THE "ACCOUNT OF CARNOT'S THEORY OF THE MOTIVE POWER OF HEAT"--TRANSITION
TO THE DYNAMICAL THEORY OF HEAT
The meeting of Thomson and Joule at Oxford in 1847 was fraught with
important results to the theory of heat. Thomson had previously become
acquainted with Carnot's essay, most probably through Clapeyron's
account of it in the _Journal de l'École Polytechnique_, 1834, and had
adopted Carnot's view that when work was done by a heat engine heat was
merely let down from a body at one temperature to a body at a lower
temperature. Joule apparently knew nothing of Carnot's theory, and had
therefore come to the consideration of the subject without any
preconceived opinions. He had thus been led to form a clear notion of
heat as something which could be transformed into work, and _vice
versa_. This was the root idea of his attempt to find the dynamical
equivalent of heat. It was obvious that a heat engine took heat from a
source and gave heat to a refrigerator, and Joule naturally concluded
that the appearance of the work done by the engine must be accompanied
by the disappearance of a quantity of heat of which the work done was
the equivalent. He carried this idea consistently through all his work
upon energy-changes, not merely in heat engines but in what might be
called electric engines. For he pointed out that the heat produced in
the circuit of a voltaic battery was the equivalent of the
energy-changes within the battery, and that, moreover, when an
electromagnetic engine was driven by the current, or when
electrochemical decomposition was effected in a voltameter in the
circuit, the heat evolved in the circuit for a given expenditure of the
materials of the battery was less than it would otherwise have been, by
the equivalent of the work done by the engine, or of the chemical
changes effected in the voltameter. Thus Joule was in possession at an
earlier date than Thomson of the fundamental notion upon which the true
dynamical theory of heat engines is founded. Thomson, on the other hand,
as soon as he had received this idea, was able to add to it the
conception, derived from Carnot, of a reversible engine as the engine of
greatest efficiency, and to deduce in a highly original manner all the
consequences of these doctrines which go to make up the ordinary
thermodynamics even of the present time. Though Clausius was the first,
as we shall see, to deduce various important theorems, yet Thomson's
discussion of the question had a quality peculiarly its own. It was
marked by that freedom from unstated assumptions, from extraneous
considerations, from vagueness of statement and of thought, which
characterises all his applications of mathematics to physics. The
physical ideas are always set forth clearly and in such a manner that
their quantitative representation is immediate: we shall have an example
of this in the doctrine of absolute temperature. In most of the
thermodynamical discussions which take the great memoir of Clausius as
their starting point, temperature is supposed to be given by a
hypothetical something which is called a perfect gas, and it is very
difficult, if not impossible, to gather a precise notion of the
properties of such a gas and of the temperature scale thereon founded.
Thomson's scale enables a perfect gas to be defined, and the deviations
of the properties of ordinary gases from those of such a gas to be
observed and measured.
The idea, then, which Joule had communicated to Section A, when Thomson
interposed to call attention to its importance, was that work spent in
overcoming friction had its equivalent in the heat produced, that, in
fact, the amount of heat generated in such a case was proportional to
the work spent, quite irrespective of the materials used in the process,
provided no change of the internal energy of any of them took place so
as to affect the resulting quantity of heat. This forced upon physicists
the view pointed to by the doctrine of the immateriality of heat,
established by the experiments of Rumford and Davy, that heat itself was
a form of energy; and thus the principle of conservation of energy was
freed from its one defect, its apparent failure when work was done
against friction.
