Robert D. Whitaker University of South Florida Tampa. 33620
I
The Early Development of Kinetic Theory
The standard historical allusion to the kinetic theory of gases merely indicates that the theory was originated by Daniel Bernoulli in the early 18th century and eventually refined and extended by Clausius, Maxwell, and Boltzmann in the latter part of the 19th century. This paper reviews the nature of Bernoulli's work and other contributors to kinetic theory prior to the work of Clausius, Maxwell, and Boltzmann. One especially interesting and significant point is that the most outstanding work in this early period was done by a little-known Scotsman, John J. Waterston. Advancement of the kinetic theory was hampered by competition from the caloric theory, which imagined heat to be an indestructible fluid. With the support of Lavoisier, caloric theory was nourished primarily on the continent of Europe hy savants such as Laplace and Poisson. Among other things, this idea was used to explain the pressure of a gas on the basis of repulsions among particles. Caloric was assumed to be mutually repulsive and presumably was attracted by and surrounded each molecule of the gas. John Dalton, for example, adopted this idea in his work on the atmosphere. Opposition to kinetic theory was also sustained hy t h e . general suspicion with which atomic theory was viewed in the early 19th century. Although speculation concerning atoms had been prevalent for centuries, people were skeptical when it came to attributing to these hypothetical particles such definitie properties as mass and volume. The kinetic theory involves a literal belief in atoms that many scientists of that period preferred not to embrace ( I ) . Daniel Bernoulli, John Herapath, and John Waterston emerge as the principal workers prior to 1850 who attempted a more or less complete mathematical treatment of gases based on a set of kinetic molecular postulates. We shall consider Bernoulli's actual derivation (2)in more detail than that of Herapath (3) or Waterston, (4) both of which have been dealt with previously. The essential results of all three will be presented and briefly compared. Bernoulli-Early Derivation of ihe Kinetic Theory Daniel Bernoulli (1700-82)was a member of an illustrious Swiss family. He was born while his family was living in Holland, where his father was teaching mathematics. However, the family returned to Basel in 1705, where Daniel spent most of his life. Bernoulli studied medicine and was a practicing physician for a short time, but he soon turned to his real interests-mathematics and physics. For eight years be served as a professor of mathematics in St. Petersburg, Russia. In 1733, he became aprofessor of anatomy and hotany in Basel. He published "Hydrodynamica" in 1738. His kinetic theory represents only a very small part of this work, which deals with many aspects of the physics of fluids. Bernoulli's original symholism is somewhat confusing, so we shall use a more consistent and simpler set of symbols in the following derivation. The discussion, however, closely follows Bernoulli's reasoning. Bernoulli begins by considering a cylinder of unit height, fitted with a movable piston with a weight representing the force of the atmosphere (see figure). He makes several assumptions. 1) The particles in the cylinder are in rapid motion and sustain the weight by their repeated impacts. 2) The particles are very tiny spheres. 3) The particles are practically infinite in number and represent ordinary air.
Diagram of piston In
cylinder of air usad for Bernoulli's calculations
4) A greater weight will cause the piston ta move to the heights (figure),the velocity of the particles remaining the same.
Bernoulli fmt calculates the pressure necessary to compress the air. He asserts that the pressure of the air is ~ r o ~ o r t i o n a l to two factors, which we s h d call f, and fi, where?" ripresents the pressure effect due to the number of particles that are very near the piston, and f i represent the average frequency 01 collisions between the particles near the piston and the piston face. Bernoulli now sets 1 f"
= (pistonheight)2/3
and 1
h - -D i , - d where Dh is the mean distance between the centers of the particles a t any arbitrary piston height h, and d is the diameter of the spherical particles. We shall examine the rationale behind f. later on. We now let PI represent the atmospheric pressure (that is, the pressure of the air when the piston is a t unit height), and P, represent the pressure of the air when the piston is at the heights. Multiplying the above proportionalities and taking the ratio PJPI, we find Pa- 1 D l - d
6-w.m
Bernoulli states that Dh = (piston height)'". This proportionality is reasonable if we assume that the average distance between the centers of the particles is directly related to the cylinder volume. In particular, the volume and a sphere of radius Dh should be proportional to the cylinder volume, and hence the cube root of (Dh)3would he proportional to the cube root of the piston height. If we let D represent the mean distance between the center of the particles when the piston is at unit height, thenD1 = D(l)Im = D and D, = D(s)'l3. Bernoulli considers the relationshiv between DA and d hv assuming that if the air were compressed sufficknt~y, thbparticks would come intocontact when the piston height reachea Volume 56, Number 5, May 1979 1 315
some very small value m (Fig. 1). We thus can write Dm = d = D(m)'l3. Substitution of this value for d along with the above values for D l and D, into the expression for PSIPI yields
-.
