**
Rudolf Clausius - the specialist in thermodynamics through all ages**

Rudolf Clausius was born on 2^{ND}
January 1822 in Koszalin, Poland and died on 24^{TH}
August 1888 in Bonn. He went to a private school belonging to his father who was
a Pastor and a regional legal adviser and later to a grammar school in Szczecin.
In 1840 he started his studies at Berlin University at the same time as teaching
at Friedrich-Wanderschen grammar school in Berlin. In 1847 he got a PhD in
Philosophy Sciences in Halle as a result of his studies on rainbow and other
optical phenomena in the atmosphere. After publishing his famous work in 1850 he
worked first at the Royal School of Artillery and Engineering in Berlin and from
1855 at the Higher Confederate Technical School (ETH) in Zurich, where he
productively worked as a professor of mathematical physics for twelve years.
During years 1867-1869 he worked at the University in
Würzburg.
Next he moved to Bonn, where he first worked as a professor and then even as a
rector of the university.

He married Adelaide Rimpau (1833-1875) and during this happy marriage he had two sons and four daughters. After the 1875 his scientific work decreased significantly due to the death of his wife who died after the birth of their sixth child. He married again in 1886 (two years before his death) to Sofia Sack (1862-1911) who gave birth to their son.

Today Clausius remains the main creator of the foundations of classical thermodynamics. He is rightly regarded as the creator of the conception of entropy and the Second Law of Thermodynamics. In this publication we would like to show that he was also the discoverer of the First Law of Thermodynamics for reversible Carnot cycles and for cycles using reversible phase transitions and a discoverer of a complicated state equation many years before the discovery of van der Waals equation.

A two-part article entitled :"About the moving power of heat
and the laws that can be derived from it for a science about heat"^{1}
was published in a volume No.79 (1850) of *Annalen der Physik *: a private
periodical of Poggendorff . Its author was Rudolf Clausius: a 28 years old
assistant professor of Berlin University and at the same time Instructor of
Nature Sciences in the Prussian School of Artillery. It was his third scientific work after graduation. His first publication was
his doctorate printed in 1847 in a private mathematical Crelle’s periodical and
his second work about the conditions of waves’ propagation in the crystalline
bodies was published in *Annalen der Physik* in 1849.

Science historians agree that Clausius’ third work was the
beginning of a new age in the history of physics. If thermodynamics is supposed
to be, using Rankine’s argument, a science about thorough foundations, exact
definitions and formulas and a science having an application in clearly
delimited circumstances^{2},
then the answer to the questions: who created these foundations and who set up
definitions and delimited applications; can be only one : Rudolf Clausius. He
was the first one who analyzed the heat engine of Carnot with a new perspective
which appeared in thermodynamics because of the discovery of the Energy
Conservation Law. The results of his work from 1850 turned out to be very
successful. They gave scientific foundations to the thermodynamics that had been
until then a mixture of false and true propositions. Clausius’ work which
defined the role of the Energy Conservation Law in the structure of the First
Law of Thermodynamics brought into being the science which we know today as *
Classical Thermodynamics*.

Clausius had two co-workers in his struggles with the
coherent foundations of thermodynamics: W.Rankine (1820-1872): a two years older
builder of roads and bridges and Sir William Thomson (1824-1907): a two years
younger *spiritus movens* of British science. Clausius refers many times to
the results of their work and vice versa, both those scientists quote Clausius’
works pointing out the important and inspiring questions but also carrying on a
controversy with his solutions. Significantly we don’t find the contemporaries
of Clausius (Herman Helmholtz (1821-1894) and Gustav Robert Kirchhoff
(1824-1879)) in the group of the founders of thermodynamics. Both scientists
appreciated the pioneer achievements of Clausius.

