**p. **71

The first column** **is
the result of the direct experiments of MM. Delaroche and Berard
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 an 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. (See table page 72.)

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

p. 72

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 64 a second theory, which will serve as a corollary
to that just demonstrated.

Table

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 65, 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

}

**p. **73*
*

to those of the piston cd, that we have made it occupy
successively the positions c'd' corresponding to cd, and ef*
*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.* 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 64, 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 law of Mariotte, which is here made the foundation
upon which to establish our demonstration, is one of the best
authenticated physical laws. It has served as a basis to many
theories verified by experience, and which in turn verify all
the laws on which they are founded. We can cite also, as a valuable
verification of Mariotte's law and also of that of MM. Gay-Lussac
and Dalton, for a great difference of temperature, the experiments
of MM. Dulong and Petit. (See *Annals *de *Chimie *et
*Physique , *Feb. 1818, t. vii. p.122.)

The more recent experiments of Davy and Faraday can also be cited.

The theories that we deduce here would not perhaps
be exact if applied outside of certain limits either of density
or temperature. They should be regarded as true only within the
limits in which the laws of Mariotte and of MM. Gay-Lussac and
Dalton are themselves proven.

p. 74

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 in-

**p. ***75
*

crease 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.*

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 Berard on the specific heat of air taken at different densities.
(See the Memoire in the *Annals de Chimie*,* *t. 'XXXV.
pp.72,224.)

The two theorems explained on pp.67 and 74 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.

*** When the volume is reduced 1/116, that is, when
it becomes 115/116 of what it was at first, the temperature rises
one degree. Another reduction of 1/116 carries the volume to (115/116)^{2}:,
and the temperature should rise another degree. After x similar
reductions the volume becomes (115/116)^{x}, and the temperature
should be raised x degrees. If we suppose (115/116)^{x}-=1/14
and if we take the logarithms of both, we find**

**x = about 300**

**If we suppose (115/116)^{x} =1/2, we
find**

**x = 80^{0};**

**which shows that air compressed one half rises
80^{0}**

**All this is subject to the hypothesis that the
specific heat of air does not change, although the volume diminishes.
But if, for the reasons hereafter given (pp.77,79), we regard
the specific heat of air compressed one half as reduced in the
relation of 700 to 616, the number 80^{0} must be multiplied
by 700/616, which raises it to 90^{0.
}**

p. 76

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 page
65, in-stead 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 bodyA
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 dilation;
(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

I

**p. 77
**

or rather,

Now, according to the theorem given on page 74, 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 fulfill 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.
*

p. 78

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 + 2h for a density equal to one eighth, a + 3h; 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 70, 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.*

** MM. Gay~lussac and Welter have found by direct
experiments, cited in the Mecanique Celeste and in the
Annales & Chimie et de Physiquc, July, 1822, p.267,
that the ratio between the specific heat at constant pressure
and the specific heat at constant volume varies very little with
the density of the gas, According to what we have just seen, the
difference should remain constant,*

p. 79

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 Berard: in fact these physicists have found
0.967 for the specific heat of air under the pressure of 1 metre
of mercury (see Memoire already cited), taking for the unit the
specific heat of the same weight of air under the pressure of
O^{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 Berard which has just been given, we have prepared
the following table of the specific heat of air under different
pressures:

and not the ratio. As, further, the specific heat of gases for a given weight varies very little with the density, it is evident that the ratio itself experiences but alight changes.

The ratio between the specific heat of atmospheric
air at constant pressure and at constant volume is, according
to MM. Gay~lussac and Welter, 1.3748, a number almost constant**
***for *all press'ures, and even for all temperatures.
We

__267 + 116__

have come, through other considerations, to the number = 1.44,

257

which differs from the former 1/20, and we have used
this number to prepare a table of the specific heats of gases
at constant volume. So we need not regard this table as very exact,
any more than the table given on p.80. These tables are mainly
intended to demonstrate the laws governing specific heats of aeriform
fluids.

- 80

**Table
**

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 m 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

p. 81

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 74, ought to be represented by an expression of the form

**I)**

*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

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*

ing it *t, *we shall have

t= __A__ __+Blogv__

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

*p. 82 *

*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.**

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

*~= A+Blogv A' + *B'logv

should be applicable to it, and be found to accord with the table of tensions derived from the direct experiments of M. Dalton.