"
\...,
\",
Bernoulli then argues that for ordinary air, or in fact any gas less dense than air, m is very much smaller than unity or s. Aooroximatelv then. m = 0. so that PJP? = 11s. Pressure is inversely proportionh to the cylinder height akd thus (since the cross-section of the cvlinder is constant) to volume. as required by Boyle's law. Boorse and Motz ( 5 ) have shown that Bernoulli's relationship for the factor f, results from considering the numher of particles contained in a small cylinder of height Dh and cross-sectional area the same as that of the piston. In essence, f, is proportional toDh, to the density of the air, p, expressed as numher of particles per unit volume, and to the area of the piston face, A.
f. " DhpA Since P = NIA (oiston heieht). where N reoresents the total number of p&icles in the cylinder, add Dh o; (piston hei~ht)'/~, Bernoulli's expression for .f,.. follows for a fixed value of N. When we consider the real significance off, and fi, the actual basis of Bernoulli's approach becomes apparent. Iff, rn DhpA and f; l/@h - d ) , and if total pressure, P = f,fi, then
where A has been dropped because the area of the piston is constant. and D(m)1/3has been substituted for d. Under the assumption, rn = 0. the above expression reduces to P p. ('onseuuently, what Hernoulli really did in his expressions which we ha;e called f, and f ; was simply require that pressure be proportional to density! Bernoulli started from kinetic molecular &sumptions, hut his calculation of pressure does not directly involve these assumptions. The obvious mechanical . orooerties of velocitv and mass of the narticles are . not involved. Aooarentlv Bernoulli thoueht that deviations from Bovle's .. law might readily he observed for gases more dense than air. He felt that the value of m could he determined from his pressure equation if careful exprimenu were carried out with gases at hirh " .uressures. Hernoulli did not sureest -- that deviations from ideality might also arise from intermolecualr attractions. Bernoulli finally considers the effect of temperature chanaes. Without any mathematical derivation, he declared of agas by heating it a t constant that increasing the volume indicates a "more intense motion in the particles of air." According to Bernoulli. the oressure is orooortional to the square of The velocity of theLparticlesbecake as their velocitv increases. hoth the numher of impacts and their intensity increase equally. Bernoulli thus correctly argued that there is a relationshio between the square of molecular speed and temperature. Experimental work with gases had not progressed sufficiently,however, to allow identification of an absolute temperature scale. His final equation becomes PJP1 = u2/s, where v is the velocity of the particles. Pressure is thus directly proportional to the square of molecular velocity and inversely proportional to volume. Bernoulli concludes his treatment with a discussion of a numher of subjects such as the problems of defining a quantitative temperature scale, gaseous effusion, and the variation of pressure and temperature with height in the atmosphere.