Rudolf Clausius didn’t belong to the group of the versatile
physicists of nineteenth century. Physicists like Kirchhoff and Clerk-Maxwell
worked with now unheard of facility in hydrodynamics, theory of elasticity,
thermodynamics, optics, acoustics and other sections of the field theory.
Clausius after his successful work in fixing the bases of Classical
Thermodynamics started to engaged himself in the bases of the kinetic theory of
gases. His first work in this field was entitled "About the nature of movement
which is called heat"^{3}.
The work concerned the problem formulated earlier by Rankine^{4}
of a derivation of the state equations and expressions concerning specific heat
from the kinetics of movement of separated gas molecules. Although the work
includes mostly a verbal description of "the movement which
is called heat", it became an initial work for the present-day, still
dynamically developing model of the kinetic theory of gases. Many hypothesis,
for example those concerning evaporation and condensation or the state equation
of the gas mixture are still successfully used in almost unchanged form in
modern thermodynamics. Also the very beginnings of a mathematical model,
developing Krönig’s
way of reasoning and exposing the kinetic energy of the translational motion of
the molecules, identified with the measure of phenomenological temperature are
up-to-date. This and Clausius’ others works^{5}
had powerful influence. Even such original British scientists as Maxwell and
Tait accepted and excellently developed this way of reasoning, although
Rankine’s previous model said that the temperature is a rotational kinetic
energy of molecules. This is a certain success, especially in a situation when
Clausius himself analyzed the rotational motion of the molecules and Rankine’s
work and stated that a main contribution to the state equation of perfect gas
comes from the kinetic energy of the translational motion and that the ratio of
the* vis viva *of the translational motion to the whole *vis viva* for
the air amounts 0.6315 (§ 20).

Another important concept introduced to the kinetic theory by
Clausius is the "virial" concept. According to this concept there is a relation
between an average *vis viva* and the sum of the products of the forces and
displacements of each molecule^{6}.
That work was a* spiritus movens* of the whole series of the works of
Boltzmann, Maxwell, Tait, Gibbs, Bodaszewski, Smoluchowski, Gąsiewski. It would
be difficult to imagine the present-day powerful structure of the kinetic theory
of gases without that work and without the virial theorem.

Maybe encouraged by Maxwell’s and Kirchhoff’s successes,
55-years old Clausius, being at the height of his creative possibilities, took
up the previously unknown to him theory of electromagnetism. His approach to it
is very original. He wants to examine the thermodynamic aspects of phenomena
like the passage of current and the magnetization of solids and gases^{7}.
He partly comes back to the subject of his doctorate and his handbook about the
role of potentials in mathematical physics,
especially in gravitation and electricity. The series of
works concerning electrodynamics is crowned by the new editions of Clausius’
books about the Mechanical Theory of Heat and about Thermoelectromagnetism.
Today Clausius’ achievements in this field, except Clausius-Mossotti formula,
are forgotten, although they were important at the moment of their formation.
Even Gibbs writes that the conception of chemical potential, which is
fundamental in phase transition theory, came to his mind during studding
Clausius’ books^{8}.
Clausius’ works can’t be measured by the quantity of his publications or
published books. However it doesn’t mean that he wasn’t creative and
hardworking, the catalogue of the works of the Royal Academy in London contains
a list of 77 works which were written by Clausius during years 1847-1875, and 25
of them were published in the most important at that time Poggendorff periodical
"Annalen der Physik". Its importance can be confirmed by the fact that his
articles were obligatorily translated into English and French and printed in *
Philosophical Magazine* and *Journal de Phisique*.

Around 1840 scientists were convinced that there is a relation or even a unity between the "forces of nature": an identity saying that "motion, heat, light, electricity, magnetism and chemical changes are the signs of one common force". Today the problem of "the unity of the forces of nature" which was the source of the studies on the uniform model of field theory is regarded as the one of those badly formulated problems which are rather dismissed than solved.

