac, ¹⁄₂₆₇₀₀₀ of
the elastic force of the gas, or very nearly ¹⁄₂₆₇₀₀₀ of the atmospheric
pressure.
The atmospheric pressure balances at 10.40 metres head of water;
¹⁄₂₆₇₀₀₀ of this pressure equals ¹⁄₂₆₇₀₀₀ × 10^m.40 of head of water.
As to the increase of volume, it is, by supposition, ¹⁄₁₁₆ + ¹⁄₂₆₇ of
the original volume, that is, of the volume occupied by one kilogram of
air at zero, a volume equal to 0^{mc}.77, allowing for the specific
weight of the air. So then the product,
(¹⁄₁₁₆ + ¹⁄₂₆₇) × 0.77 × ¹⁄₂₆₇₀₀₀ × 10.40,
will express the motive power developed. This power is estimated here in
cubic metres of water raised one metre.
If we carry out the indicated multiplications, we find the value of the
product to be 0.000000372.
Let us endeavor now to estimate the quantity of heat employed to give
this result; that is, the quantity of heat passed from the body _A_ to
the body _B_.
The body _A_ furnishes:
(1) The heat required to carry the temperature of one kilogram of air
from zero to 0°.001;
(2) The quantity necessary to maintain at this temperature the
temperature of the air when it experiences a dilatation of
¹⁄₁₁₆ + ¹⁄₂₆₇.
The first of these quantities of heat being very small in comparison
with the second, we may disregard it. The second is, according to the
reasoning on page 74, equal to that which would be necessary to increase
one degree the temperature of one kilogram of air subjected to
atmospheric pressure.
According to the experiments of MM. Delaroche and Bérard on the specific
heat of gases, that of air is, for equal weights, 0.267 that of water.
If, then, we take for the unit of heat the quantity necessary to raise 1
kilogram of water 1 degree, that which will be required to raise 1
kilogram of air 1 degree would have for its value 0.267. Thus the
quantity of heat furnished by the body _A_ is
0.267 units.
This is the heat capable of producing 0.000000372 units of motive power
by its fall from 0°.001 to zero.
For a fall a thousand times greater, for a fall of one degree, the
motive power will be very nearly a thousand times the former, or
0.000372.
If, now, instead of 0.267 units of heat we employ 1000 units, the motive
power produced will be expressed by the proportion
(0.267)/(0.000372) = (1000)/(x), whence x = (372)/(267) = 1.395.
Thus 1000 units of heat passing from a body maintained at the
temperature of 1 degree to another body maintained at zero would
produce, in acting upon the air,
1.395 units of motive power.
We will now compare this result with that furnished by the action of
heat on the vapor of water.
[Illustration:
FIG. 4.
]
Let us suppose one kilogram of liquid water enclosed in the cylindrical
vessel _abcd_ (Fig. 4), between the bottom _ab_ and the piston _cd_. Let
us suppose, also, the two bodies _A_, _B_ maintained each at a constant
temperature, that of _A_ being a very little above that of _B_. Let us
imagine now the following operations:
(1) Contact of the water with the body _A_, movement of the piston from
the position _cd_ to the position _ef_, formation of steam at the
temperature of the body _A_ to fill the vacuum produced by the extension
of volume. We will suppose the space _abef_ large enough to contain all
the water in a state of vapor.
(2) Removal of the body _A_, contact of the vapor with the body _B_,
precipitation of a part of this vapor, diminution of its elastic force,
return of the piston from _ef_ to _ab_, liquefaction of the rest of the
vapor through the effect of the pressure combined with the contact of
the body _B_.
(3) Removal of the body _B_, fresh contact of the water with the body
_A_, return of the water to the temperature of this body, renewal of the
former period, and so on.
The quantity of motive power developed in a complete cycle of operations
is measured by the product of the volume of the vapor multiplied by the
difference between the tensions that it possesses at the temperature of
the body _A_ and at that of the body _B_. As to the heat employed, that
is to say, transported from the body _A_ to the body _B_, it is
evidently that which was necessary to turn the water into vapor,
disregarding always the small quantity required to restore the
temperature of the liquid water from that of _B_ to that of _A_.
Suppose the temperature of the body _A_ 100 degrees, and that of the
body _B_ 99 degrees: the difference of the tensions will be, according
to the table of M. Dalton, 26 millimetres of mercury or 0^m.36 head of
water.
The volume of the vapor is 1700 times that of the water. If we operate
on one kilogram, that will be 1700 litres, or 1^{mc}.700.