**I 4
**

**MOTIVE POWER OF HEAT **83

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.

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 a priori. We will investigate its reality by exact reasoning.

Let us imagine that the operations described on page 65 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

**84 REFLECTIONS ON 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 anyone 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 one 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

**I
**

**I**

**4
**

*****

{þÿ____

**MOTIVE POWER OF HEAT **85

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 10 to **1000 **when
it occupies the space abef.

a is the quantity of heat required to cause the gas
to rise from 10 to **1000 **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 Berard, already cited on page 78, 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 **10 **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.
*

86 **REFLECTIONS ON THE
**

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^{0}
and 100^{0}.

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 Berard, 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. 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 65. We will make the following hypotheses: The air is taken
under atmospheric pressure. The temperature of the body A is 1/1000

**MOTIVE POWER OF HEAT **87

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 65) 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^{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 X lO^{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 O^{me}.77,
allowing for the specific weight of the air. So then the product,

(1/116 + 1/267 ) X 0.77 X 1/267000 X 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
em-

88 **REFLECTIONS ON THE
**

ployed 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^{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 68, 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
Berard 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^{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.

**MOTIVE POWER OF HEAT **89

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.

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 *Fig. 4*

large enough to contain all the water in a state of vapor.

- 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*.*

90 REFLECTIONS ON THE

(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 O^{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 l^{me}.700.

Thus the value of the motive power developed is the
product

1.700 x 0.36 = 0.611 units,

of the kind of which we have previously made use.

The quantity of heat employed is the quantity required
to turn into vapor water already heated to 100^{0}. This
quantity is found by experiment. We have found it equal to 550^{0}**,
**or, to speak more exactly, to 550 of our
units of heat.

Thus 0.611 units of motive power result from the
employment of 550 units of heat. The quantity of motive power
result-

ing from 1000 units of beat will be given by the
proportion

550/0.611 = 1000/x, whence x = 611/550 = 1.112.

Thus 1000 units of heat transported from one body kept at 100 degrees to another kept at 99 degrees will produce, acting upon vapor of water, 1.112 units of motive power.

The number 1.112 differs by about 1/4 from the number
1.395 previously found for the value of the motive power developed
by 1000 units of heat acting upon the air; but it should be observed
that in this case the temperatures of the bodies A* *and
B* *were 1 degree and zero, while here they are 100 degrees
and 99 degrees. The difference is much the same; but it is not
found at the same height in the thermometric scale. To make an
exact comparison, it would have been necessary to estimate the
motive power developed by the steam formed at 1 degree and condensed
at zero. It would also have been necessary to know the quantity
of heat contained in the steam formed at one degree.

The law of MM. Clement and Desormes referred to on page 82 gives us this datum. The constituent heat of vapor of water being always the same at any temperature at which vaporization takes place, if 550 degrees of heat are required to vaporize water already brought up to 100 degrees, 550 + 100 or 650 will be required to vaporize the same weight of water taken at zero.

Making use of this datum and reasoning exactly as we did for water at 100 degrees, we find, as is easily seen,

for the motive power developed by 1000 units of heat acting upon the vapor of water between one degree and zero. This num-

`92 `**REFLECTIONS ON THE
**

ber approximates more closely than the first to

1.395.

It differs from it only 1/13, an error which does
not exceed probable limits, considering the great number of data
of different sorts of which we have been obliged to make use in
order to arrive at this approximation. Thus is our fundamental
law verified a* *special case.*

We will examine another case in which vapor of alcohol
is acted upon by heat. The reasoning is precisely the same as
for the vapor of water. The data alone are changed. Pure alcohol
boils under ordinary pressure at 78^{0}.7 Centigrade.
One kilogram absorbs, according to MM. Delaroche and Berard, 207
units of heat in undergoing transformation into vapor at this
same temperature, 78.^{0}.7.

The tension of the vapor of alcohol at one degree
below the boiling-point is found to be diminished 1/25. It is
1/25 less than the atmospheric pressure; at least, this is the
result of the experiment of M. Betancour reported in the second
part of *l'Architecture hydraulique *of M. Prony, pp.180,
195.

