Chapter IV: Part 4

11. _Specific Heats of Gases._--In order to estimate the quantities of heat concerned in experiments with gases, it was necessary in the first instance to measure their specific heats, which presented formidable difficulties. The earlier attempts by Lavoisier and others, employing the ordinary methods of calorimetry, gave very uncertain and discordant results, which were not regarded with any confidence even by the experimentalists themselves. Gay-Lussac (_Memoires d'Arcueil_, 1807) devised an ingenious experiment, which, though misinterpreted at the time, is very interesting and instructive. With the object of comparing the specific heats of different gases, he took two equal globes A and B connected by a tube with a stop-cock. The globe B was exhausted, the other A being filled with gas. On opening the tap between the vessels, the gas flowed from A to B and the pressure was rapidly equalized. He observed that the fall of temperature in A was nearly equal to the rise of temperature in B, and that for the same initial pressure the change of temperature was very nearly the same for all the gases he tried, except hydrogen, which showed greater changes of temperature than other gases. He concluded from this experiment that equal volumes of gases had the same capacity for heat, except hydrogen, which he supposed to have a larger capacity, because it showed a greater effect. The method does not in reality afford any direct information with regard to the specific heats, and the conclusion with regard to hydrogen is evidently wrong. At a later date (_Ann. de Chim._, 1812, 81, p. 98) Gay-Lussac adopted A. Crawford's method of mixture, allowing two equal streams of different gases, one heated and the other cooled about 20 deg. C., to mix in a tube containing a thermometer. The resulting temperature was in all cases nearly the mean of the two, from which he concluded that equal volumes of all the gases tried, namely, hydrogen, carbon dioxide, air, oxygen and nitrogen, had the same thermal capacity. This was correct, except as regards carbon dioxide, but did not give any information as to the actual specific heats referred to water or any known substance. About the same time, F. Delaroche and J. E. Berard (_Ann. de chim._, 1813, 85, p. 72) made direct determinations of the specific heats of air, oxygen, hydrogen, carbon monoxide, carbon dioxide, nitrous oxide and ethylene, by passing a stream of gas heated to nearly 100 deg. C. through a spiral tube in a calorimeter containing water. Their work was a great advance on previous attempts, and gave the first trustworthy results. With the exception of hydrogen, which presents peculiar difficulties, they found that equal volumes of the permanent gases, air, oxygen and carbon monoxide, had nearly the same thermal capacity, but that the compound condensible gases, carbon dioxide, nitrous oxide and ethylene, had larger thermal capacities in the order given. They were unable to state whether the specific heats of the gases increased or diminished with temperature, but from experiments on air at pressures of 740 mm. and 1000 mm., they found the specific heats to be .269 and .245 respectively, and concluded that the specific heat diminished with increase of pressure. The difference they observed was really due to errors of experiment, but they regarded it as proving beyond doubt the truth of the calorists' contention that the heat disengaged on the compression of a gas was due to the diminution of its thermal capacity.

Dalton and others had endeavoured to measure directly the rise of temperature produced by the compression of a gas. Dalton had observed a rise of 50 deg. F. in a gas when suddenly compressed to half its volume, but no thermometers at that time were sufficiently sensitive to indicate more than a fraction of the change of temperature. Laplace was the first to see in this phenomenon the probable explanation of the discrepancy between Newton's calculation of the velocity of sound and the observed value. The increase of pressure due to a sudden compression, in which no heat was allowed to escape, or as we now call it an "adiabatic" compression, would necessarily be greater than the increase of pressure in a slow isothermal compression, on account of the rise of temperature. As the rapid compressions and rarefactions occurring in the propagation of a sound wave were perfectly adiabatic, it was necessary to take account of the rise of temperature due to compression in calculating the velocity. To reconcile the observed and calculated values of the velocity, the increase of pressure in adiabatic compression must be 1.410 times greater than in isothermal compression. This is the ratio of the adiabatic elasticity of air to the isothermal elasticity. It was a long time, however, before Laplace saw his way to any direct experimental verification of the value of this ratio. At a later date (_Ann. de chim._, 1816, 3, p. 238) he stated that he had succeeded in proving that the ratio in question must be the same as the ratio of the specific heat of air at constant pressure to the specific heat at constant volume.

In the method of measuring the specific heat adopted by Delaroche and
Berard, the gas under experiment, while passing through a tube at
practically constant pressure, contracts in cooling, as it gives up
its heat to the calorimeter. Part of the heat surrendered to the
calorimeter is due to the contraction of volume. If a gramme of gas at
pressure p, volume v and temperature T abs. is heated 1 deg. C. at
constant pressure p, it absorbs a quantity of heat S = .238 calorie
(according to Regnault) the specific heat at constant pressure. At the
same time the gas expands by a fraction 1/T of v, which is the same as
1/273 of its volume at 0 deg. C. If now the air is suddenly compressed
by an amount v/T, it will be restored to its original volume, and its
temperature will be raised by the liberation of a quantity of heat R',
the latent heat of expansion for an increase of volume v/T. If no heat
has been allowed to escape, the air will now be in the same state as
if a quantity of heat S had been communicated to it at its original
volume v without expansion. The rise of temperature above the original
temperature T will be S/s degrees, where s is the specific heat at
constant volume, which is obviously equal to S - R'. Since p/T is the
increase of pressure for 1 deg. C. rise of temperature at constant
volume, the increase of pressure for a rise of S/s degrees will be
[gamma]p/T, where [gamma] is the ratio S/s. But this is the rise of
pressure produced by a sudden compression v/T, and is seen to be
[gamma] times the rise of pressure p/T produced by the same
compression at constant temperature. The ratio of the adiabatic to the
isothermal elasticity, required for calculating the velocity of sound,
is therefore the same as the ratio of the specific heat at constant
pressure to that at constant volume.