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ther's malt husiness. Early in life he turned to mathematics as a pastime and read the works of Newton and Laplace. Herapath gave up the malt husiness in 1815 and taught mathematics for the next 17 years, first in Bristol and later in London. During this period, he published his kinetic theory in a series of papers in the "Annals of Philosophy." He had first submitted his work to the Royal Society, hut Humphrey D a w and other referees reieded it. ostensiblv because it was n t the railroad business in too ipeculative. ~ e r a ~ a t h w einto 1832 and edited a railwav maeazine for several vears. In 1847. he published two volume; of a h o o k , " ~ a t h e m a ~ cl'rinciples al of Natural I'hilowr~h\." . . (I?,, in which he rurther exolained his kinetic theory. Herapath derived Bovle's law using an argument reminis. path saidthst the pressure cent d ;hat used hy ~ e & u l l ierap of a gas is proportiona1 hoth to 11 lr).'nnd to I 1 \'I :'so that the prcssure i&ersely proportional to the volume. He did not employ any cunsideration ol'the finite aizeof the particlesas Hernoulli had done. 1)ut in a later iectiun of his trx~k.Herapath does discuss ekpected deviations from Boyle's iaw hecause of the finite size of molecules. Herapath correctly deduced that pressure is proportional to the number of particles per unit volume, p, their mass, M, and the square of their velocity. However, he assumed that the temperature was equal to the momentum of the particles. Consequently, his ideal gas law took the form
Heraoath called T the "true tem~erature"and understood it to represent what today we call absolute temperature. Using ' Gav-Lussac's data for the exoansion of eases as a function of temperature, Herapath caiculated an absolute zero of -488"F.I He defined his "true temperature" by the equation 171
--where F is temoerature on the Fahrenheit scale. The factors 1000 and 480 enter because Herapath decided to make the "true temperature" of the ice ooint (32°F) equal to 1000. The square rodt arises from his incbrrect formation of the ideal gas law. Herapath attempted to apply his kinetic theory to various phenomena such as phase changes and latent heats, diffusion of gases, and the velocity of sound in gases. He succeeded in influencing a t least one other scientist. In 1848, J. P. Joule oresented a oaner . . which later was oublished (1851) in the :.?nemoirs uf the Manchester l.iteraiY and ~'hiluso~h'ical Soof I ~ P I R D Rtotal~~" c ~ r t v "in which h(. used the "hvuotheiis ". culate the velocity of hydrogen molecules from experimental densities. Joule's method of calculation can he summed up hv the expression for the root mean square velocity, u = where P is pressure and d is densitv in mass oer unit volume. Even though he did not explicitly use this equation, he arrived at the equivalent result in his areument. Joule also calculated the speiific heat of hydrogen and other diatomic gases. His results were incorrect, however, because he considered only translational degrees of freedom. The diatomic nature of most elemental gases was not realized at that time. This paper was later published in the Philosophical Magazine (8). Joule correctly interpreted absolute temperature and clearly stated that it is proportional to the square of the velocity of the gas molecules. The calculations and reasoning of Joule have been analyzed by Brush (9).
m-,
Waterston-Advancement
of the Kinetic Theory
John J. Waterston (1811-83) was horn and grew up in Edinburgh. His father had a successful business manufacturing sealing wax and other writing supplies. John was educated at
'Herapath-Contribution to the Kinetic Theory
John Herapath (1790-1868) was a self-educated Englishman. He was horn in Bristol and for a time worked in his fa316 1 Journal of Chemical Education
'
The data used by Herapath were somewhat in error. The modern value of absolute zero on the Fahrenheit scale is -459'.
the very fine Edinburgh High School and later attended lectures on a variety of sihjecis a t the university. He hecame a civil engineer and went to London in 1832 to pursue his career. In 1839, Waterston became naval instructor to the East India Comoanv's cadets a t Bombav. He remained in this nosition untii18i7 when he resignedand returned to s cot land. He activelv owsued scientific interests until a few vears before his death.u~aterstonnever married, hut he came from a large family with whom he remained in close contact. He mysteriously vanished in 1883 while on his morning outing. His nephew later reported that his uncle liked to walk on a new breakwater a t Leith and assumed that he had lost his footing and been carried out to sea by the swift tide. During the years in India, Waterston worked out his kinetic theory. He puhlished a book anonymously in 1843 with the stranee title. "Thouehts on the Mental Functions." Buried within this h w k on physiology is a discussion of kinetic theory. Two years later, through a friend, Waterston transmitted a paper to the Royal Society, "On the Physics of Media that are Composed of Free and Perfectly Elastic Molecules in a State of Motion" (10). The paper was read before the Royal Society in 1846, hut it was rejected for publication in the Transactions. An ahstract was printed in the Proceedings (I]), hut the paper itself remained entombed in the archives until Lord Ravleieh ran across it vears later arraneed for its nuhlication in i895 (12). waters& attempted tocall attedtion to his Daner bv nrivatelv circulatine an outline (131. In 1851. he Bubmitted a paper to the ~ r y t i s h~ssociatiohfor the k d vancement of Science, and a brief ahstract of the paper was puhlished in the Reports of the British Association (14). Waterston derived Boyle's law and the perfect gas equation. He proved that "the molecular uis uiua of a medium is equal to its tension actinr throuph three times its volume" (15).The term kinetic energy had not yet come into use, and energy of motion was commonly referred to as uis uiua (living force), mass times velocity squared. Using modern symbolism, Waterstnn's statement can he summarized as NMu" = P(3V), which can he rearranged to the familiar equation