If* *we use these data, we find that, in acting
upon one kilogram of** **alcohol
at the temperatures of 78^{0}.7 and
77^{0}.7, the motive power developed will be 0.251 units.

This results from the employment of 207 units of
heat. For 1000 units the proportion must be

*We** **find *(Annales de Chimie et de Physique,
July, *1818, p.294) in a memoir of M. Petit an estimate of
the motive power of heat applied to air and to vapor of water.
This estimate leads us to attribute a great advantage to atmospheric
air, but it is derived by a method of considering the action of
heat which is quite imperfect.

**MOTIVE POWER OF HEAT **93

This** **number
is a little more than the 1.112 resulting from the use of the
vapor of water at the temperatures 100^{0}**
**and 99^{0}; if we suppose the
vapor of water used at the temperatures 78^{0}
and 77^{0} we find, according to the
law of MM. Clement and Desorme, 1.212 for the motive power due
to 1000 units of heat. This latter number approaches, as we see,
very nearly to 1.230. There is a difference of ouly 1/50.

We should have liked to be able to make other approximations of this sort-to be able to calculate, for example, the motive power developed by the action of heat on solids and liquids, by the congelation of water, and so on; but Physics as yet refuses us the necessary data.*

The fundamental law that we propose to confirm seems to us to require, however, in order to be placed beyond doubt, new verifications. It is based upon the theory of heat as it is understood to-day, and it should be said that this foundation does not appear to be of unquestionable solidity. New experiments alone can decide the question. Meanwhile we can apply the theoretical ideas expressed above, regarding them as exact, to the examination of the different methods proposed up to date, for the realization of the motive power of heat.

It has sometimes been proposed to develop motive
power by the action of heat on solid bodies. The mode of procedure
which naturally first occurs to the mind is to fasten immovably
a solid body-a metallic bar, for example by one of its extremities;
to attach the other extremity to a movable part of the machine;

* Those that we need are the expansive force acquired
by Solids and liquids by a given increase of temperature, and
the quantity of heat absorbed or relinquished in the changes of
volume of these bodies.

94 **REFLECTIONS ON THE
**

then, by successive heating and cooling, to cause the length of the bar to vary, and so to produce motion. Let us try to decide whether this method of developing motive power can be advantageous. We have shown that the condition of the most effective employment of heat in the production of motion is, that all changes of temperature occurring in the bodies should be due to changes of volume. The nearer we come to fulfilling this condition the more fully will the heat be utilized. Now, working in the manner just described, we are very far from fulfilling this condition: change of temperature is not due here to change of volume; all the changes are due to contact of bodies differently heated-to the contact of the mettallic bar, either with the body charged with furnishing heat to it, or with the body charged with carrying it off.

The only means of fulfilling the prescribed condition would be to act upon the solid body exactly as we did on the air in the operations described on page 82. But for this we must be able to produce, by a single change of volume of the solid body, considerable changes of temperature, that is, if we should want to utilize considerable falls of caloric. Now this appears impracticable. In short, many considerations lead to the conclusion that the changes produced in the temperature of solid or liquid bodies through the effect of compression and rarefaction would be but slight.

(1) We often observe in machines (particularly in
steam-engines) solid pieces which endure considerable strain in
one way or another, and although these efforts may be sometimes
as great as the nature of the substances employed permits, the
variations of temperature are scarcely perceptible.

**MOTIVE POWER OF HEAT **95

(2) In the action of striking medals, in that of
the rolling-mill, of the draw-plate, the metals undergo the greatest
compression to which we can submit them, employing the hardest
and strongest tools. Nevertheless the elevation of temperature
is not great. If it** **were,
the pieces of steel used in these operations Would soon lose their
temper.

(3) We know that it would be necessary to exert on
solids and liquids a very great strain in order to produce in
them a reduction of volume comparable to that which they experience
in cooling (cooling from 100^{0}**
**to zero, for example). Now the cooling
requires a greater abstraction of caloric than would simple reduction
of volume. If this reduction were produced by mechanical means,
the heat set free would not then be able to make the temperature
of the body vary as many degrees as the cooling makes it vary.
It would, however, necessitate the employment of a force undoubtedly
very considerable.

