Rudolf Clausius

Rudolf Julius Emanuel Clausius (2 January 1822 –24 August 1888) was a German physicist and mathematician. He is considered one of the founders of the science of thermodynamics. By his restatement of Sadi Carnot's principle known as the Carnot cycle, he provided a more fundamental foundation for the theory of heat. His most important paper, On the Moving Force of Heat (1850) was first to declare the second law of thermodynamics. He introduced the concept of entropy in 1865, and the virial theorem for heat in 1870.

The Mechanical Theory of Heat (1867)
T. Archer Hirst, F.R.S.


 * My memoirs "On the Mechanical Theory of Heat" are of different kinds. Some are devoted to the development of the general theory and to the application thereof to those properties of bodies which are usually treated of in the doctrine of heat. Others have reference to the application of the mechanical theory of heat to electricity. ...Other memoirs... have reference to the conceptions I have formed of the molecular motions which we call heat. These conceptions, however, have no necessary connexion with the general theory, the latter being based solely on certain principles which may be accepted without adopting any particular view as to the nature of molecular motions. I have therefore kept the consideration of molecular motions quite distinct from the exposition of the general theory.
 * Preface (August, 1864)


 * The steam-engine having furnished us with a means of converting heat into a motive power, and our thoughts being thereby led to regard a certain quantity of work as an equivalent for the amount of heat expended in its production, the idea of establishing theoretically some fixed relation between a quantity of heat and the quantity of work which it can possibly produce, from which relation conclusions regarding the nature of heat itself might be deduced, naturally presents itself. Already, indeed, have many successful efforts been made with this view; I believe, however, that they have not exhausted the subject, but that, on the contrary, it merits the continued attention of physicists... The most important investigation in connexion with this subject is that of S. Carnot.
 * First Memoir. On the Moving Force of Heat and the Laws which may be Deduced Therefrom


 * The careful experiments of Joule, who developed heat in various ways by the application of mechanical force, establish almost to a certainty, not only the possibility of increasing the quantity of heat, but also the fact that the newly-produced heat is proportional to the work expended in its production.
 * First Memoir.

These circumstances, of which Carnot was also well aware, and the importance of which he expressly admitted, pressingly demand a comparison between heat and work, to be undertaken with reference to the divergent assumption that the production of work is not only due to an alteration in the distribution of heat, but to an actual consumption thereof; and inversely, that by the expenditure of work, heat may be produced.
 * Many facts have lately transpired which tend to overthrow the hypothesis that heat is itself a body, and to prove that it consists in a motion of the ultimate particles of bodies. If this be so, the general principles of mechanics may be applied to heat; this motion may be converted into work, the loss of  in each particular case being proportional to the quantity of work produced.
 * First Memoir.

1. The energy of the universe is constant. 2. The entropy of the universe tends to a maximum.
 * If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat.
 * Ninth Memoir. On Several Convenient Forms of the Fundamental Equations of the Mechanical Theory of Heat.

Quotes about Clausius

 * In The Kind of Motion We Call Heat, Clausius had shown how to relate the temperature and pressure of a volume of gas to the motion of the atoms, and was able to deduce their average speed. ...That calculation drew a quick response from the Dutch meteorologist Christopher Buys Ballot. ...It atoms were really flying through the air at hundreds of meters per second, shouldn't the fragrant vapors of a hot dinner race through the room...? In figuring out the answer... Clausias added a fundamentally new innovation to gas theory. Atoms... banged into each other a good deal. ...battling through all the other atoms ...What mattered was the average distance between collisions. This turned out to be an all-important quantity... and Clausius gave it the name mean free path.
 * David Lindley, Boltzmann's Atom: The Great Debate that Launched a Revolution in Physics (2001)


 * In their calculations, Clausius (and Waterston, for that matter) had imagined all atoms in a gas moving at the same speed. They knew this wasn't true... but they didn't have the mathematical sophistication to tackle the full problem. Maxwell... defined a mathematical function called the distribution of velocities, which kept track of how many atoms were moving at any particular speed relative to the average, and by dealing in terms of this distribution... was able to give his calculations a precision that those of Clausius lacked.
 * David Lindley, Boltzmann's Atom: The Great Debate that Launched a Revolution in Physics (2001)

Clausius pointed out that Carnot's perfect heat engine was rather an abstraction...The entropy... tended to increase in spontaneous natural processes, not to remain constant as in the perfect heat engine.
 * The views of Joule, Mayer, and others were assimilated into the theory of heat engines by Kelvin at Glasgow and Rudolph Clausius at Berlin. They noted that when gases and vapours expanded against an opposing force and performed mechanical work they lost heat. ...the law was put forward as a general principle by Clausius and Kelvin in 1851. Whilst the amount of heat decreased during the cycle of operations of the Carnot heat engine, it was seen that there was a quantity which remained constant throughout the cycle. The amount of heat given out was smaller than that taken in by the heat engine, but the quantity of heat taken in divided by the temperature of the heat source had quantitatively the same value as the amount of heat given out divided by the temperature of the heat sink. Clausius in 1865 termed this quotient, the entropy.
 * Stephen F. Mason, A History of the Sciences (1956)