12. _Experimental Verification of the Ratio of Specific Heats._--This
was a most interesting and important theoretical relation to discover,
but unfortunately it did not help much in the determination of the
ratio required, because it was not practically possible at that time
to measure the specific heat of air at constant volume in a closed
vessel. Attempts had been made to do this, but they had signally
failed, on account of the small heat capacity of the gas as compared
with the containing vessel. Laplace endeavoured to extract some
confirmation of his views from the values given by Delaroche and
Berard for the specific heat of air at 1000 and 740 mm. pressure. On
the assumption that the quantities of heat contained in a given mass
of air increased in direct proportion to its volume when heated at
constant pressure, he deduced, by some rather obscure reasoning, that
the ratio of the specific heats S and s should be about 1.5 to 1,
which he regarded as a fairly satisfactory agreement with the value
[gamma] = 1.41 deduced from the velocity of sound.

The ratio of the specific heats could not be directly measured, but a
few years later, Clement and Desormes (_Journ. de Phys._, Nov. 1819)
succeeded in making a direct measurement of the ratio of the
elasticities in a very simple manner. They took a large globe
containing air at atmospheric pressure and temperature, and removed a
small quantity of air. They then observed the defect of pressure p0
when the air had regained its original temperature. By suddenly
opening the globe, and immediately closing it, the pressure was
restored almost instantaneously to the atmospheric, the rise of
pressure p0 corresponding to the sudden compression produced. The air,
having been heated by the compression, was allowed to regain its
original temperature, the tap remaining closed, and the final defect
of pressure p^1 was noted. The change of pressure for the same
compression performed isothermally is then p0 - p^1. The ratio p0/(p0
- p^1) is the ratio of the adiabatic and isothermal elasticities,
provided that p0 is small compared with the whole atmospheric
pressure. In this way they found the ratio 1.354, which is not much
smaller than the value 1.410 required to reconcile the observed and
calculated values of the velocity of sound. Gay-Lussac and J. J.
Welter (_Ann. de chim._, 1822) repeated the experiment with slight
improvements, using expansion instead of compression, and found the
ratio 1.375. The experiment has often been repeated since that time,
and there is no doubt that the value of the ratio deduced from the
velocity of sound is correct, the defect of the value obtained by
direct experiment being due to the fact that the compression or
expansion is not perfectly adiabatic. Gay-Lussac and Welter found the
ratio practically constant for a range of pressure 144 to 1460 mm.,
and for a range of temperature from -20 deg. to +40 deg. C. The
velocity of sound at Quito, at a pressure of 544 mm. was found to be
the same as at Paris at 760 mm. at the same temperature. Assuming on
this evidence the constancy of the ratio of the specific heats of air,
Laplace (_Mecanique celeste_, v. 143) showed that, if the specific
heat at constant pressure was independent of the temperature, the
specific heat per unit volume at a pressure p must vary as
p^(1/[gamma]), according to the caloric theory. The specific heat per
unit mass must then vary as p^(1/[gamma]-1) which he found agreed
precisely with the experiment of Delaroche and Berard already cited.
This was undoubtedly a strong confirmation of the caloric theory.
Poisson by the same assumptions (_Ann. de chim._, 1823, 23, p. 337)
obtained the same results, and also showed that the relation between
the pressure and the volume of a gas in adiabatic compression or
expansion must be of the form pv^[gamma] = constant.

P. L. Dulong (_Ann. de chim._, 1829, 41, p. 156), adopting a method
due to E. F. F. Chladni, compared the velocities of sound in different
gases by observing the pitch of the note given by the same tube when
filled with the gases in question. He thus obtained the values of the
ratios of the elasticities or of the specific heats for the gases
employed. For oxygen, hydrogen and carbonic oxide, these ratios were
the same as for air. But for carbonic acid, nitrous oxide and olefiant
gas, the values were much smaller, showing that these gases
experienced a smaller change of temperature in compression. On
comparing his results with the values of the specific heats for the
same gases found by Delaroche and Berard, Dulong observed that the
changes of temperature for the same compression were in the inverse
ratio of the specific heats at constant volume, and deduced the
important conclusion that "_Equal volumes of all gases under the same
conditions evolve on compression the same quantity of heat_." This is
equivalent to the statement that the difference of the specific heats,
or the latent heat of expansion R' per 1 deg., is the same for all
gases if equal volumes are taken. Assuming the ratio [gamma] = 1.410,
and taking Delaroche and Berard's value for the specific heat of air
at constant pressure S = .267, we have s = S/1.41 = .189, and the
difference of the specific heats per unit mass of air S - s = R' =
.078. Adopting Regnault's value of the specific heat of air, namely, S
= .238, we should have S - s = .069. This quantity represents the heat
absorbed by unit mass of air in expanding at constant temperature T by
a fraction 1/T of its volume v, or by 1/273rd of its volume 0 deg. C.