Mayer and Joule were also looking for the quantity laws of
the mutual changeability of the factors of nature, they created the concept of
energy and showed that two forms of energy, work and heat, are equivalent. If we
lift a heavy body and then let it fall, it can give back the work that we put
into it while lifting it. Also a compressed spring, a bent rod, compressed air
and electric current can do immense work. We can abstractly call this reserve of
work energy; independently whether it is used with utility or dissipated. The
concept of energy goes far beyond the concept of force known from Newton
dynamics and its development is partly implied by a change of views concerning
the nature of heat. Leibnitz called the product *mv*^{2}*
vis viva *and Coriolis proposed to call half of this product alive power.
Rankine proposed the name of actual energy for *1/2mv*^{2},
and Thomson called it kinetic energy. The conception of work also evolved from
Euler’s "force’s effort" to Joule’s "mechanical power".

When Clausius graduated in 1847 the Energy Conservation Law was already found to be a law of nature. Moreover the equivalence of heat and work announced in 1842 in a sacristy of St. Anna Church in Manchester by the producer of beer J.P.Joule was confirmed in academic circles. Around 1845, when Joule already doesn’t care about the new proofs for the equivalence of heat and work and focuses only on the more and more precise measurements, theorists: Helmholtz(1847), Clausius and Rankine(1850) and W.Thomson(1851) take up the work on the experimental material. It is the time of the formation of thermodynamics, whose task is the description of the conversion of heat energy into mechanical energy and vice versa. It is based upon two pillars: the First and the Second Laws of Thermodynamics.

We all know (thus begins the work of Sadi Carnot (1824))
that heat can be the cause of movement, that it even has the great moving
"power". The phenomenon that fascinated the 26 years old Frenchman, the creator
of heat engines theory, isn’t at all less mysterious in present times . The
conversion of heat energy into work is a problem not fully recognized or modeled
until today. The Carnot "moving power" as we see it today, is that part of
mechanical work which can be received from heat in the reversible way by the
Joule heat equivalent *"J"* , which means, as Carnot says, that "
everywhere where a difference of temperatures exists the moving power can be
produced and vice versa, everywhere where this power can be used it is possible
to create a difference of temperatures". The irreversibility which accompanies
every real engine is perceived by Carnot as "every equalization of temperatures
which occurs without generation of the moving power must be treated as an
essential loss".

The mysterious element responsible for the generation of work is "calorique" which alongside the specific volume is responsible for the energy conversion. However, as the Carnot principle says "this generation of moving power is due not to real usage of calorique but to moving it from a hot to a cold body". This is why calorique taken by the working substance from a heater is completely given to a cooler.

Carnot uses an analogy comparing the moving power of heat to
the moving power of a waterfall. Since the moving power of a waterfall depends
on its height and the quantity of liquid, so the moving power of heat depends,
by analogy, on a difference of temperatures and the quantity of "calorique". We
only don’t know , as Carnot says at the end of his work^{9}
, if a fall of calorique from 100°C
to 50°C
gives us less or more moving power than a fall of the same calorique from 50°C
to 0°C.

The Carnot principle, saying that calorique taken by a working substance is completely given to a cooler, became a source of doubt, especially among those research workers who tried to adjust it with the energy conservation law crystallized twenty years later. After the research of Joule, Mayer and Regnault it was already clear that everywhere, where moving power is used, calorique is produced simultaneously in the quantity directly proportional to the quantity of used moving power. And inversely, everywhere where heat is annihilated moving power is generated. Energy exists in nature in an unchanging quantity . It can never be annihilated or created; it can only be converted. The conversion of energy as a sign of an obedience and a controllability of nature shows us also another, still not subjugated power of nature: a continuos degradation of energy and its inevitable dissipation. Although the energy itself can’t be degraded a useful part of it ( the available energy) proceeds to decrease.

Viscosity, diffusion, friction, conductivity, crushing, scattering and dissipation are the names of the phenomena accompanying the degradation of energy. Conversion and degradation of energy are the two basic faces of nature.