The quantity T is the kinetic energy of the system... that part of the energy which is due to the motion of the parts of the system. ...In the second term, r is the distance between any two particles, and R is the attraction between them. ...The quantity &frac12;Rr or half the product of the attraction into the distance across which the attraction is exerted is defined by Clausius as the virial of the attraction. &sum;&sum;(&frac12;Rr)... indicates that the value of &frac12;Rr is to be found for every pair of particles and the results added together. Clausius has established this equation by a very simple mathematical process... it indicates two causes which may affect the pressure of the fluid on the vessel which contains it... We may therefore attribute the pressure of a fluid either to the motion of its particles or to a repulsion between them.
 * The equation of Clausius to which I must now call your attention is of the following form: $$pV=\frac{2}{3}T-\frac{2}{3}\sum\sum(\frac{1}{2}Rr).$$ Here p denotes the pressure of a fluid, and V the volume of the vessel which contains it. The product pV, in the case of gases at constant temperature, remains, as Boyle's Law tells us, nearly constant for different volumes and pressures. ...The other member of the equation consists of two terms, the first depending on the motion of the particles, and the second on the forces with which they act on each other.
 * James Clerk Maxwell, On the Dynamical Evidence of the Molecular Constitution of Bodies (1875) Nature Vol. XI as quoted in The Scientific Papers of James Clerk Maxwell (1890)


 * To him we are indebted for the conception of the mean length of the path of a molecule of a gas between its successive encounters with other molecules. As soon as it was seen how each molecule, after describing an exceedingly short path, encounters another, and then describes a new path in a quite different direction, it became evident that the rate of diffusion of gases depends not merely on the velocity of the molecules, but on the distance they travel between each encounter.
 * James Clerk Maxwell, On the Dynamical Evidence of the Molecular Constitution of Bodies (1875) Nature Vol. XI


 * He opened up a new field of mathematical physics by shewing how to deal mathematically with moving systems of innumerable molecules.
 * James Clerk Maxwell, On the Dynamical Evidence of the Molecular Constitution of Bodies (1875) Nature Vol. XI


 * Carnot's annunciation of his theory was defective in that it took no notice of the fact that the hot body gives out more heat than the cold one receives from it, and that it regarded as equal the amount of heat received upon one isothermal side of a cycle and that emitted from the other side; a principle that may hold good for infinitely small cycles, but not for larger ones, in which a difference exists between the thermic quantities proportioned to the size of the cycle. This error and the true condition as pointed out by Clausius are defined by Prof. Rankine, who says, in his paper "On the Economy of Heat in Expansive Machines": "Carnot was the first to assert the law that the ratio of the maximum mechanical effect to the whole heat expended in an expansive machine is a function solely of the two temperatures at which the heat is respectively received and emitted, and is independent of the nature of the working substance. But his investigations, not being based on the principle of the dynamic convertibility of heat, involve the fallacy that power can be produced out of nothing. The merit of combining Carnot's law, as it is termed, with that of the convertibility of heat and power, belongs to Mr. Clausius and Prof. William Thomson; and, in the shape in which they have brought it, it may be stated thus: The maximum proportion of heat converted into expansive power by any machine is a function solely of the temperatures at which heat is received and emitted by the working substance, which function for each pair of temperatures is the same for all substances in nature." The law as thus modified and newly expressed might, as M. Langlois remarks, be designated as the equation of Clausius. But Clausius himself, acknowledging the influence which the Frenchman's ideas had exercised upon him, called it the theorem of Carnot.
 * William John Macquorn Rankine "On the Economy of Heat in Expansive Machines" Transactions of the Royal Society of Edinburgh (1853) Vol. 20 as quoted in "Sketch of Rudolf Clausius," The Popular Science Monthly (1889) Vol. 35.

This is the only basic law of physics that distinguishes the past from the future. None of the others do. Not Newton's laws governing... mechanics... not the equations for electricity and magnatism... by Maxwell. Not Einstein's on relativistic gravity, nor those of quantum mechanics... by Heisenberg, Schrödinger, and Dirac. Not those for elementary particles... by twentieth-century physicists. Not one of these distinguishes... past from... future. If a sequence of events is allowed by these equations, so is the same sequence run backward in time.
 * Sadi’s pamphlet finds its way into the hands of... Rudolf Clausius. It is he who grasps the fundamental issue at stake, formulating a law that was destined to become famous: if nothing else around it changes, heat cannot pass from a cold body to a hot one. ...[A] ball may fall, but it can also come back up, by rebounding... Heat cannot.
 * Carlo Rovelli, The Order of Time (2018) 2. Loss of Direction. Where Does the Eternal Current Come From?


 * The essential feature of Maxwell's work was showing that the properties of gases made sense not if gas molecules all flew around at a similar "average" velocity, as Clausius had surmised, but only if they moved at all sorts of speeds, most near the average, but some substantially faster or slower, and a few very fast or slow. ...Just as Quetelet's average man was fictitious, and key insights into society came from analyzing the spread of features around the average, understanding gases meant figuring out the range and distribution of molecular velocities around the average. And that distribution, Maxwell calculated, matched the bell-shaped curve describing the range of measurement errors.
 * Tom Siegfried, A Beautiful Math (2006) Ch. 7: Quetelet's Statistics and Maxwell's Molecules, p. 139.


 * The name and fame of Professor Clausius stand as high in this country as in his own. ...his writings ...fell into my hands at a time when I knew but little of the Mechanical Theory of Heat. In those days their author was my teacher; and in many respects I am proud to acknowledge him as my teacher still.
 * John Tyndall (May, 1867) Introduction to The Mechanical Theory of Heat: With Its Applications to the Steam-engine and to the Physical Properties of Bodies (1867) by Rudolf Clausius, ed. T. Archer Hirst