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the velocity of sound determination for “vapour of mercury” by Kundt and Warburg: Maxwell’s stated result for that situa- tion is that the speed of sound becomes c = (51/2/3)vrms, where vrms = √<v2> is the root-mean-square (rms) velocity from Maxwell (1860). His result was based on the assumption of “no movement of rotation” caused by molecular “encoun- ters” (i.e., collisions) between the spherical molecules. Max- well’s result may be arrived at as follows. (1) The measure- ments of Kundt and Warburg were consistent with γ = 5/3, which from Equation 2 requires β = 1, corresponding to no rotational kinetic energy. (2) In modern notation, Maxwell’s (1860) result for the average translational kinetic energy per molecule is such that (M/2)<v2> = (3/2)kBT, where kB is now known as Boltzmann’s constant. (3) Also in modern nota- tion, Boyle’s law gives (P/ρ) = kBT/M. These relationships combine to give Maxwell’s asserted relationship between the speed of sound (c) and vrms. Preston restated this relation- ship in a paper in Nature in 1878, adding that Maxwell’s ex- pression requires a slight additional correction in the case of most gases because of the movements of rotation devel- oped at the collision of molecules. Maxwell noted in his 1879 analysis of thermal transpiration that Kundt and Warburg’s result indicated that molecules of mercury gas “do not take up any sensible amount of energy in the form of internal motion” (Niven, 1890). His attention to their result is also implied in Campbell and Garnett’s overview (1882, p. 569) of Maxwell’s research on gases.
Preston’s 1876 letter to Maxwell and his 1877 publication are also important because he explicitly mentions Waterston’s (1858) qualitative attempt to relate the propagation of sound in gases with an earlier more qualitative approach to the ki- netic theory of gases. This appears to be the only clear indi- cation that Maxwell ever became aware of Waterston’s prior interest in kinetic theory. It is noteworthy, however, that from the abstract of Maxwell’s initial presentation for the 1859 BAAS, he planned to apply his kinetic theory of gases to the propagation of sound. That application was, howev- er, not mentioned in his associated publication (Maxwell, 1860). It is plausible that because Maxwell couldn’t resolve the aforementioned difference between the kinetic theory prediction for γ and the value implied by measured sound speeds in air (typically close to γ of 1.41), he neglected to pursue that application. In 1878, in one of his final papers, Maxwell gave a rigorous derivation of his energy equiparti- tion theorem for gases independent of particular properties assumed of molecules (Niven, 1890).
After Maxwell: Applications
and New Physics
In an early advance after Maxwell’s death, Hendrik A. Lo- rentz (1853-1928) considered sound propagation in gases in 1880 by examining the near-equilibrium molecular distribu- tion function (Kox, 1990). Lorentz obtained Laplace’s result (c = √γP/ρ) and introduced a new transport coefficient cor- responding to the one eventually known as the bulk viscos- ity. Maxwell’s proofs of the energy equipartition theorem remained controversial, although by 1900, Rayleigh strongly supported Maxwell’s derivation of 1878. Given the diffi- culties in predicting the heat capacities of common gases, Rayleigh’s support appears to have contributed to Kelvin’s concerns in his famous 1901 address, “Nineteenth century clouds over the dynamical theory of heat and light” (Gar- ber, 1978). Eventually, the incorporation of rotational and vibrational motion of diatomic gases (and other molecular gases) required quantum mechanical reasoning and the rec- ognition of characteristic rotational and vibrational activa- tion temperatures typically lying respectively far below and far above room temperature (Rushbrooke, 1949). In 1916- 1917, Sidney Chapman and David Enskog extended the ap- proach of Maxwell (1867) to different molecular force laws. Molecular velocity measurements with molecular beams in a vacuum in around 1930 eventually confirmed Maxwell’s predicted velocity distribution. Herzfeld and Litovitz (1959) described the advances in the understanding of absorption and dispersion of sound in gases in the 1950s as a significant “generalization” of Maxwell’s 1867 analysis of relaxation; the inclusion of timescales associated with the relaxation of ro- tational and vibrational excitations was also needed. By the 1970s, energy equipartition between the modes of complex macroscopic dynamical systems was found useful for “sta- tistical energy analysis” in acoustics. In other developments, eventually b db dφ in Equation 3 became known as a differ- ential molecular scattering cross section, and by the 1930s, there was interest in their quantum mechanical evaluation using scattering phase shifts. Partially analogous expressions have been recently reintroduced in acoustics for the radia- tion force on spheres (Zhang and Marston, 2016). In Max- well’s January 1874 anonymous book review, he examined the consequences of an instantaneous reversal of the direc- tion of motion of “every particle in the universe” so that “ev- erything would run backwards.” That thought experiment, commonly associated with an 1876 publication of Josef Los- chmidt, was also considered in February 1874 by William
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