In 1820 in England John Herapath, born 1790 at Bristol, critisized the derivation of the gas laws by Laplace and gave an account of the kinetic theory, but his paper was not accepted for publication in the Philosophical Transactions of the Royal Society because it was considered to be too speculative. It finally appeared in 1821 in the Philosoph ical Transactions and would have been forgotten if James Prescott Joule (1818-89) would not have been in uenced by it and by Herapath's book on Mathematical Physics published in 1847.

In 1847 Joule published a paper in the Manchester Courier with the title "On Matter, Living Force, and Heat" where he stated the principle of conservation of energy. From his experiments he concluded in this paper that heat is not a substance but a form of energy: "Experiment has shown that whenever living force [kinetic energy] is apparently destroyed or absorbed, heat is produced. The most frequent way in which living force is thus converted into heat is by means of friction." Furthermore he gave the amount of heat equivalent to the converted kinetic energy. A year later he read another paper in Manchester where he used Herapath's kinetic theory. It was not published until 1851 and at first got very little attention until Clausius quoted it in 1857.

The real breakthrough for the kinetic theory started in Germany after Karl Kronig (1822-79) published a paper about it in Poggendorfs Annalen der Physik in 1856. Contrary to the widespread belief that molecules of gas merely oscillate around definite positions of equilibrium he assumed that they move with constant velocity in straight lines until they strike against other molecules, or against the surface of the container. Kr.onig was actually a chemist but he had a great reputation because he was editor of Fortschritte der Physik, an annual review of physics, and he had great in uence in the German Physical Society. Kronig's paper apparently motivated Rudolf Clausius (1822-88) to publish on the kinetic theory. In fact Clausius writes that already before his first paper on heat in 1850 he had a very similar conception of heat as Kronig but in his former papers he intentionally avoided mentioning this conception, because his conclusions were deducible from general principles and did not depend on these special conceptions. Clausius was already well known from his papers on thermodynamics. In 1850 he had given his verbal formulation of the second law that there exists no thermo dynamic transformation whose sole effect is to extract a quantity of heat from a colder reservoir and to deliver it to a hotter reservoir. In the years until 1854 he had worked out its mathematical formulation. Since 1855 he was professor at the ETH in Zurich. When Clausius started to work on the kinetic theory it became fashionable. In 1857 he published his first paper on the kinetic theory with the title "The Nature of the Motion which we call Heat" where he quoted the papers of Kronig and of Joule. The English translation of his paper appeared in the same year in the Philosophical Magazine. Two important arguments against the kinetic theory of heat were the following:

How can heat traverse a vacuum if it is just irregular motion of matter particles? There is no matter in the vacuum which could propagate heat while the particles of caloric could easily penetrate through the vacuum.

C.H.D. Buys-Ballot (1817-90) argued that since gas particles in the kinetic theory move with velocities of a few hundred meters per second one would expect that gases diffuse and mix much more rapidly than observed.

In 1858 Clausius published a paper in which he could cope with the second of these objections by introducing the mean free path of a gas molecule. Gas molecules move at speeds of a few hundred meters an hour but they undergo collisions with other gas molecules which change their direction after a very short time of ight. The actual distance they can move on the average freely along a straight line in one direction is the mean free path l given by

where n is the number density of gas molecules and ÿ is the diameter of the hard sphere particles which approximate the gas molecules. For his estimate of the mean free path Clausius made the drastic approximation that only one particle is moving and all others are at rest. His result differs less than 10 per cent from the result in Eq. obtained by Maxwell one year later from a much more rened derivation. Clausius who in 1865 introduced the concept of entropy continued to work on the kinetic theory.

James Clerk Maxwell (1831-79), best known from his electromagnetic field theory which he developed in the years from 1855 to 1873, read his first paper on the kinetic theory in 1859 at a Meeting of the British Association at Aberdeen. With the title "Illustration of the Dynamical Theory of Gases" it appeared 1860 in print in the Philo sophical Magazine. While in earlier treatments the absolute value of the velocities of the molecules was considered to be rather uniform he was the first to assume a random motion for the molecules. For thermal equilibrium he could then derive from symmetry considerations his famous velocity distribution function which in modern notation is given by

where v^-> is the velocity and n the density of the molecules, m their mass, k Boltzmann's constant and T the absolute temperature. For the mean free path he then obtained

and for the viscosity of a dilute gas

where v^-> is the mean absolute value of the velocity. Inserting l into the last equation he obtained a value independent of the density and because of v^-> proportional to the square root of the absolute temperature

The density independence of the viscosity was quite unexpected since for a uid the viscosity in general increases with increasing density. After its experimental verification this result served as a strong argument in favour of the kinetic theory.