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where P is the pressure (tension), V is the volume, N represents the total numher of molecules, M is the mass of each molecule, and uZ is the mean square velocity. Waterston clearly spoke of mean molecular velocities. Waterston identified absolute temperature with the mean square molecular velocity and defined a correct ahsolute t&nprrature wale. He co~kidereda mixture ofgasrs and deduced that [he nvrrage molenllnr i'is iruo of all species is equal under eq~lilihriumrunrlit~ons. He mlculated the rutin of spt.c~ficheat at constant p r e s s ~ ~tn r r that at constant volume. ? . (onaiderine unls tmnslatirmal dvgrees ot' rreedmn. His method would have produced the correct value of 1.67 for monatomic gases, but an arithmetic error in his calculations resulted in the ratio 1.33. Interestingly, Waterston's incorrect value corresponded fairly well with data then available for diatomic gases. In "Thoughts on the Mental Functions," he suggested the idea of mean free path, which he called the "impinging distance" and reasoned correctly that i t is inMany chemists are familiar with Stanislao Cannizarro's eloquent presentation of Avogadro's ideas to the Karlsruhe Congress in 1860. This presentation is generally considered to mark the beginning of a clear understanding by chemists of the diatomic nature of most elemental gases and the means of determining accurate atomic and molecular weights from gas density measurements. Most chemists do not realize that Clausius convincingly argued for the existence of elemental diatomic molecules in his 1857 paper. Clausius based his opinion both on chemical evidence and on heat capacity data. He reasoned that rotational as well as translational degrees of freedom are necessary to explain heat capacities of gases. Since a simple, spherical molecule has no rotational degrees of freedom, Clausius maintained that the "simple" molecules must consist of two atoms bound together.
versely proportional to the numher of molecules per unit volume and the square of the "diameter of molecular surface of impact" (16). Following his unfortunate experience with the Royal Society, Waterston turned to other scientific interests and was deeply involved in the physical chemistry of liquids and various topics in astronomy by the time he returned from India. Kinetic theory lay dormant until a short paper by August Kronig (17) appeared in 1856. Kronig derived an incorrect perfect gas equation. and briefly discussed al~soluttttemperature and specific heat. 'I'he follotving vear, 1~11dolf ('lausius published a beautifully clear and cmscisc: paprr ( 1 8 ) that marks the heginning of modern kinetic theory.? Clausius credited Kr6nig's paper of the prrvious year with spurring his own rfforts, "to publish thuse parts uf my o w views which I have not [Found expressed h\f Krmiel." \Vaterston's work thus ma\, srem tu have had no eifect onihe subsequent development i f kinetic theory. This conclusion is not comnletelv, . iustified. however. Waterston's t+l;,rt did not remain totally unnoticed. W . J . bl.Hankine was aware of at bast sonic ot' Watrrstun's work and suecificallv equation referred to Waterston's derivation of the perfect (19). E. E. Dauh (20)is of the opinion that Waterston's ideas had a profound influence upon Kronig, who was editor of Die Fortschritte der Physik in the 1850's. Dauh believes that Kronig probably read the ahstract of Waterston's 1851 paper (141. . . A renort hv Helmholtz of this ahstract anneared in Knmig's journal, so it is likely that thr ahstract irselt would have n a s s d thrnueh the hands of th~,editor.I)auh bases his opinion on these facts: (a) the ideas expressed in Kronig's oaoer oarallel those in Waterston's abstract: (bl Kronie's k k h e k a t i c a l ability was demonstrably weak: h e could &t correctly derive the ~ e r f e cgas t equation (which was not explicitly given in the abstract); and he seriously misunderstood the effects of intermolecular cdlisions. The likelihood is slim that Kronig could have derived even his incorrect perfect gas equation without prior knowledge of the variables involved; and (c) Krouig attempted uusuccessfully to derive the harometric formula along the lines suggested in the Waterston ahstract. If Daub's belief is correct, Kronig becomes merely a rather faulty "conduit" for the transmission of kinetic ideas from Waterston to Clausius. ~
~
..
Conclusion Prior to 1850, three persons worked out mathematical treatments of eases based on kinetic molecular assumntions. Ihniel Brrnoulli's ronrril~utionii interesting mainly hecause