If, instead of taking unit mass, we take a volume v0 = 22.30 litres at
0 deg. C. and 760 mm. being the volume of the molecular weight of the
gas in grammes, the quantity of heat evolved by a compression equal to
v/T will be approximately 2 calories, and is the same for all gases.
The work done in this compression is pv/T = R, and is also the same
for all gases, namely, 8.3 joules. Dulong's experimental result,
therefore, shows that the heat evolved in the compression of a gas is
proportional to the work done. This result had previously been deduced
theoretically by Carnot (1824). At a later date it was assumed by
Mayer, Clausius and others, on the evidence of these experiments, that
the heat evolved was not merely proportional to the work done, but was
equivalent to it. The further experimental evidence required to
justify this assumption was first supplied by Joule.

Latent heat of expansion R' = .069 calorie per gramme of air, per 1
deg. C.
= 2.0 calories per gramme-molecule of any
gas.
Work done in expansion R = .287 joule per gramme of air per 1 deg. C.
= 8.3 joules per gramme-molecule of any
gas.

13. _Carnot: On the Motive Power of Heat._--A practical and theoretical question of the greatest importance was first answered by Sadi Carnot about this time in his _Reflections on the Motive Power of Heat_ (1824). How much motive power (defined by Carnot as weight lifted through a certain height) can be obtained from heat alone by means of an engine repeating a regular succession or "cycle" of operations continuously? Is the efficiency limited, and, if so, how is it limited? Are other agents preferable to steam for developing motive power from heat? In discussing this problem, we cannot do better than follow Carnot's reasoning which, in its main features could hardly be improved at the present day.

Carnot points out that in order to obtain an answer to this question, it is necessary to consider the essential conditions of the process, apart from the mechanism of the engine and the working substance or agent employed. Work cannot be said to be produced _from heat alone_ unless nothing but heat is supplied, and the working substance and all parts of the engine are at the end of the process in precisely the same state as at the beginning.[3]

_Carnot's Axiom._--Carnot here, and throughout his reasoning, makes a fundamental assumption, which he states as follows: "When a body has undergone any changes and after a certain number of transformations is brought back identically to its original state, considered relatively to density, temperature and mode of aggregation, it must contain the same quantity of heat as it contained originally."[4]

Heat, according to Carnot, in the type of engine we are considering, can evidently be a cause of motive power only by virtue of changes of volume or form produced by alternate heating and cooling. This involves the existence of cold and hot bodies to act as boiler and condenser, or source and sink of heat, respectively. Wherever there exists a difference of temperature, it is possible to have the production of motive power from heat; and conversely, production of motive power, from heat alone, is impossible without difference of temperature. In other words the production of motive power from heat is not merely a question of the consumption of heat, but always requires transference of heat from hot to cold. What then are the conditions which enable the difference of temperature to be most advantageously employed in the production of motive power, and how much motive power can be obtained with a given difference of temperature from a given quantity of heat?

_Carnot's Rule for Maximum Effect._--In order to realize the maximum effect, it is necessary that, in the process employed, there should not be any direct interchange of heat between bodies at different temperatures. Direct transference of heat by conduction or radiation between bodies at different temperatures is equivalent to wasting a difference of temperature which might have been utilized to produce motive power. The working substance must throughout every stage of the process be in equilibrium with itself (i.e. at uniform temperature and pressure) and also with external bodies, such as the boiler and condenser, at such times as it is put in communication with them. In the actual engine there is always some interchange of heat between the steam and the cylinder, and some loss of heat to external bodies. There may also be some difference of temperature between the boiler steam and the cylinder on admission, or between the waste steam and the condenser at release. These differences represent losses of efficiency which may be reduced indefinitely, at least in imagination, by suitable means, and designers had even at that date been very successful in reducing them. All such losses are supposed to be absent in deducing the ideal limit of efficiency, beyond which it would be impossible to go.

14. _Carnot's Description of his Ideal Cycle._--Carnot first gives a rough illustration of an incomplete cycle, using steam much in the same way as it is employed in an ordinary steam-engine. After expansion down to condenser pressure the steam is completely condensed to water, and is then returned as cold water to the hot boiler. He points out that the last step does not conform exactly to the condition he laid down, because although the water is restored to its initial state, there is direct passage of heat from a hot body to a cold body in the last process. He points out that this difficulty might be overcome by supposing the difference of temperature small, and by employing a series of engines, each working through a small range, to cover a finite interval of temperature. Having established the general notions of a perfect cycle, he proceeds to give a more exact illustration, employing a gas as the working substance. He takes as the basis of his demonstration the well-established experimental fact that a gas is heated by rapid compression and cooled by rapid expansion, and that if compressed or expanded slowly in contact with conducting bodies, the gas will give out heat in compression or absorb heat in expansion while its temperature remains constant. He then goes on to say:--

"This preliminary notion being settled, let us imagine an elastic
fluid, atmospheric air for example, enclosed in a cylinder _abcd_,
fig. 4, fitted with a movable diaphragm or piston cd. Let there also
be two bodies A, B, each maintained at a constant temperature, that of
A being more elevated than that of B. Let us now suppose the following
series of operations to be performed:

"1. Contact of the body A with the air contained in the space _abcd_,
or with the bottom of the cylinder, which we will suppose to transmit
heat easily. The air is now at the temperature of the body A, and _cd_
is the actual position of the piston.