If we replace, as it was done in the middle of the nineteenth
century, the word "calorique" by the word "heat", then the Carnot principle,
saying that in the perfect Carnot cycle heat is only taken and completely given,
will contradict the Energy Conservation Law. Clausius’ work from 1850 rejects
this part of the Carnot principle and says that " it’s not the base of the
Carnot principle that contradicts the Energy Conservation Law but only its part
saying that heat doesn’t get lost". "The remaining part of Carnot reasoning can
be adjusted with the principle of the equivalence of heat and work"^{10}.

A mathematical part of his work from 1850 begins with an
analysis of the state equation of a perfect gas *pv = R(a + t) *known
earlier in Mariotte and Gay-Lussac particular forms^{11}.
This equation expresses the perfect conversion of energy by combining pressure
*(p)*, specific volume* (v)* and temperature *(t)* of the working
substance. The gas constant *R* (like Regnault) constitutes the elastic gas
properties, and the constant (a) defines the temperature of absolute zero which
Clausius computed according to Magnus’ data as a=273°C.
The central mathematical element of Clausius’ idea, directed at proving that the
Carnot cycle satisfies the Energy Conservation Law, is an introduction of an
infinitesimal Carnot cycle containing many different and still in use "crossed
d", "d prim" and other signs for infinitesimal increments. The results of
Clausius reasoning, marked by formulas (II.) and (IIa.) are two standard
elements of every thermodynamics handbook even today. Only the original verbal
interpretation of the formula (IIa.), where Clausius says that heat during
conversion is divided into compensated heat and uncompensated heat is not used.

Clausius dedicates a great part of his work to the cycle, where the working substance is subject to a multiple number phase transition - best known as the evaporation of water and condensation of water vapor (now the Clausius-Rankine cycle). The other part of his work from 1850 Clausius dedicates to the derivation of the second, beside R, elastic constant of the perfect gas which is the working substance. The specific heats at constant volume and constant pressure and their ratio, called the Poisson coefficient, are discussed as an example. Equations (13)-(19) of Clausius’ work analyze four processes of the Clausius cycle which, if we consider the difficulties in expression of the Poisson adiabatic process without using the conception of entropy, requires many skills. Equation (IV.) is an expression for the function postulated by Clausius describing the change in heat caused by the change in volume. Its definition alongside (IIc.) makes it possible to describe the series of Clapeyron’s experiments documenting a need for a new state equation, better than the state equation of the perfect gas. This equation was derived by Clausius (formula No.(27) ). It is far more complicated than van der Waals famous equation from 1878.

Equation (24) for the latent heat of phase transition is also
up-to-date, although it is based on the old Regnault experimental data. The last
ten pages of his work Clausius dedicates to the analytic derivation of the
mechanical heat equivalent *J=1/A* . He shows that the constant *J*
derived with the help of the two elastic constants of the perfect gas (J=370) is
different from the value based on the Joule experiment (J=425). To solve this
problem he proposes using his new state equation, which has four elastic
constants and from the final calculation he arrives at J=421. We think that this
spectacular result was the reason why Clausius’ contemporaries paid attention to
his work. It is strange however that they didn’t connect this analytic
confirmation of the *J *value with Clausius’ new state equation, which was
certainly overlooked.