Since solid bodies are susceptible of little change of temperature through changes of volume, and since the condition of the most effective employment of heat for the development of motive power is precisely that all change of temperature should be due to a change of volume, solid bodies appear but ill fitted to realize this power.

The same remarks apply to liquids. The same reasons may be given for rejecting them.*

We are not speaking now of practical difficulties. They will be numberless. The motion produced by the dilatation and com-

- The recent experiments of M. Oerstedt on the
compressibility of water have shown that, for a pressure of five
atmospheres, the temperature of this liquid exhibits no appreciable
change. (See Annals de
*Chimie*et deFeb. 1823, p.192.)**,**

`96 `REFLECTIONS ON THE

pression of solid or liquid bodies would only be very slight. In order to give them sufficient amplitude we should be forced to make use of complicated mechanisms. It would be necessary to employ materials of the greatest strength to transmit enormous pressure; finally, the successive operations would be executed very slowly compared to those of the ordinary steam-engine, so that apparatus of large dimensions and heavy cost would produce but very ordinary results.

The elastic fluids, gases or vapors, are the means really adapted to the development of the motive power of heat. They combine all the conditions necessary to fulfill this office. They are easy to compress; they can be almost infinitely expanded; variations of volume occasion in them great changes of temperature; and, lastly, they are very mobile, easy to heat and cool, easy to transport from one place to another, which enables them to produce rapidly the desired effects. We can easily conceive a multitude of machines fitted to develop the motive power of heat through the use of elastic fluids; but in whatever way we look at it, we should not lose sight of the following principles:

(1) The temperature of the fluid should be made as high as possible, in order to obtain a great fall of caloric, and consequently a large production of motive power.

(2) For the same reason the cooling should be carried as far as possible.

(3) It should be so arranged that the passage of
the elastic fluid from the highest to the lowest temperature should
be due to increase of volume; that is, it should be so arranged
that the cooling of the gas should occur spontaneously as the
effect of rarefaction. The limits of the temperature to which
it is possible

to bring the fluid primarily, are simply the limits
of the temperature* *obtainable by combustion; they are very
high.

The limits of cooling are found in the temperature of the coldest body of which we can easily and freely make use; this body is usually the water of the locality.

As to the third condition, it involves difficulties in the realization of the motive power of heat when the attempt is made to take advantage of great differences of temperature, to utilize great falls of heat. In short, it is necessary then that the gas, by reason of its rarefaction, should pass from a very high temperature to a very low one, which requires a great change of volume and of density, which requires also that the gas be first taken under a very heavy pressure, or that it acquire by its dilatation an enormous volume-conditions both difficult to fulfill. The first necessitates the employment of very strong vessels to contain the gas at a very high temperature and under very heavy pressure. The second necessitates the use of vessels of large dimensions. These are, in a word, the principal obstacles which prevent the utilization in steam-engines of a great part of the motive power of the heat. We are obliged to limit ourselves to the use of a slight fall of caloric, while the combustion of the coal furnishes the means of procuring a very great one.

It is seldom that in steam-engines the elastic fluid
is produced under a higher pressure than six atmospheres- pressure
corresponding to about 160^{0 }Centigrade, and it is seldom
that condensation takes place at a temperature much under 40^{0}.
The fall of caloric from 160^{0} to^{ }40^{0}
is 120^{0}, while by combustion we can procure a fall
of 1000^{0}** **to
2000^{0}**.
**

98 **REFLECTIONS ON THE
**

In order to comprehend this more clearly, let us
recall what we have termed the fall of caloric. This is the passage
of the heat from one body, A*, *having an elevated temperature,
to another, B*, *where it is lower. We say that the fall
of the caloric is 100^{0}** **or
1000^{0}** **when
the difference of temperature between the bodies A* *and
B is 100^{0} or 1000^{0}**.**

