indi an Juurnal of Chemical Tech nology VoL II. March :2004. pp. :275-2::\5

~-

John Herapath-The beginning of the kinetic theory of gas(}S -.. .-.0-- -'

\ J~e Wisniak*

( Department of Chemical Engineering, Ben-Gurion University of the Negev. Beer-Sheva. Israel 84 10S J \ /--"

,-/John Herapath ( 1790-1 868) , a self-taught scientist, was the first to present i.l comprehensi ve description of the structure of a gas. in which the molecules moved and co llided wi th each other. instead of vi brating in a fixed iatti ce. His mode l made the wrong assumption that the momentum of the molecules , instead of thei r ki net ic energy meas ur~d

the lCmperature. He used his mode l to ca lculate for the first tim~ the ve loci ty of a mol ecu le. the veloc it y of sound and to describe effus ion phenomena through small pores , and adiabatic compress ion. Hi s speculati ve approach put him at odds with the empiri cal approach of the British sc ientifi c estab li shment of his time.

Keywords: Kinetic theory of gases

Life and career John Herapath was born on May 30, 1790, in

Bristol , England, the son of a malster and cousin of William Bird Herapath ( 1820-1 868), the chemist for whom the compound herapathite [quinine iodosulfate, used by Edwin Land (1909-1991) in the early form s of Polaroid] is named.

Most of what is now known about Herapath is contai ned in the ob ituary notices that appeared in Herapafh 's Raihmy }our/w( He was large ly se l f­educated, and did not attend a University for sc ient ific education . Early in life he learned French, turned to mathematics as a pastime, and read the works of Isaac Newton (1642-1727) and of the great French mathematicians of the late eighteenth century, such as Pierre-Simon Laplace ( 1749- i 827). From them he captured the specu lati ve approach to science, which led him to deve lop his model of a kinetic theory for gases, alld at the same time put him in conflict with th e empirical approach of th e Briti sh sc ientific establishment of his time.

He worked in hi s father 's malt business until his marriage in 18 15 ; afte rwards he started a private "school for mathematics and for preparing young men for the na vy" , first in Bristo l and then in London. Accord i ng to Brush 2 3 th is venture was not very successful because of Herapath 's clash with the Royal

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Society after Humphry Davy ( 1778- 1819) and other refe rees rejected hi s memoir on a kinetic theory of gases, allegedly because it was too specu lat ive (see below). His family responsibilities increased great ly during this period since by 1837 he had eleven children. ranging in age from one to twenty- two. During thi s period he published his kinetic theory in a se ries of papers in the A I/I/als ot' Philosoph.\', a second line journal.

By 1811 Herapath was engaged in resea rches on the theory of lunar motion attempting to reconcile lsaac Newton 's ( 1642-1727) calculati ons with observati on. Having fail ed to und e rstand th e ca li se o f these di screpancies, he turned to Newton's opinion regarding the cause of gravitation : Throughout all space is diffused a very subtle and highly elastic medium, which Newton called a~ th er, composed of ex tremely sma ll and act ive particles. This medium is of the same nature as air or any other gas, and can be expanded or contracted by an increase or diminution of heat or temperature . Herapath reasoned that if the a~ th e r were of the sa me nature as the atmosphere and other gaseous bodies then its variation in density mi ght be due to th e hi gh temperature of the sun and to a lesser degree of the planets 4 . He agreed that the principle of equal ity between action and reaction was perfect ly valid when considering homogeneous bodies of the same temperature but that this was not the case when the bodies had different composition or were at different temperatures , a difference should be observed in the ir attract ive forces

276 I DIAN J. CHEM. TECH TOL.. MARCH 2004

proporti onal to the greatness of the di ssimilarity. Other th ings being equal, the attracti ve forces would be proporti onal to the temperature, th at is, an increase in tempe rature woul d res ul t in an augment ati on of attraction. To justify hi s claim he qu oted the fac t that a reg ular di minut ion of the attrac ti ve fo rces of the pla nets reckonin g fro m the sun had been observed, whi ch showed that the co lder pl anets had less attraction than the warmer. The fac t that the pl anet Uranus did not sa tisfy thi s rule was not a negati on of hi s thes is because it cou ld we ll be that the constituti on of th is pla net was substantiall y different fro m the res t. If the attraction fo rce was proporti onal to the temperature it followed that since the po lar reg ions of the earth were co lder than the eq uatorial, the attracti on of the whole earth varied from weaker values at the poles to a stronger one at the equator and , therefore, the isochronal pendu lum mu st be shorter under the poles th an at the equator. Not onl y th at, if the temperature in the southern hemisphere was lower enough than that in the northern onc, it would affect the lunar theory, it would accelerate the lunar moti on in certain positi ons and retard it in others; since the earth's action on the moon is greater when the earth is nea rer to the su n.