Rumford had noted the very great evolution of heat when gun-metal was
rubbed by a blunt borer, and had come to the reasonable conclusion that
what was evolved in apparently unlimited quantity by the abrasion or
cutting down of a negligible quantity of materials could not be a
material substance. He had also made a rough estimate of the relation
between the work spent in driving the borer by horse-power and the heat
generated. Joule's method of determining the work-equivalent of heat was
a refinement of Rumford's, but differed in the all-important respect
that accurate means were employed for measuring the expenditure of work
and the gain of heat. He stirred a liquid, such as water or mercury, in
a kind of churn driven by a falling weight. The range of descent of the
weight enabled the work consumed to be exactly estimated, and a
sensitive thermometer in the liquid measured the rise of temperature;
thus the heat produced was accurately determined. The rise of
temperature was very slight, and the change of state of the liquid, and
therefore any possible change in its internal energy, was infinitesimal.
The experiments were carried out with great care, and included very
exact measurements of the various corrections--for example, the amount
of work spent at pulleys and pivots without affecting the liquid, and
the loss of heat by radiation. The experiments proved that the work
spent on the liquid and the heat produced were in direct proportion to
one another. He found, finally, in 1850, that 772 foot-pounds of work at
Manchester generated one British thermal unit, that is, as much heat as
sufficed to raise a pound of water from 60° F. to 61° F. An
approximation to this conclusion was contained in the paper which he
communicated to the British Association at Oxford in 1847.
The results of a later determination made with an improved apparatus,
and completed in 1878, gave a very slightly higher result. When
corrected to the corresponding Fahrenheit degree on the air thermometer
it must be increased by somewhat less than one per cent. The exact
relation has been the subject during the last twenty years of much
refined experimental work, but without any serious alteration of the
number indicated above.
It is probable that in consequence of the conference which he had with
Joule at Oxford Thomson had his thoughts turned for some time almost
exclusively to the dynamical theory of heat engines. He worked at the
subject almost continuously for a long time, sending paper after paper
to the Edinburgh Royal Society. As we have seen, he had given Joule a
description of Carnot's essay on the Motive Power of Heat and the
conclusions, or some of them, therein contained. Joule's result, and the
thermodynamic law which it established, gave the key to the correction
of Carnot's theory necessary to bring it into line with a complete
doctrine of energy, which should take account of work done against
frictional resistances.
Mayer of Heilbronn had endeavoured to determine the dynamical equivalent
of heat in 1842, by calculating from the knowledge available at the time
of the two specific heats of air--the specific heat at constant pressure
and the specific heat at constant volume--the heat value of the work
spent in compressing air from a given volume to a smaller one. The
principle of this determination is easily understood, but it involves an
assumption that is not always clearly perceived. Let the air be imagined
confined in a cylinder closed by a frictionless piston, which is kept
from moving out under the air pressure by force applied from without.
Let heat be given to the air so as to raise its temperature, while the
piston moves out so as to keep the pressure constant. If the pressure be
p and the increase of volume be dv, the work done against the external
force is pdv. Let the rise of temperature be one degree of the
Centigrade scale, and the mass of air be one gramme, the heat given to
the gas is the specific heat Cp of the gas at constant pressure, for
there is only slight variation of specific heat with temperature. But if
the piston had been fixed the heat required for the same rise of
temperature would have been Cv, the specific heat at constant volume.
Now Mayer assumed that the excess of the specific heat Cp above Cv was
the thermal equivalent of the work pdv done in the former case. Thus he
obtained the equation J(Cp － Cv) = pdv, where J denotes the dynamical
equivalent of heat and Cp, Cv are taken in thermal units. But if a be
the coefficient of expansion of the air under constant pressure (that is
1⧸273), and v₀ be the volume of the air at 0° C., we have dv = av₀,
so that J(Cp － Cv) = apv₀. Now if p be one atmosphere, say 1.014 × 10^6
dynes per square centimetre, and the temperature be the freezing point
of water, the volume of a gramme of air is 1⧸.001293 in cubic
centimetres. Hence
J(Cp － Cv) = (1.014 × 10^6)⧸(273 × .001293)
from which, if Cp － Cv is known, the value of J can be found.