In a* *steam-engine which works under a pressure
of six* *atmospheres the temperature of the boiler is 160^{0}.
This is the body A*. *It is kept, by contact with the furnace,
at the constant temperature of 160^{0}**,
**and continually furnishes the heat necessary
for the formation of steam. The condenser is the body B*. *By
means of a current of cold water it is kept at a nearly constant
temperature of 40^{0}. It absorbs continually the caloric
brought from the body A* *by the steam. The difference of
temperature between these two bodies is** **160^{0 }-
40^{0}, or 120^{0.}. Hence we say that the fall
of caloric is here** **120^{0}**.**

Coal being capable of producing, by its combustion,
a temperature higher than 1000^{0}**,
**and the cold water, which is generally
used in our climate, being at about 10^{0}, we can easily
procure a fall of caloric of 1000^{0}**,
**and of this only 120^{0} are utilized
by steam-engines. Even these 120^{0}**
**are not wholly utilized. There is always
considerable loss due to useless re-establishments of equilibrium
in the caloric.

It is easy to see the advantages possessed by high-pressure
machines over those of lower pressure. *This superiority lies
essentially in the power of utilizing a greater fall of caloric*.
The steam produced under a higher pressure is found also at a
higher temperature, and as, further, the temperature of condensation
remains always about the same, it is evident that the fall of

**I
**

**99 MOTIVE POWER OF HEAT
**

caloric is more considerable. But to obtain from
high-pressure engines really advantageous results, it is necessary
that the fall of caloric should be most profitably utilized. It
is not enough that the steam be produced at a high temperature:
it is also necessary that by the expansion of its volume its temperature
should become sufficiently low. A good steam-engine, therefore,
should not only employ steam under heavy pressure, but *under
successive and variable pressures, differing greatly from one
another, and progressively decreasing.*

In* *order to understand in some sort a* *posteriori*
*the advantages of high-pressure engines, let us sup-pose steam
to be formed under atmospheric pressure and introduced into the
cylindrical vessel abcd (fig.5), under the piston cd, which at
first touches the bottom ab. The steam, after having moved the
piston from ab to cd, will continue finally to produce its results
in a manner with which we will not concern ourselves.

Let us suppose that the piston having moved to cd*
*is forced downward to ef*, *without the steam being allowed
to escape, or any portion of its caloric to be lost. It will*
*be driven back into the space abef*, *and will* *increase
at the same time in* *density, elastic force, and temperature.
If* *the steam, instead of being produced under atmospheric
pressure, had been produced just when it was being forced back
into abef*, *and so that after its introduction into the
cylinder it had made the piston move from ab* *to ef*,
*and had moved it simply by its extension of volume, from ef*
*to cd*, *the motive power produced would have been more
considerable than in the first* *case. In* *fact, the
movement of the

`100 `**REFLECTIONS ON THE
**

piston, while equal in extent, would have taken place under the action of a greater pressure, though variable, and though progressively decreasing.

The steam, however, would. have required for its formation exactly the same quantity of caloric, only the caloric would have been employed at a higher temperature.

It is considerations of this nature which have led
to the making of double-cylinder engines-engines invented by Mr.
Hornblower, improved by Mr. Woolf, and which, as regards economy
of the combustible, are considered the best. They consist of a
small cylinder, which at each pulsation is filled more or less
(often entirely) with steam, and of a second cylinder having usually
a capacity quadruple that of the first, and which receives no
steam except that which has already operated in the first cylinder.
Thus the steam when it ceases to act has at least quadrupled in
volume. From the second cylinder it is carried directly into the
condenser, but it is conceivable that it might be carried into
a third cylinder quadruple the second, and in which its volume
would have become sixteen times the original volume. The principal
obstacle to the use of a third cylinder of this sort is the capacity
which it would be necessary to give it, and the large dimensions
which the opening for the passage of the steam must have. We will
say no more on this subject, as we do not propose here to enter
into the details of construction of steam-engines. These details
call for a work devoted specially to them, and which does not
yet exist, at least in France.*

*We find in the work called *De la**
Richesse Minerale, *by
M. Heron de ville-fosse, vol. ii

**MOTIVE POWER OF HEAT **101

If the expansion of steam is mainly limited by the dimensions of the vessels in which the dilatation must take place, the degree of condensation at which it is possible to use it at first is limited only by the resistance of the vessels in which it is produced, that is, of the boilers.