A contemporary estimation of Clausius’ work
would be in our opinion as follows: Remembering, thanks to Kuhn’s works, that at
least 12 research workers formulated the Energy Conservation Law at the same
time (Mayer, Joule, Colding, Helmholtz, Carnot, Seguin, Holtznamm, Hirn, Mohr,
Grave, Faraday and Liebing), we think that Clausius should be included in this
group. He was the first one who indicated the mathematical method of transition
from the local form of the energy balance to the integral form for the Carnot
cycle. It allowed, for the first time in literature, derivation of the
mechanical heat equivalent, *J *as a function of the thermal and calorific
state equation. Unfortunately this formulation of the First Law holds only for
completely reversible processes and we are not sure even today if the balance of
energy for a reversible cycle *W=JQ* is still true for an irreversible
cycle (for example if *W<JQ* is not true). The specialists in
thermodynamics are convinced that the First Law of Thermodynamics should be
comprised both reversible and irreversible processes. There is only no
compromise with reference to the integral form of this law for the Carnot cycle.
Let’s notice that if for irreversible (real) cycles,* W**≠**Q*
then the traditional way of counting *J* as *W/Q* for a measured value
of work, *W* and heat, *Q* is very doubtful. But in 1850 Clausius had
not yet taken into consideration yet the irreversibility of real cycles so he
looked for the reasons for a difference between the measured and analytically
derived value of* J* in too poor a state equation (only reversible, elastic
gas properties). The new state equation (eq.27) proposed by him crossed the
limits of the perception of his contemporaries. It was "discovered" 28 years
later by van der Waals in a much poorer version. However in our opinion the
problem of the right form of the energy balance equation and the First Law of
Thermodynamics is still open. Although there is a consensus concerning the local
form of the energy balance there is no agreement concerning the energy balance
in the irreversible Carnot cycles (e.g.
Δø=JQ-W,
where Δø
is an increment of Rayleigh dissipative function). It turns out that the
extension of the First Law of Thermodynamics for the irreversible cycles is not
so obvious case.

The work from 1850 has one especially important element for
the technics. Apart from the Carnot cycle using alternately two isothermal and
two adiabatic processes, it proposes a cycle for saturated steam based on a
reversible twofold phase transition: evaporation and condensation. These cycles
gave bases to the modern steam cycles used as the heat engines (right-way
Clausius-Rankine cycles^{12})
and as the heat pumps and coolers (left-way Linde cycles). The original Clausius
cycle for saturated steam has its only application today in nuclear power
stations, where a source of heat energy (nuclear reactor) has a limited
temperature and can’t be used for the production of properly superheated steam.
Yet, the Clausius cycles brought to perfection, with a resuperheating of steam
are the basis of the Polish power industry based on steam turbines. Today the
Clausius-Rankine cycle (the isobaric-adiabatic one) is different than the
original one (the isothermal-adiabatic one). But the dream of designers of the
greatest turbine companies is a return to this perfect engine by so-called
carnotization of Clausius-Rankine cycle. To do this they use the multistage,
regenerative preheating of a feeding water by trapped steam and other ideas
giving maximum efficiency.

The concept of entropy was formed in quite complicated circumstances, in the course of the evolution of ideas of three great pioneers of classical thermodynamics: Rankine, Clausius and Thomson. Rankine introduced entropy (he called it a metamorphic function) even in his first works about the rotative elasticity of gases (1851) and used it during the whole period of his activity only with reference to the reversible phenomena and the Second Law of Thermodynamics for the reversible phenomena. Thomson (1852) during the analysis of the irreversible Carnot cycles postulated an introduction of the Second Law of Thermodynamics in the form of energetic inequality D>0, where D is the dissipated energy, useless from the point of view of the possibilities of doing work in a cycle. Clausius (1854), contrary to Rankine analyzed the necessary conditions for the irreversibility of a cycle and the special forms of the conditions on which a cycle becomes reversible. That’s why his solution is complete, it includes the Second Law for both reversible and irreversible cycles.

But only in his work from 1862^{13}
Clausius gave both versions (reversible and irreversible) of the Second Law of
Thermodynamics. These expressions are (eq. I. and Ia.)

∫dQ/T=0 and ∫dQ/T≥0

described as:* The algebraic sum of the entropy of all
processes in the cycle can only be positive or in the extreme (reversible) case
can be equal to zero.*