In Mayer's time the difference of the specific heats of air was
imperfectly known, and so J could not be found with anything like
accuracy. From Regnault's experiments on the specific heat at constant
pressure, and from the known ratio of the specific heats as deduced
from the velocity of sound combined with Regnault's result, the value of
Cp － Cv may be taken as .0686. Thus J works out to 42.2 × 10^6,
in ergs per calorie, which is not far from the true value. Mayer
obtained a result equivalent to 36.5 × 10^6 ergs per calorie.
The assumption on which this calculation is founded is that there is no
alteration of the internal energy of the gas in consequence of
expansion. If the air when raised in temperature, and at the same time
increased in volume, contained less internal energy than when simply
heated without alteration of volume, the energy evolved would be
available to aid the performance of the work done against external
forces, and less heat would be required, or, in the contrary case, more
heat would be required, than would be necessary if the internal energy
remained unaltered. Thus putting dW for pdv, the work done, e for the
internal energy before expansion, and dH for the heat given to the gas,
we have obviously the equation
JdH = de + dW
where de is the change of internal energy due to the alteration of
volume, together with the alteration of temperature. If now the
temperature be altered without expansion, no external work is done and
dW for that case is zero. Let ∂e and ∂H be the energy change and the
heat supplied, then in this case
J∂H = ∂e + O
Thus
J(dH － ∂H) = de － ∂e + dW
and the assumption is that de = ∂e, so that dW = J(dH － ∂H); that
is, dW = J(Cp － Cv), when the rise of temperature is 1° C. and the
mass of air is one gramme. This assumption requires justification, and
by an experiment of Joule's, which was repeated in a more sensitive form
devised by Thomson, it was shown to be a very close approximation to
the truth. Joule's experiment is well known: the explanation given
above may serve to make clear the nature of the research undertaken
later by Thomson and Joule conjointly.
The inverse process, the conversion of heat into work, required
investigation, and it is this that constitutes the science of
thermodynamics. It was the subject of the celebrated _Réflexions sur la
Puissance Motrice du Feu, et sur les Machines Propres à Développer cette
Puissance_, published in 1824 by Sadi Carnot, an uncle of the late
President of the French Republic. Only a few copies of this essay were
issued, and its text was known to very few persons twenty-four years
later, when it was reprinted by the Academy of Sciences. Its methods and
conclusions were set forth by Thomson in 1849 in a memoir which he
entitled, "An Account of Carnot's Theory of the Motive Power of Heat."
Numerical results deduced from Regnault's experiments on steam were
included; and the memoir as a whole led naturally in Thomson's hands to
a corrected theory of heat engines, which he published in 1852. Carnot's
view of the working of a heat engine was founded on the analogy of the
performance of work by a stream of water descending from a higher level
to a lower. The same quantity of water flows away in a given time from a
water wheel in the tail-race as is received in that time by the wheel
from the supply stream. Now a heat engine receives heat from a supplying
body, or source, at one temperature and parts with heat to another body
(for example, the condenser of a steam engine) at a lower temperature.
This body is usually called the refrigerator. According to Carnot these
temperatures corresponded to the two levels in the case of the water
wheel; the heat was what flowed through the engine. Thus in his theory
as much heat was given up by a heat engine to the body at the lower
temperature as was received by it from the source. The heat was simply
transferred from the body at the higher temperature to the body at the
lower; and this transference was supposed to be the source of the
work.[17]
The first law of thermodynamics based on Joule's proportionality of heat
produced to work expended, and the converse assumed and verified _a
posteriori_, showed that this view is erroneous, and that the heat
delivered to the refrigerator must be less in amount than that received
from the source, by exactly the amount which is converted into work,
together with the heat which, in an imperfect engine, is lost by
conduction, etc., from the cylinder or other working chamber. This
change was made by Thomson in his second paper: but he found the ideas
of Carnot of direct and fruitful application in the new theory. These
were the cycle of operations and the ideal reversible engine.