In this respect we have by no means attained the best possible results. The arrangement of the boilers generally in use is entirely faulty, although the tension of the steam rarely exceeds from four to six atmospheres. They often burst and cause severe accidents. It will undoubtedly be possible to avoid such accidents, and meantime to raise the steam to much greater pressures than is usually done.

Besides the high-pressure double-cylinder engines
of which we have spoken, there are also high-pressure engines
of one cylinder. The greater part of these latter have been constructed
by two ingenious English engineers, Messrs. Trevithick and Vivian.
They employ the steam under a very high pressure, sometimes eight
to ten atmospheres, but they have no condenser. The steam, after
it has been introduced into the cylinder, undergoes therein a
certain increase of volume, but preserves always a pressure higher
than atmospheric. When it has fulfilled its office it is thrown
out into the atmosphere. It is evident that this mode of working
is fully equivalent, in respect to the motive power produced,
to condensing the steam at** **100^{0}**,
**and that a portion of the useful effect
is lost. But the engines working thus dispense with condenser
and air-pump. They are less costly than the others, less complicated,
occupy less space, and can be used in places where there is not
sufficient water for condensation. In such places they are of
inestimable advantage, since no others could take their place.
These engines are principally employed

102 REFLECTIONS ON THE

in England to move coal-wagons on railroads laid either in the interior of mines or outside of them.

We have, further, only a few remarks to make upon the use of permanent gases and other vapors than that of water in the development of the motive power of heat.

Various attempts have been made to produce motive power by the action of heat on atmospheric air. This gas presents, as compared with vapor of water, both advantages and disadvantages, which we will proceed to examine.

(1) It presents, as compared with vapor of water, a notable advantage in that, having for equal volume a much less capacity for heat, it would cool more rapidly by an equal increase of volume. (This fact is proved by what has already been stated.) Now we have seen how important it is to produce by change of volume the greatest possible changes of temperature.

(2)Vapors of water can be formed only through the intervention of a boiler, while atmospheric air could be heated directly by combustion carried on within its own mass. Considerable loss could thus be prevented, not only in the quantity of heat, but also in its temperature. This advantage belongs exclusively to atmospheric sir. Other gases do not possess it. They would be even more difficult to heat than vapor of water.

- In order to give to air great increase in volume,
and by that expansion to produce a great change of temperature,
it must first be taken under a sufficiently high pressure; then
it must be compressed with a pump or by some other means before
heating it. This operation would require a special apparatus,
an apparatus not found in steam-engines. In the latter, water
is in

**MOTIVE POWER OF HEAT **`103
`

a liquid state when injected into the boiler, and to introduce it requires but a small pump.

(4) The condensing of the vapor by contact with the refrigerant body is much more prompt and much easier than is the cooling of air. There might, of course, be the expedient of throwing the latter out into the atmosphere, and there would be also the advantage of avoiding the use of a refrigerant, which is not always available, but it would be requisite that the increase of the volume of the air should not reduce its pressure below that of the atmosphere.

(5) One of the gravest inconveniences of steam is that it cannot be used at high temperatures without necessitating the use of vessels of extraordinary strength. It is not so with air for which there exists no necessary relation between the elastic force and the temperature. Air, then, would seem more suitable than steam to realize the motive power of falls of caloric from high temperatures. Perhaps in low temperatures steam may be more convenient. We might conceive even the possibility of making the same heat act successively upon air and vapor of water. It would be only necessary that the air should have, after its use, an elevated temperature, and instead of throwing it out immediately into the atmosphere, to make it envelop a steam-boiler, as if it issued directly from a furnace.

The use of atmospheric air for the development of
the motive power of heat presents in practice very great, but
perhaps not insurmountable, difficulties. If we should succeed
in overcoming them, it would doubtless offer a notable advantage
over vapor of water.

104 REFLECTIONS ON THE

As to the other permanent gases, they should be absolutely rejected. They have all the inconveniences of atmospheric air, with none of its advantages. The same can be said of other vapors than that of water, as compared with the latter.

If we could find an abundant liquid body which would vaporize at a higher temperature than water, of which the vapor would have, for the same volume, a less specific heat, which would not attack the metals employed in the construction of machines, it would undoubtedly merit the preference. But nature provides no such body.