"2. The piston is gradually raised, and takes the position _ef_. The
air remains in contact with the body A, and is thereby maintained at a
constant temperature during the expansion. The body A furnishes the
heat necessary to maintain the constancy of temperature.

"3. The body A is removed, and the air no longer being in contact with
any body capable of giving it heat, the piston continues nevertheless
to rise, and passes from the position _ef_ to _gh_. The air expands
without receiving heat and its temperature falls. Let us imagine that
it falls until it is just equal to that of the body B. At this moment
the piston is stopped and occupies the position _gh_.

"4. The air is placed in contact with the body B; it is compressed by
the return of the piston, which is brought from the position _gh_ to
the position _cd_. The air remains meanwhile at a constant
temperature, because of its contact with the body B to which it gives
up its heat.

"5. The body B is removed, and the compression of the air is
continued. The air being now isolated, rises in temperature. The
compression is continued until the air has acquired the temperature of
the body A. The piston passes meanwhile from the position _cd_ to the
position _ik_.

"6. The air is replaced in contact with the body A, and the piston
returns from the position _ik_ to the position _ef_, the temperature
remaining invariable.

"7. The period described under (3) is repeated, then successively the
periods (4), (5), (6); (3), (4), (5), (6); (3), (4), (5), (6); and so
on.

"During these operations the air enclosed in the cylinder exerts an
effort more or less great on the piston. The pressure of the air
varies both on account of changes of volume and on account of changes
of temperature; but it should be observed that for equal volumes, that
is to say, for like positions of the piston, the temperature is higher
during the dilatation than during the compression. Since the pressure
is greater during the expansion, the quantity of motive power produced
by the dilatation is greater than that consumed by the compression. We
shall thus obtain a balance of motive power, which may be employed for
any purpose. The air has served as working substance in a heat-engine;
it has also been employed in the most advantageous manner possible,
since no useless re-establishment of the equilibrium of heat has been
allowed to occur.

"All the operations above described may be executed in the reverse
order and direction. Let us imagine that after the sixth period, that
is to say, when the piston has reached the position _ef_, we make it
return to the position _ik_, and that at the same time we keep the air
in contact with the hot body A; the heat furnished by this body during
the sixth period will return to its source, that is, to the body A,
and everything will be as it was at the end of the fifth period. If
now we remove the body A, and if we make the piston move from _ik_ to
_cd_, the temperature of the air will decrease by just as many degrees
as it increased during the fifth period, and will become that of the
body B. We can evidently continue in this way a series of operations
the exact reverse of those which were previously described; it
suffices to place oneself in the same circumstances and to execute for
each period a movement of expansion in place of a movement of
compression, and vice versa.

"The result of the first series of operations was the production of a
certain quantity of motive power, and the transport of heat from the
body A to the body B; the result of the reverse operations is the
consumption of the motive power produced in the first case, and the
return of heat from the body B to the body A, in such sort that these
two series of operations annul and neutralize each other.

"The impossibility of producing by the agency of heat alone a quantity
of motive power greater than that which we have obtained in our first
series of operations is now easy to prove. It is demonstrated by
reasoning exactly similar to that which we have already given. The
reasoning will have in this case a greater degree of exactitude; the
air of which we made use to develop the motive power is brought back
at the end of each cycle of operations precisely to its initial state,
whereas this was not quite exactly the case for the vapour of water,
as we have already remarked."

15. _Proof of Carnot's Principle._--Carnot considered the proof too obvious to be worth repeating, but, unfortunately, his previous demonstration, referring to an incomplete cycle, is not so exactly worded that exception cannot be taken to it. We will therefore repeat his proof in a slightly more definite and exact form. Suppose that a reversible engine R, working in the cycle above described, takes a quantity of heat H from the source in each cycle, and performs a quantity of useful work W_r. If it were possible for any other engine S, working with the same two bodies A and B as source and refrigerator, to perform a greater amount of useful work W_s per cycle for the same quantity of heat H taken from the source, it would suffice to take a portion W_r of this motive power (since W_s is by hypothesis greater than W_r) to drive the engine R backwards, and return a quantity of heat H to the source in each cycle. The process might be repeated indefinitely, and we should obtain at each repetition a balance of useful work W_s - W_r, _without taking any heat from the source_, which is contrary to experience. Whether the quantity of heat taken from the condenser by R is equal to that given to the condenser by S is immaterial. The hot body A might be a comparatively small boiler, since no heat is taken from it. The cold body B might be the ocean, or the whole earth. We might thus obtain without any consumption of fuel a practically unlimited supply of motive power. Which is absurd.

_Carnot's Statement of his Principle._[5]--If the above reasoning be admitted, we must conclude with Carnot that _the motive power obtainable from heat is independent of the agents employed to realize it_. _The efficiency is fixed solely by the temperatures of the bodies between which, in the last resort, the transfer of heat is effected._ "We must understand here that each of the methods of developing motive power attains the perfection of which it is susceptible. This condition is fulfilled if, according to our rule, there is produced in the body no change of temperature that is not due to change of volume, or in other words, if there is no direct interchange of heat between bodies of sensibly different temperatures."

