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.
That a quantity of heat equivalent to the work performed actually
disappears when a gas does work in expansion, was first shown by Joule
in the paper on condensation and rarefaction of air (1845) already
referred to. At the conclusion of this paper he felt justified by direct
experimental evidence in reasserting definitely the hypothesis of Seguin
(_loc. cit._ p. 383) that "the steam while expanding in the cylinder
loses heat in quantity exactly proportional to the mechanical force
developed, and that on the condensation of the steam the heat thus
converted into power is not given back." He did not see his way to
reconcile this conclusion with Clapeyron's description of Carnot's
cycle. At a later date, in a letter to Professor W. Thomson (Lord
Kelvin) (1848), he pointed out that, since, according to his own
experiments, the work done in the expansion of a gas at constant
temperature is equivalent to the heat absorbed, by equating Carnot's
expressions (given in S 17) for the work done and the heat absorbed, the
value of Carnot's function F't must be equal to J/T, in order to
reconcile his principle with the mechanical theory.
Professor W. Thomson gave an account of Carnot's theory (_Trans. Roy.
Soc. Edin._, Jan. 1849), in which he recognized the discrepancy between
Clapeyron's statement and Joule's experiments, but did not see his way
out of the difficulty. He therefore adopted Carnot's principle
provisionally, and proceeded to calculate a table of values of Carnot's
function F't, from the values of the total-heat and vapour-pressure of
steam-then recently determined by Regnault (_Memoires de l'Institut de
Paris_, 1847). In making the calculation, he assumed that the specific
volume v of saturated steam at any temperature T and pressure p is that
given by the gaseous laws, pv = RT. The results are otherwise correct so
far as Regnault's data are accurate, because the values of the
efficiency per degree F't are not affected by any assumption with regard
to the nature of heat. He obtained the values of the efficiency F't over
a finite range from t to 0 deg. C., by adding up the values of F't for
the separate degrees. This latter proceeding is inconsistent with the
mechanical theory, but is the correct method on the assumption that the
heat given up to the condenser is equal to that taken from the source.
The values he obtained for F't agreed very well with those previously
given by Carnot and Clapeyron, and showed that this function diminishes
with rise of temperature roughly in the inverse ratio of T, as suggested
by Joule.
R. J. E. Clausius (_Pogg. Ann._, 1850, 79, p. 369) and W. J. M. Rankine
(_Trans. Roy. Soc. Edin._, 1850) were the first to develop the correct
equations of thermodynamics on the mechanical theory. When heat was
supplied to a body to change its temperature or state, part remained in
the body as intrinsic heat energy E, but part was converted into
external work of expansion W and ceased to exist as heat. The part
remaining in the body was always the same for the same change of state,
however performed, as required by Carnot's fundamental axiom, but the
part corresponding to the external work was necessarily different for
different values of the work done. Thus in any cycle in which the body
was exactly restored to its initial state, the heat remaining in the
body would always be the same, or as Carnot puts it, the quantities of
heat absorbed and given out in its diverse transformations are exactly
"compensated," so far as the body is concerned. But the quantities of
heat absorbed and given out are not necessarily equal. On the contrary,
they differ by the equivalent of the external work done in the cycle.
Applying this principle to the case of steam, Clausius deduced a fact
previously unknown, that the specific heat of steam maintained in a
state of saturation is negative, which was also deduced by Rankine (loc.
cit.) about the same time. In applying the principle to gases Clausius
assumes (with Mayer and Holtzmann) that the heat absorbed by a gas in
isothermal expansion is equivalent to the work done, but he does not
appear to be acquainted with Joule's experiment, and the reasons he
adduces in support of this assumption are not conclusive. This being
admitted, he deduces from the energy principle alone the propositions
already given by Carnot with reference to gases, and shows in addition
that the specific heat of a perfect gas must be independent of the
density. In the second part of his paper he introduces Carnot's
principle, which he quotes as follows: "The performance of work is
equivalent to a transference of heat from a hot to a cold body without
the quantity of heat being thereby diminished." This is not Carnot's way
of stating his principle (see S 15), but has the effect of exaggerating
the importance of Clapeyron's unnecessary assumption. By equating the
expressions given by Carnot for the work done and the heat absorbed in
the expansion of a gas, he deduces (following Holtzmann) the value J/T
for Carnot's function F't (which Clapeyron denotes by 1/C). He shows
that this assumption gives values of Carnot's function which agree
fairly well with those calculated by Clapeyron and Thomson, and that it
leads to values of the mechanical equivalent not differing greatly from
those of Joule. Substituting the value J/T for C in the analytical
expressions given by Clapeyron for the latent heat of expansion and
vaporization, these relations are immediately reduced to their modern
form (see THERMODYNAMICS, S 4). Being unacquainted with Carnot's
original work, but recognizing the invalidity of Clapeyron's description
of Carnot's cycle, Clausius substituted a proof consistent with the
mechanical theory, which he based on the axiom that "heat cannot of
itself pass from cold to hot." The proof on this basis involves the
application of the energy principle, which does not appear to be
necessary, and the axiom to which final appeal is made does not appear
more convincing than Carnot's. Strange to say, Clausius did not in this
paper give the expression for the efficiency in a Carnot cycle of finite
range (Carnot's Ft) which follows immediately from the value J/T assumed
for the efficiency F't of a cycle of infinitesimal range at the
temperature t C or T Abs.
Rankine did not make the same assumption as Clausius explicitly, but
applied the mechanical theory of heat to the development of his
hypothesis of molecular vortices, and deduced from it a number of
results similar to those obtained by Clausius. Unfortunately the paper
(loc. cit.) was not published till some time later, but in a summary
given in the _Phil. Mag._ (July 1851) the principal results were
detailed. Assuming the value of Joule's equivalent, Rankine deduced the
value 0.2404 for the specific heat of air at constant pressure, in place
of 0.267 as found by Delaroche and Berard. The subsequent verification
of this value by Regnault (_Comptes rendus_, 1853) afforded strong
confirmation of the accuracy of Joule's work. In a note appended to the
abstract in the _Phil. Mag._ Rankine states that he has succeeded in
proving that the maximum efficiency of an engine working in a Carnot
cycle of finite range t1 to t0 is of the form (t1 - t0)/(t1 - k), where
k is a constant, the same for all substances. This is correct if t
represents temperature Centigrade, and k = -273.
Professor W. Thomson (Lord Kelvin) in a paper "On the Dynamical Theory
of Heat" (_Trans. Roy. Soc. Edin._, 1851, first published in the _Phil.
Mag._, 1852) gave a very clear statement of the position of the theory
at that time. He showed that the value F't = J/T, assumed for Carnot's
function by Clausius without any experimental justification, rested
solely on the evidence of Joule's experiment, and might possibly not be
true at all temperatures. Assuming the value J/T with this reservation,
he gave as the expression for the efficiency over a finite range t1 to
t0 C., or T1 to T0 Abs., the result,