The use of the vapor of alcohol has been proposed. Machines have even been constructed for the purpose of using it, by avoiding the mixture of its vapor with the water of condensation, that is, by applying the cold body externally instead of introducing it into the machine. It has been thought that a remarkable advantage might be secured by using the vapor of alcohol in that it possesses a stronger tension than the vapor of water at the same temperature. We can see in this only a fresh obstacle to be overcome. The principal defect of the vapor of water is its excessive tension at an elevated temperature; now this defect exists still more strongly in the vapor of alcohol. As to the relative advantage in a greater production of motive power, -an advantage attributed to it, -we know by the principles above demonstrated that it is imaginary.

It is thus upon the use of atmospheric air and vapor of water that subsequent attempts to perfect heat-engines should be based. It is to utilize by means of these agents the greatest possible falls of caloric that all efforts should be directed.

**I
**

**MOTIVE POWER OF HEAT **105

Finally, we will show how far we are from having realized, by any means at present known, all the motive power of combustibles.

One kilogram of carbon burnt in the calorimeter furnishes a quantity of heat capable of raising one degree Centigrade about 7000 kilograms of water, that is, it furnishes 7000 units of heat according to the definition of these units given on page 88.

The greatest fall of caloric attainable is measured
by the difference between the temperature produced by combustion
and that of the refrigerant bodies. It is difficult to perceive
any other limits to the temperature of combustion than those in
which the combination between oxygen and the combustible may take
place. Let us assume, however, that 1000^{0} may be this
limit, and we shall certainly be below the truth. As to the temperature
of the refrigerant, let us suppose it .0^{0}
We estimated approximately (page 91) the quantity
of motive power that 1000 units of heat develop between 100^{0}and
99^{0}. We found it to be 1.112 units of power, each equal
to 1 metre of water raised to a height of 1 metre.

If the motive power were proportional to the fall
of caloric, if it were the same for each thermometric degree,
nothing would be easier than to estimate it from 1000^{0}
to **0^{0}. **Its
value would be

But as this law is only approximate, and as possibly
it deviates much from the truth at high temperatures, we can only
make a very rough estimate. We will suppose the number 1112 reduced
one-half, that is, to 560.

`106 `REFLECTIONS ON THE

Since a kilogram of carbon produces 7000 units of
heat, and since the number 560 is relatively 1000 units, it must
be multiplied by 7, which gives

This is the motive power of 1 kilogram of carbon. In order to compare this theoretical result with that of experiment, let us ascertain how much motive power a kilogram of carbon actually develops in the best-known steam-engines.

The engines which, up to this time, have shown the best results are the large double-cylinder engines used in the drainage of the tin and copper mines of Cornwall. The best results that have been obtained with them are as follows:

65 millions of lbs. of water have been raised one English foot by the bushel of coal burned (the bushel weighing 88 lbs.). This is equivalent to raising, by a kilogram of coal, 195 cubic metres of water to a height of 1 metre, producing thereby 195 units of motive power per kilogram of coal burned.

195 units are only the twentieth of 3920, the theoretical maximum; consequently 1/20 only of the motive power of the combustible has been utilized.

We have, nevertheless, selected our example from among the best steam-cngines known.

Most engines are greatly inferior to these. The old engine of Charnot, for example, raised twenty cubic metres of water thirty-three metres, for thirty kilograms of coal consumed, which amounts to twenty-two units of motive power per kilogram, -a result nine times less than given above, and one hundred and eighty times less than the theoretical maximum.

4

107

We should not expect ever to utilize in practice all the motive power of combustibles. The attempts made to attain this result would be far more hurtful than useful if they caused other important considerations to be neglected. The economy of the combustible is only one of the conditions to be fulfilled in heat-engines. In many cases it is only secondary. It should often give precedence to safety, to strength, to the durability of the engine, to the small space which it must occupy, to small cost of installation, etc. To know how to appreciate in each case, at their true value, the considerations of convenience and economy which may present themselves; to know how to discern the more important of those which are only accessories; to balance them properly against each other, in order to attain the best results by the simplest means: such should be the leading characteristics of the man called to direct, to co-ordinate among themselves the labors of his comrades, to make them co-operate towards one useful end, of whatsoever sort it may be.