It is characteristic of a state of frictionless mechanical equilibrium that an indefinitely small difference of pressure suffices to upset the equilibrium and reverse the motion. Similarly in thermal equilibrium between bodies at the same temperature, an indefinitely small difference of temperature suffices to reverse the transfer of heat. Carnot's rule is therefore the criterion of the reversibility of a cycle of operations as regards transfer of heat. It is assumed that the ideal engine is mechanically reversible, that there is not, for instance, any communication between reservoirs of gas or vapour at sensibly different pressures, and that there is no waste of power in friction. If there is equilibrium both mechanical and thermal at every stage of the cycle, the ideal engine will be perfectly reversible. That is to say, all its operations will be exactly reversed as regards transfer of heat and work, when the operations are performed in the reverse order and direction. On this understanding Carnot's principle may be put in a different way, which is often adopted, but is really only the same thing put in different words: _The efficiency of a perfectly reversible engine is the maximum possible, and is a function solely of the limits of temperature between which it works_. This result depends essentially on the existence of a state of thermal equilibrium defined by equality of temperature, and independent, in the majority of cases, of the state of a body in other respects. In order to apply the principle to the calculation and prediction of results, it is sufficient to determine the manner in which the efficiency depends on the temperature for one particular case, since the efficiency must be the same for all reversible engines.

16. _Experimental Verification of Carnot's Principle._--Carnot
endeavoured to test his result by the following simple calculations.
Suppose that we have a cylinder fitted with a frictionless piston,
containing 1 gram of water at 100 deg. C., and that the pressure of
the steam, namely 760 mm., is in equilibrium with the external
pressure on the piston at this temperature. Place the cylinder in
connexion with a boiler or hot body at 101 deg. C. The water will then
acquire the temperature of 101 deg. C., and will absorb 1 gram-calorie
of heat. Some waste of motive power occurs here because heat is
allowed to pass from one body to another at a different temperature,
but the waste in this case is so small as to be immaterial. Keep the
cylinder in contact with the hot body at 101 deg. C. and allow the
piston to rise. It may be made to perform useful work as the pressure
is now 27.7 mm. (or 37.7 grams per sq. cm.) in excess of the external
pressure. Continue the process till all the water is converted into
steam. The heat absorbed from the hot body will be nearly 540
gram-calories, the latent heat of steam at this temperature. The
increase of volume will be approximately 1620 c.c., the volume of 1
gram of steam at this pressure and temperature. The work done by the
excess pressure will be 37.7 X 1620 = 61,000 gram-centimetres or 0.61
of a kilogrammetre. Remove the hot body, and allow the steam to expand
further till its pressure is 760 mm. and its temperature has fallen to
100 deg. C. The work which might be done in this expansion is less
than 1/1000th part of a kilogrammetre, and may be neglected for the
present purpose. Place the cylinder in contact with the cold body at
100 deg. C., and allow the steam to condense at this temperature. No
work is done on the piston, because there is equilibrium of pressure,
but a quantity of heat equal to the latent heat of steam at 100 deg.
C. is given to the cold body. The water is now in its initial
condition, and the result of the process has been to gain 0.61 of a
kilogrammetre of work by allowing 540 gram-calories of heat to pass
from a body at 101 deg. C. to a body at 100 deg. C. by means of an
ideally simple steam-engine. The work obtainable in this way from 1000
gram-calories of heat, or 1 kilo-calorie, would evidently be 1.13
kilogrammetre (= 0.61 X 1000/540).

Taking the same range of temperature, namely 101 deg. to 100 deg. C.,
we may perform a similar series of operations with air in the
cylinder, instead of water and steam. Suppose the cylinder to contain
1 gramme of air at 100 deg. C. and 760 mm. pressure instead of water.
Compress it without loss of heat (adiabatically), so as to raise its
temperature to 101 deg. C. Place it in contact with the hot body at
101 deg. C., and allow it to expand at this temperature, absorbing
heat from the hot body, until its volume is increased by 1/374th part
(the expansion per degree at constant pressure). The quantity of heat
absorbed in this expansion, as explained in S 14, will be the
difference of the specific heats or the latent heat of expansion R' =
.069 calorie. Remove the hot body, and allow the gas to expand further
without gain of heat till its temperature falls to 100 deg. C.
Compress it at 100 deg. C. to its original volume, abstracting the
heat of compression by contact with the cold body at 100 deg. C. The
air is now in its original state, and the process has been carried out
in strict accordance with Carnot's rule. The quantity of external work
done in the cycle is easily obtained by the aid of the indicator
diagram ABCD (fig. 5), which is approximately a parallelogram in this
instance. The area of the diagram is equal to that of the rectangle
BEHG, being the product of the vertical height BE, namely, the
increase of pressure per 1 deg. at constant volume, by the increase of
volume BG, which is 1/273rd of the volume at 0 deg. C. and 760 mm., or
2.83 c.c. The increase of pressure BE is 760/373, or 2.03 mm., which
is equivalent to 2.76 gm. per sq. cm. The work done in the cycle is
2.76 X 2.83 = 7.82 gm. cm., or .0782 gram-metre. The heat absorbed at
101 deg. C. was .069 gram-calorie, so that the work obtained is
.0782/.069 or 1.13 gram-metre per gram-calorie, or 1.13 kilogrammetre
per kilogram-calorie. This result is precisely the same as that
obtained by using steam with the same range of temperature, but a very
different kind of cycle. Carnot in making the same calculation did not
obtain quite so good an agreement, because the experimental data at
that time available were not so accurate. He used the value 1/267 for
the coefficient of expansion, and .267 for the specific heat of air.
Moreover, he did not feel justified in assuming, as above, that the
difference of the specific heats was the same at 100 deg. C. as at the
ordinary temperature of 15 deg. to 20 deg. C., at which it had been
experimentally determined. He made similar calculations for the vapour
of alcohol, which differed slightly from the vapour of water. But the
agreement he found was close enough to satisfy him that his
theoretical deductions were correct, and that the resulting ratio of
work to heat should be the same for all substances at the same
temperature.