In the Carnot cycle the working substance--which might be a gas or a
vapour, or a liquid, or a vapour and its liquid in contact: it did not
matter what for the result--was supposed to be put through a succession
of changes in which the final state coincided with the initial. Thus the
substance having been brought back to the same physical condition as it
had when the cycle began, has the same internal energy as it had at the
beginning, and in the reckoning of the work done by or against external
forces, nothing requires to be set to the account of the working
substance. This is the first great advantage of the method of reasoning
which Carnot introduced.
The ideal engine was a very simple affair: but the notion of
reversibility is difficult to express in a form sufficiently definite
and precise. Carnot does not attempt this; he merely contents himself
with describing certain cycles of operations which obviously can be
carried through in the reverse order. Nor does Thomson go further in his
"Account of Carnot's Theory," though he states the criterion of a
perfect engine in the words, "A perfect thermodynamic engine is such
that, whatever amount of mechanical effect it can derive from a certain
thermal agency, if an equal amount be spent in working it backwards, an
equal reverse thermal effect will be produced." This proposition was
proved by Carnot: and the following formal statement in the essay is
made: "La puissance motrice de la chaleur est independante des agents
mis en œuvre pour la réaliser: sa quantité est fixée uniquement par
les temperatures des corps entre lesquels se fait, en dernier résultat,
le transport du calorique." The result involved in each, that the work
done in a cycle by an ideal engine depends on the temperatures between
which it works and not at all on the working substance, is, as we shall
see, of the greatest importance. The proof of the proposition, by
supposing a more efficient engine than the ideal one to exist, and to be
coupled with the latter, so that the more efficient would perform the
cycle forwards and the ideal engine the same cycle backwards, is well
known. In Carnot's view the former would do more work by letting down a
given quantity of heat from the higher to the lower temperature than was
spent on the latter in transferring the same quantity of heat from the
lower to the higher temperature, so that no heat would be taken from or
given to source or refrigerator, while there would be a gain of work on
the whole. This would be equivalent to admitting that useful work could
be continually performed without any resulting thermal or other change
in the agents performing the work. Even at that time this could not be
admitted as possible, and hence the supposition that a more efficient
engine than the reversible one could exist was untenable.
Carnot showed that the work done by an ideal engine, in transferring
heat from one temperature to another, was to be found by means of a
certain function of the temperature, hence called "Carnot's function."
The corresponding function in the true dynamical theory is always called
Carnot's. A certain assignment of value to it gave, as we shall see,
Thomson's famous absolute thermodynamic scale of temperature.
In the light of the facts and theories which now exist, and are almost
the commonplaces of physical text books, it is very interesting to
review the ideas and difficulties which occurred to the founders of the
science of heat sixty years ago. For example, Thomson asks, in his
"Account of Carnot's Theory," what becomes of the mechanical effect
which might be produced by heat which is transferred from one body to
another by conduction. The heat leaves one body and enters another and
no mechanical effect results: if it passed from one to the other through
a heat engine, mechanical effect would be produced: what is produced in
place of the mechanical effect which is lost? This he calls a very
"perplexing question," and hopes that it will, before long, be cleared
up. He states, further, that the difficulty would be entirely avoided by
abandoning Carnot's principle that mechanical effect is obtained by "the
transference of heat from one body to another at a lower temperate."
Joule urges precisely this solution of the difficulty in his paper, "On
the Changes of Temperature produced by the Rarefaction and Condensation
of Air" (_Phil. Mag._, May 1845). Thomson notes this, but adds, "If we
do so, however, we meet with innumerable other difficulties--insuperable
without further experimental investigation, and an entire reconstruction
of the theory of heat from its foundation. It is in reality to
experiment that we must look, either for a verification of Carnot's
axiom, and an explanation of the difficulty we have been considering, or
for an entirely new basis of the Theory of Heat."