Because of the lack of space, we can not take the reader through
all Clausius’ reasoning; we can only try a contemporary interpretation. Moreover
at the first reading it can appear that Clausius doesn’t postulate the Second
Law but only does complicated transformations on the formulas coming from the
First Law. But, as we conclude from his original reasoning, the Second Law for
the reversible cycles is needed as a supplementary condition in order that the
working substance could come to the same thermodynamic state after passing
through the whole cycle. It turns out that the equation of the initial and final
internal energy is not sufficient. Generally
internal energy, U is a function of two parameters, so after
passage through the whole cycle we can maintain the same value of the energy, U
although the parameters now have different values. So a condition fixing one of
those parameters should be added. Fortunately for the development of
thermodynamics Clausius chose the entropy dS=dQ/T which total change for
reversible cycles equals zero. So the Carnot cycle will be reversible if after
its passage the internal energy of the working substance is the same and one of
its parameters: specific entropy or specific volume stays unchanged. If Clausius
had chosen a specific volume then the Second Law based on it would be still
up-to-date, although that version also would have been attacked by Thomson and
Tait preferring energetic inequality. Clausius introduced the designation of
entropy as the letter S in his next work^{14},
where in an explicit way he decomposes entropy into compensated and dissipated
parts (eq.63).

S=Y+Z (*)

In the cases of reversible cycles a dissipated part is equal
to zero (Z=0) and entropy includes only reversible effects so it can increase as
well as decrease^{15}.
The famous equation (*) is also known in incremental form dS=dY+dZ=d_{r}S+d_{i}S,
where d_{r}S,
d_{i}S
are contemporary designations introduced by Prigogine and signifying reversible
and irreversible entropy increments. Today the quantity d_{i}S
is called the production of entropy, so not an original designation nor a name
proposed by Clausius was left unchanged. The Second Law is best described by the
last sentence of §11 of the work from 1865:* The uncompensated processes can
be only positive - d*_{i}*S*≥*0.*
The Clausius principle is most often used in this version, today under the name
of Clausius - Duhem inequality. Its efficacy comes from the fact that it is
important both for the cycles and for the local phenomena having place in the
working substance. Moreover it has the qualities of a general law defining how
the arrow of time and irreversibility in nature is perceived by us. Clausius
realized it too, for he ended his work in 1865 with conclusions coming from the
First and Second Laws for the whole Universe:

*1. The energy of the Universe is constant
2. The entropy of the Universe approaches the maximum value*

The last thesis has deep implications for the thermodynamics
of the Universe, maybe reaching even further than its author intended. The
evolution of an isolated system, as the Universe probably is, approaching to the
state of maximum entropy, will end in thermodynamic equilibrium which in
ultimate terms means the thermal death of the Universe. Nature in a state like
this loses the ability to change and the ability to control the changes on the
behalf of uncontrollable phenomena and, as Prigogine^{16}
says, a spontaneous and intrinsic activity which nature shows when we try to
subjugate it disappears. Complete equilibrium on all scales also means a loss of
energy able for further conversion and further dissipation. A complete lack of
gradients and any differences will lead to a standstill and maybe a not
completely cooled down universe will freeze in perfect equilibrium.

Clausius contemporaries took up his hypothesis as the first. On the British Islands W. Thomson and E. Tyndall were its apologists. Then this hypothesis was developed by W. Gibbs, J. Jeans, B. Steward and B.Russel. Philosophical and religious implications of this hypothesis were written and commented on by Prigogine and Stengers in a book "From chaos to order" and to this book we would like to direct an interested reader.

This question comes from William Thomson, the second (after
Rankine) Britisher competing with Clausius. He was well known to the majority
because of the loud excesses on the 126-ton yacht *Lalla Rookh*, received
from Queen Victoria alongside the title of Lord Kelvin. Thomson presented the
idea that the Second Law of Thermodynamics should rather describe the
degradation of energy than the condition of constantly positive production of
entropy. Against Clausius’ Second Law of Thermodynamics affirming that *heat
itself (without compensation) can’t pass from a cooler to a warmer body *he
gave the following argumentation. Clausius’ Second Law refers to the kind of
energy conversion which is called the passage of heat. In this way it is
distinguished from a different kind of energy conversion which is called work.
But according to the molecular heat theory, even the one developed by Clausius,
the only difference between those two forms of energy conversion is that the
movements and displacements of the molecules which are responsible for the heat
are so numerous and irregular, and their individual values are so small that
they are beyond our observation. Yet, the movements and displacements of greater
quantity of the molecules moving together are observed as work. However if we
imagine that we could bring our instruments to perfection and that they would be
able to observe each molecule with ease, then the difference between work and
heat will disappear and the passage of energy as heat will be perceived as a
transfer of work. Therefore the Second Law of Thermodynamics, in Thomson’s
opinion, should concern the degradation of all forms of energy^{17}.
For this purpose he created the concept of disposed energy^{18};
more useful, in his opinion, in the description of irreversible cycles.