17. _Carnot's Function. Variation of Efficiency with Temperature._--By
means of calculations, similar to those given above, Carnot
endeavoured to find the amount of motive power obtainable from one
unit of heat per degree fall at various temperatures with various
substances. The value found above, namely 1.13 kilogrammetre per
kilo-calorie per 1 deg. fall, is the value of the efficiency per 1
deg. fall at 100 deg. C. He was able to show that the efficiency per
degree fall probably diminished with rise of temperature, but the
experimental data at that time were too inconsistent to suggest the
true relation. He took as the analytical expression of his principle
that the efficiency W/H of a perfect engine taking in heat H at a
temperature t deg. C., and rejecting heat at the temperature 0 deg.
C., must be some function Ft of the temperature t, which would be the
same for all substances. The efficiency per degree fall at a
temperature t he represented by F't, the derived function of Ft. The
function F't would be the same for all substances at the same
temperature, but would have different values at different
temperatures. In terms of this function, which is generally known as
Carnot's function, the results obtained in the previous section might
be expressed as follows:--

"The increase of volume of a mixture of liquid and vapour per
unit-mass vaporized at any temperature, multiplied by the increase of
vapour-pressure per degree, is equal to the product of the function
F't by the latent heat of vaporization.

"The difference of the specific heats, or the latent heat of expansion
for any substance multiplied by the function F't, is equal to the
product of the expansion per degree at constant pressure by the
increase of pressure per degree at constant volume."

Since the last two coefficients are the same for all gases if equal
volumes are taken, Carnot concluded that: "The difference of the
specific heats at constant pressure and volume is the same for equal
volumes of all gases at the same temperature and pressure."

Taking the expression W = RT log _e r for the whole work done by a gas
obeying the gaseous laws pv = RT in expanding at a temperature T from
a volume 1 (unity) to a volume r, or for a ratio of expansion r, and
putting W' = R log _e r for the work done in a cycle of range 1 deg.,
Carnot obtained the expression for the heat absorbed by a gas in
isothermal expansion

H = R log_e r/F't. (2)

He gives several important deductions which follow from this formula,
which is the analytical expression of the experimental result already
quoted as having been discovered subsequently by Dulong. Employing the
above expression for the latent heat of expansion, Carnot deduced a
general expression for the specific heat of a gas at constant volume
on the basis of the caloric theory. He showed that if the specific
heat was independent of the temperature (the hypothesis already
adopted by Laplace and Poisson) the function F't must be of the form

F't = R/C(t + t0) (3)

where C and t0 are unknown constants. A similar result follows from
his expression for the difference of the specific heats. If this is
assumed to be constant and equal to C, the expression for F't becomes
R/CT, which is the same as the above if t0 = 273. Assuming the
specific heat to be also independent of the volume, he shows that the
function F't should be constant. But this assumption is inconsistent
with the caloric theory of latent heat of expansion, which requires
the specific heat to be a function of the volume. It appears in fact
impossible to reconcile Carnot's principle with the caloric theory on
any simple assumptions. As Carnot remarks: "The main principles on
which the theory of heat rests require most careful examination. Many
experimental facts appear almost inexplicable in the present state of
this theory."

Carnot's work was subsequently put in a more complete analytical form by B. P. E. Clapeyron (_Journ. de l'Ec. polytechn._, Paris, 1832, 14, p. 153), who also made use of Watt's indicator diagram for the first time in discussing physical problems. Clapeyron gave the general expressions for the latent heat of a vapour, and for the latent heat of isothermal expansion of any substance, in terms of Carnot's function, employing the notation of the calculus. The expressions he gave are the same in form as those in use at the present day. He also gave the general expression for Carnot's function, and endeavoured to find its variation with temperature; but having no better data, he succeeded no better than Carnot. Unfortunately, in describing Carnot's cycle, he assumed the caloric theory of heat, and made some unnecessary mistakes, which Carnot (who, we now know, was a believer in the mechanical theory) had been very careful to avoid. Clapeyron directs one to compress the gas at the lower temperature in contact with the body B _until the heat disengaged is equal to that which has been absorbed at the higher temperature_.[6] He assumes that the gas at this point contains the same quantity of heat as it contained in its original state at the higher temperature, and that, when the body B is removed, the gas will be restored to its original temperature, when compressed to its initial volume. This mistake is still attributed to Carnot, and regarded as a fatal objection to his reasoning by nearly all writers at the present day.