The experiments here asked for had already, as was soon after perceived
by Thomson, been made by Joule, not merely in his determinations of the
dynamical equivalent of heat, but in his exceedingly important
investigation of the energy changes in the circuit of a voltaic cell, or
of a magneto-electric machine. Moreover, the answer to this "very
perplexing question" was afterwards to be given by Thomson himself in
his paper, "On a Universal Tendency in Nature to the Dissipation of
Mechanical Energy," published in the Edinburgh Proceedings in 1852.
Again, we find, a page or two earlier in the "Account of Carnot's
Theory," the question asked with respect to the heat evolved in the
circuit of a magneto-electric machine, "Is the heat which is evolved in
one part of the closed conductor merely transferred from those parts
which are subject to the inducing influence?" and the statement made
that Joule had examined this question, and decided that it must be
answered in the negative. But Thomson goes on to say, "Before we can
finally conclude that heat is absolutely generated in such operations,
it would be necessary to prove that the inducing magnet does not become
lower in temperature and thus compensate for the heat evolved in the
conductor."
Here, apparently, the idea of work done in moving the magnet, or the
conductor in the magnetic field, is not present to Thomson's mind; for
if it had been, the idea that the work thus spent might have its
equivalent, in part, at least, in heat generated in the circuit, would
no doubt have occurred to him and been stated. This idea had been used
just a year before by Helmholtz, in his essay "Die Erhaltung der Kraft,"
to account for the heat produced in the circuit by the induced current,
that is, to answer the first question put above in the sense in which
Joule answered it. The subject, however, was fully worked out by Thomson
in a paper published in the _Philosophical Magazine_ for December 1851,
to which we shall refer later.
Tables of the work performed by various steam engines working between
different stated temperatures were given at the close of the "Account,"
and compared with the theoretical "duty" as calculated for Carnot's
ideal perfect engine. Of course the theoretical duty was calculated from
the temperatures of the boiler and condenser; the much greater fall of
temperature from the furnace to the boiler was neglected as inevitable,
so that the loss involved in that fall is not taken account of. Carnot's
theory gave for the theoretical duty of one heat unit (equivalent to
1390 foot-pounds of work) 440 foot-pounds for boiler at 140° C. and
condenser at 30° C.; and the best performance recorded was 253
foot-pounds, giving a percentage of 57.5 per cent. The worst was that of
common engines consuming 12 lb. of coal per horse-power per hour, and
gave 38.1 foot-pounds, or a percentage of 8.6 per cent. These
percentages become on the dynamical theory 68 and 10.3, since the true
theoretical duty for the heat unit is only 371 foot-pounds.
It is worthy of notice that the indicator-diagram method of graphically
representing the changes in a cycle of operations is adopted in
Thomson's "Account," but does not occur in Carnot's essay. The cycles
consist of two isothermal changes and two adiabatic changes; that is,
two changes at the temperatures of the source and refrigerator
respectively, and two changes--from the higher to the lower temperature,
and from the lower to the higher. These changes are made subject to the
condition in each case that the substance neither gains nor loses energy
in the form of heat, but is cooled in the one case by expansion and
heated in the other by compression. The indicator diagram was due not to
Thomson but to Clapeyron (see p. 99 above), who used it to illustrate an
account of Carnot's theory.
There appeared in the issue of the Edinburgh _Philosophical
Transactions_ for January 2, 1849, along with the "Account of Carnot's
Theory," a paper by James Thomson, entitled, "Theoretical Considerations
on the Effect of Pressure in Lowering the Freezing Point of Water." The
author predicted that, unless the principle of conservation of energy
was at fault, the effect of increase of pressure on water in the act of
freezing would be to lower the freezing point; and he calculated from
Carnot's theory the amount of lowering which would be produced by a
given increment of pressure. The prediction thus made was tested by
experiments carried out in the Physical Laboratory by Thomson, and the
results obtained completely confirmed the conclusions arrived at by
theory. This prediction and its verification have been justly regarded
as of great importance in the history of the dynamical theory of heat;
and they afford an excellent example of the predictive character of a
true scientific theory. The theory of the matter will be referred to in
the next chapter.