The kinetic theory of gases and especially its branch
developed by Krönig,
Clausius and Maxwell met even greater critique by Thomson. The greatest defect
of the kinetic approach was an unclear passage from a microscopic description; a
completely reversible motion defined by the laws of Newton dynamics; to an
irreversible degradation of energy observed on a macroscopic level. This
objection Thomson raises
in a mathematical way, affirming that if the initial
equations of the molecules’ dynamics are not sensitive to the change of time (t)
for (-t), then why should the macroscopic result of averaging be sensitive to
it. Thomson maliciously says that this irreversibility with regard to time in
the kinetic theory equations can be assured only by a sorting demon and even by
a whole army of them^{19}.
Thomson tries to solve this problem in another paper^{20}
by replacing the concept of "the elastic spheres" in Clausius model of
collisions by "the sphere of activity" modeled on Boscovich atom. That
rationalization, in his opinion, could save the Maxwell model from "the invasion
of sorting demons".

Classical Thermodynamics is used to connect the Second Law of
Thermodynamics with the concept of entropy^{21}.
Rankine’s works show us that this concept can be already introduced on the
grounds of the reversible Carnot cycles: when an irreversible component of
entropy is equal to zero. Rankine calls a reversible (compensated) part of
entropy a metamorphic function and signifies it by the Greek letter ø
and the temperature related to it by
*θ*,
giving it the name of a metabatic function. The Second Law of Thermodynamics for
reversible Carnot cycles has a special form . It just confirms an existence of
entropy, thanks to which the quantity of heat energy which can be changed into
work is equal (ø_{1}-ø_{0})(θ_{1}-θ_{0})
and the efficiency of Carnot cycle (θ_{1}-θ_{0})/θ_{1}^{22}_{.}

From the time of his work in 1850 about 80 thousands papers have been written on the subject of classical thermodynamics. Hundreds of original experiments were also done and a part of them was supposed to show evidently that Clausius was wrong. So how is it possible that the laws discovered by Clausius (both known and unknown to our society) one hundred and fifty years ago are still up-to-date? The answer is simple: the reason is the genius of this modest man from Koszalin who got ahead of us all not by hundreds but maybe by thousands of years. Maybe the laws discovered by Clausius will stay true forever and then the title of "the specialist in thermodynamics through all ages" seems to be fitting.

And now, when we open our journals looking for new thermodynamic truths, we find the work of Clausius continually being enriched and confirmed everywhere.