18. _Mechanical Theory of Heat._--According to the caloric theory, the heat absorbed in the expansion of a gas became latent, like the latent heat of vaporization of a liquid, but remained in the gas and was again evolved on compressing the gas. This theory gave no explanation of the source of the motive power produced by expansion. The mechanical theory had explained the production of heat by friction as being due to transformation of visible motion into a brisk agitation of the ultimate molecules, but it had not so far given any definite explanation of the converse production of motive power at the expense of heat. The theory could not be regarded as complete until it had been shown that in the production of work from heat, a certain quantity of heat disappeared, and ceased to exist as heat; and that this quantity was the same as that which could be generated by the expenditure of the work produced. The earliest complete statement of the mechanical theory from this point of view is contained in some notes written by Carnot, about 1830, but published by his brother (_Life of Sadi Carnot_, Paris, 1878). Taking the difference of the specific heats to be .078, he estimated the mechanical equivalent at 370 kilogrammetres. But he fully recognized that there were no experimental data at that time available for a quantitative test of the theory, although it appeared to afford a good qualitative explanation of the phenomena. He therefore planned a number of crucial experiments such as the "porous plug" experiment, to test the equivalence of heat and motive power. His early death in 1836 put a stop to these experiments, but many of them have since been independently carried out by other observers.

The most obvious case of the production of work from heat is in the expansion of a gas or vapour, which served in the first instance as a means of calculating the ratio of equivalence, on the assumption that all the heat which disappeared had been transformed into work and had not merely become latent. Marc Seguin, in his _De l'influence des chemins de fer_ (Paris, 1839), made a rough estimate in this manner of the mechanical equivalent of heat, assuming that the loss of heat represented by the fall of temperature of steam on expanding was equivalent to the mechanical effect produced by the expansion. He also remarks (_loc. cit._ p. 382) that it was absurd to suppose that "a finite quantity of heat could produce an indefinite quantity of mechanical action, and that it was more natural to assume that a certain quantity of heat disappeared in the very act of producing motive power." J. R. Mayer (_Liebig's Annalen_, 1842, 42, p. 233) stated the equivalence of heat and work more definitely, deducing it from the old principle, _causa aequat effectum_. Assuming that the sinking of a mercury column by which a gas was compressed was equivalent to the heat set free by the compression, he deduced that the warming of a kilogramme of water 1 deg. C. would correspond to the fall of a weight of one kilogramme from a height of about 365 metres. But Mayer did not adduce any fresh experimental evidence, and made no attempt to apply his theory to the fundamental equations of thermodynamics. It has since been urged that the experiment of Gay-Lussac (1807), on the expansion of gas from one globe to another (see above, S 11), was sufficient justification for the assumption tacitly involved in Mayer's calculation. But Joule was the first to supply the correct interpretation of this experiment, and to repeat it on an adequate scale with suitable precautions. Joule was also the first to measure directly the amount of heat liberated by the compression of a gas, and to prove that heat was not merely rendered latent, but disappeared altogether as heat, when a gas did work in expansion.

19. _Joule's Determinations of the Mechanical Equivalent._--The honour of placing the mechanical theory of heat on a sound _experimental_ basis belongs almost exclusively to J. P. Joule, who showed by direct experiment that in all the most important cases in which heat was generated by the expenditure of mechanical work, or mechanical work was produced at the expense of heat, there was a constant ratio of equivalence between the heat generated and the work expended and vice versa. His first experiments were on the relation of the chemical and electric energy expended to the heat produced in metallic conductors and voltaic and electrolytic cells; these experiments were described in a series of papers published in the _Phil. Mag._, 1840-1843. He first proved the relation, known as Joule's law, that the heat produced in a conductor of resistance R by a current C is proportional to C^2R per second. He went on to show that the total heat produced in any voltaic circuit was proportional to the electromotive force E of the battery and to the number of equivalents electrolysed in it. Faraday had shown that electromotive force depends on chemical affinity. Joule measured the corresponding heats of combustion, and showed that the electromotive force corresponding to a chemical reaction is proportional to the heat of combustion of the electrochemical equivalent. He also measured the E.M.F. required to decompose water, and showed that when part of the electric energy EC is thus expended in a voltameter, the heat generated is less than the heat of combustion corresponding to EC by a quantity representing the heat of combustion of the decomposed gases. His papers so far had been concerned with the relations between electrical energy, chemical energy and heat which he showed to be mutually equivalent. The first paper in which he discussed the relation of heat to mechanical power was entitled "On the Calorific Effects of Magneto-Electricity, and on the Mechanical Value of Heat" (_Brit. Assoc._, 1843; _Phil. Mag._, 23, p. 263). In this paper he showed that the heat produced by currents generated by magneto-electric induction followed the same law as voltaic currents. By a simple and ingenious arrangement he succeeded in measuring the mechanical power expended in producing the currents, and deduced the mechanical equivalent of heat and of electrical energy. The amount of mechanical work required to raise 1 lb. of water 1 deg. F. (1 B.Th.U.), as found by this method, was 838 foot-pounds. In a note added to the paper he states that he found the value 770 foot-pounds by the more direct method of forcing water through fine tubes. In a paper "On the Changes of Temperature produced by the Rarefaction and Condensation of Air" (_Phil. Mag._, May 1845), he made the first direct measurements of the quantity of heat disengaged by compressing air, and also of the heat absorbed when the air was allowed to expand against atmospheric pressure; as the result he deduced the value 798 foot-pounds for the mechanical equivalent of 1 B.Th.U. He also showed that there was no appreciable absorption of heat when air was allowed to expand in such a manner as not to develop mechanical power, and he pointed out that the mechanical equivalent of heat could not be satisfactorily deduced from the relations of the specific heats, because the knowledge of the specific heats of gases at that time was of so uncertain a character. He attributed most weight to his later determinations of the mechanical equivalent made by the direct method of friction of liquids. He showed that the results obtained with different liquids, water, mercury and sperm oil, were the same, namely, 782 foot-pounds; and finally repeating the method with water, using all the precautions and improvements which his experience had suggested, he obtained the value 772 foot-pounds, which was accepted universally for many years, and has only recently required alteration on account of the more exact definition of the heat unit, and the standard scale of temperature (see CALORIMETRY). The great value of Joule's work for the general establishment of the principle of the conservation of energy lay in the variety and completeness of the experimental evidence he adduced. It was not sufficient to find the relation between heat and mechanical work or other forms of energy in one particular case. It was necessary to show that the same relation held in all cases which could be examined experimentally, and that the ratio of equivalence of the different forms of energy, measured in different ways, was independent of the manner in which the conversion was effected and of the material or working substance employed.