- R. Clausius, Ueber die bewegende Kraft der Wärme und die Gesetze, welche sichdaraus für Wä rmelehre selbest ableiten lassen, Annalen der Physik, 79, 368-397, 500-524 (1850)
- W. Rankine, Outlines of the sciences of energetics, Proc. Phil. Soc. Glasgow, 3, no6 (1856)
- R. Clausius, Ueber die Art der Bewegung welche wir Wärme nennem, Annalen der Physik 100, 353-380 (1857)
- The Rankine's development was quite different than Clausius' one. He began from molecular (kinetic) heat theory, then he passed to thermodynamics and heat engines theory. His work in 1851 entitled "About the rotational elasticity theory in applications to steams and gases" developed, after Boscovicz and MacCullagh, the idea that heat is the implicit kinetic energy of the gas molecules rotation, proportional to the absolute temperature. That reasonable, from present-day point of view, approach was not developed at all. Clausius' theory which identifies heat with the kinetic energy of the translational and not rotational motion of molecules is the most common one until today.
- e.g. R. Clausius, Ueber die mittlere Länge der Wege, welche bei Molecularbewegung gasförmigen Körper von den einzelezen Molecülen zurückgelegt werden, nebst einegen anderen Bemerkungen über mechanischen Wärmetheorie, Annalen der Physik, 105, 239-258 (1858)
- R. Clausius, Ueber einen auf die Wärme anwendbaren mechanischen Satz, Sitzun. Niedderrheinishen Gesellschaft, Bonn pp 114-119 (1870)
- R. Clausius, Ueber die Ableitung eines neuen electrodynamischen Grundgesetzes, Crelle's Journal 82, 85-140 (1877). The word "new" refers only to the constitutive equations of the polarization and magnetization vectors. The structure of Maxwell's electrodynamics in 1877 was so stable that it did not require any serious changes. In that time the lacking elements of the theory were the expressions on the momentum vector, energy flux and dyad of the momentum flux.
- W. Gibbs, a letter to sir Oiver Lodge, Rep. Brit. Ass. Adv. Sci. pp 343-346 (1888)
- Probably M. Lippmann in his paper Cours de Thermodynamique (Paris, 1889) noticed that Carnot's "calorique" should be translated as "entropy" and not, as it was in Poggendroff's translation, as "wärme". We can rely on Lippmann's opinion because he was a two-languages researcher coming from Alsace.
- We should remind here that Clausius knew only the German translation of Clapeyron's publication (1834) published in Ann. der Physik, vol 59 (1835). Clausius also didn't know what exactly the concept of calorique meant because his generation completely rejected the earlier "calorique" conception of nature of heat. It seems that only Rankine didn't explain calorique as a material notion of heat but he gave to it a meaning of "metamorphic function", the quantity which Clausius called in 1865 the entropy. Clausius uses the word "calorique" in his paper from 1850 only once; he writes: "If the hypothesis of the indestructibility of calorique is something secondary in Carnot's proving then we should recognize what changes should be introduced to Carnot's reasoning to adjust his theorem with the Low of Equivalence of Heat and Work".
- This is the equation No. (I). All more important formulas from (I.) to (V.a) Clausius signifies by Roman numbers and the remaining 35 formulas he signifies by Arabic numbers. The Marriote's Low he signifies in abbreviation by M, and the the Gay-Lussac Low by G. So the present invasion of abbreviations both in thermodynamics and in the whole field theory has a European and not an American pedigree.
- This two-part name comes from the fact that in 1850 Rankine and Clausius, not knowing anything about each other, worked on the cycles with the phase transition. They came to the surprising conclusion that the specific heat of the saturated steam is, in the range of the temperatures 0°C - 200°C, negative.
- R. Clausius, Mitt. der Natur. Gesselschaft in Zürich, vol 7, 48-86 (1862)
- R. Clausius, über verschiedene für die Anwendung bequene Formen der Hauptgleichungen der mechanischen Wärmetheorie, Annalen der Physik 125, 313-356 (1865)
- An equivalent of the decomposition S=Y+Z for the specific volume is
ν=ν
_{Y}+ν_{Z}: the division to the reversible (compensated, elastic) part and the dissipated (viscous, plastic ) part. - I. Prigogine, I. Stengers, From chaos to order, PIW,1990
- His "almighty" low of the energy dissipation Thomson introduced in the article entitled: "About the universal susceptibility of nature to the dissipation of mechanical energy". Proc. Roy. Soc. Edinburgh, (1852)
- the available energy - a quantity defined by the formula No. (8) in the work from Trans. Roy. Soc. Edinburgh, March, 1851, p.174 and signified by the letter W. Sometimes modern thermodynamics comes back to that approach.
- W. Thomson, Nautre, 9, p 441 (1874)
- W. Thomson, Nautre, 6, p 355 (1891)
- e.g. M. Planc, Treatise on Thermodynamics, Longmans, 1926, § 114.
- W. Rankine. Outlines of the sciences of energetics, Proc. Phil. Soc. Glasgow, vol III (1855)