As the result of Joule's experiments, we are justified in concluding that heat is a form of energy, and that all its transformations are subject to the general principle of the conservation of energy. As applied to heat, the principle is called the first law of thermodynamics, and may be stated as follows: _When heat is transformed into any other kind of energy, or vice versa, the total quantity of energy remains invariable; that is to say, the quantity of heat which disappears is equivalent to the quantity of the other kind of energy produced and vice versa._

The number of units of mechanical work equivalent to one unit of heat is generally called the mechanical equivalent of heat, or Joule's equivalent, and is denoted by the letter J. Its numerical value depends on the units employed for heat and mechanical energy respectively. The values of the equivalent in terms of the units most commonly employed at the present time are as follows:--

777 foot-pounds (Lat. 45 deg.) are equivalent to 1 B.Th.U. (lb. deg. Fahr.)
1399 foot-pounds " " " 1 lb. deg. C.
426.3 kilogrammetres " " 1 kilogram-deg. C. or
kilo-calorie.
426.3 grammetres " " 1 gram-deg. C. or calorie.
4.180 joules " " 1 gram-deg. C. or calorie.

The water for the heat units is supposed to be taken at 20 deg. C. or 68 deg. F., and the degree of temperature is supposed to be measured by the hydrogen thermometer. The acceleration of gravity in latitude 45 deg. is taken as 980.7 C.G.S. For details of more recent and accurate methods of determination, the reader should refer to the article CALORIMETRY, where tables of the variation of the specific heat of water with temperature are also given.

The second law of thermodynamics is a title often used to denote Carnot's principle or some equivalent mathematical expression. In some cases this title is not conferred on Carnot's principle itself, but on some axiom from which the principle may be indirectly deduced. These axioms, however, cannot as a rule be directly applied, so that it would appear preferable to take Carnot's principle itself as the second law. It may be observed that, as a matter of history, Carnot's principle was established and generally admitted before the principle of the conservation of energy as applied to heat, and that from this point of view the titles, first and second laws, are not particularly appropriate.

20. _Combination of Carnot's Principle with the Mechanical Theory._--A very instructive paper, as showing the state of the science of heat about this time, is that of C. H. A. Holtzmann, "On the Heat and Elasticity of Gases and Vapours" (Mannheim, 1845; Taylor's _Scientific Memoirs_, iv. 189). He points out that the theory of Laplace and Poisson does not agree with facts when applied to vapours, and that Clapeyron's formulae, though probably correct, contain an undetermined function (Carnot's F't, Clapeyron's 1/C) of the temperature. He determines the value of this function to be J/T by assuming, with Seguin and Mayer, that the work done in the isothermal expansion of a gas is a measure of the heat absorbed. From the then accepted value .078 of the difference of the specific heats of air, he finds the numerical value of J to be 374 kilogrammetres per kilo-calorie. _Assuming the heat equivalent of the work to remain in the gas_, he obtains expressions similar to Clapeyron's for the total heat and the specific heats. In consequence of this assumption, the formulae he obtained for adiabatic expansion were necessarily wrong, but no data existed at that time for testing them. In applying his formulae to vapours, he obtained an expression for the saturation-pressure of steam, which agreed with the empirical formula of Roche, and satisfied other experimental data on the supposition that the coefficient of expansion of steam was .00423, and its specific heat 1.69--values which are now known to be impossible, but which appeared at the time to give a very satisfactory explanation of the phenomena.

The essay of Hermann Helmholtz, _On the Conservation of Force_ (Berlin, 1847), discusses all the known cases of the transformation of energy, and is justly regarded as one of the chief landmarks in the establishment of the energy-principle. Helmholtz gives an admirable statement of the fundamental principle as applied to heat, but makes no attempt to formulate the correct equations of thermodynamics on the mechanical theory. He points out the fallacy of Holtzmann's (and Mayer's) calculation of the equivalent, but admits that it is supported by Joule's experiments, though he does not seem to appreciate the true value of Joule's work. He considers that Holtzmann's formulae are well supported by experiment, and are much preferable to Clapeyron's, because the value of the undetermined function F't is found. But he fails to notice that Holtzmann's equations are fundamentally inconsistent with the conservation of energy, because the heat equivalent of the external work done is supposed to remain in the gas.