In 1815 Herapa th submitted a short account of his work to Allilols o/ Philosophv, stating hi s results in the forlll of a probl em proposed to mathematicians , to di scover whet her others already knew hi s ideas. A sUll1m ary of hi s res ult s was publi shed in 18 16.) Herapath then worked out the detail s of hi s theory during the nex t year, applying it to the thermal properties of gases and liqu ids as well as to grav itati on, and eventually submitted his main paper to the Royal Society in 1820 after ha ving made the necessary tactful inquiries of influenti al members

23. Davies Gilbel"l ( 1767-1 839;

hi s fo rmer name was Dav ies Gidd y) , the Fe ll ow to who m he subm itted it, rep lied, "whi le I was much pleased wi th the great ingenuity displ ayed th roughout the whole . . . r entertained strong doubts on the propriety of laying before the Royal Society, anythi ng so abstruse and metaphysica l. L therefore, desi red two of the best mathematicians in London to look at the premises ; and the ir opi nion confirmed my doubts. They say, that such a work should be laid before the public in a separate fo rlll."

Herapath not only offered to expla in th e mathematics to the reviewers but also went ahead and performed experi mentai work to confilll1 hi s predictions of the fin al temperatu re atta ined during the adi abat ic mix ing of two of liquids at va ri ous temperatures . In a

letter by Herapath to Gilbert, dated September 16, 1820. Herapath described the results of his experiments ami claimed th at they confirmed his theory that predicted that the tempe rature of th e mixture \Vas nor th e arithmetical mean of the temperatures of the liqu ids mixed, but the squ are of the average of the square roots of the absolute temperatures. These ac ti ons did not change the opinion of Gilbert and hi s colleagues who answered to Herapa th th at "some members of the Coun cil , who are usu all y looked upon on such questi ons ... considered the in ve~ti gati o n s too theoretical fo r the Tmn S{{Clions , without taking on themselves to judge the mathemati cs." G il bert proposed that the rev ised paper, including the experiment. , be put in the hands of the new pres ident and counc il , to be elected a month later. Herapath ex pected to find a backer in

(,

the new President , Davy, wh o was a dec lared atomi st. Davy wrote to Herapath th at he had "reod Ih ose paris 0/ il " 'hich ore illlelligi"'e. 1l ';lhoul pr%lllld 1I"/O Ih elll ( II; C(I I SlUe/V, IVilh ({ lI enlion ; Oll e/ highh' illgeniolls, as / f Ind r Ollr I,iews. / IIIl1sl Scl\' 117 01 /

0 111 n o l impressed 11 ';lh a con vicli o ll of Ih e ;r Irl/lh ... Be!ore / relurn Ih e /)(lpel; I shull f{[k e the lihertv 01" Sl.Ihlllillill g il 10 Ih e /I /OS ! ({ct tle philosopher

. ~ 7

I know ill Ihis cOl/lllr." .. . " . After the paper was finall y rejected for publicati on

in the Philosophicol Tm llsoctioll s, Herapath withdrew it and publi shed instead in the Anllals 0/ Philosophr in 1821 . Rejecti on of Herapath 's kinetic theory by the most important Briti sh sc ienti sts was tantamount to its burial , and thus it remained in almost total obscurit y un ti I 1848 when James Prescott Joule ( 1818-1 889) adopted Herapath 's hypothes is that the pressure of a Gas could be ex plained in terms of the rec tilincar b s movement of hi s molecules .

In 1821 and 1822 Hera path publ ished two additi onal long papers, consisting partl y of numerica l tabl es to be used in co nve rtin g fro m ordina ry temperatures to the "t rue" temperatures sca le he had

, 'i.IO

developed as a corol lary to his kinetic theory Herapath 's strong personal ity and belief in hi s

ideas d id not let hi m accept th is scientifi c fai lure in silence. Five years later, in 1826, he launched a bitter and unrelenting attack on Davy in the Tim es of London (for exa mple, Ja nu ary 6, 1826), acc usin g him oj" circulating unfounded criti cisms on his experimenta l work , whic h prevented its pub licati on. Herapa th consi dered himse lf persecuted by th e "i ns id ious propagation of reports." He sent a long series o/" letters

ED CATOR 277

to the ed itor of the newspaper, giving a detailed account of all the correspondence he had had on the matter and accusing the sc ientific es tabli shment, through Davy, of lack of vision and underst anding. For reasons not completely clear Davy chose to not answer, and thus reinforced Herapath 's feelings about the whole matter. Eventually Davy resigned to the pres idency of the Royal Society ( 1827), an act attributed by Herapath as a sil ent acceptance of hi s claims.

In 1847 Herapath condensed all hi s findings in hi s two-volume book Mafh eIJ/{/fica l Prin ciples of

II .

Nafuml Philosophv , whi ch he publi shed by himself. This book gives an ex tensive descripti on of the kineti c theory and its application to a wide range of subjects, fo r exa mple, a discuss ion of Thomas Graham's ( 1805-1869) results on the diffusion of gases, of Victor Reg nault 's ( 18 10-1 878) work on the iso therm s of hydroge n, and Joul e's ex pe ri ments on ad i abatic compress ion.

In 1829, Herapath became interested in the earl y steam carri ages built by Sir Goldsworthy Gurney ( 1793-1875), which were used to run a regular service between Gloucester and Cheltenham fro m February to June 183 1. Already in 1823 W. H. James had succeeded in manufacturin g a coac h capable of carrying fifteen passengers at a speed of IS miles per hour. Herapath supported the project by publishing a letter to Arthur Welles ley (1769- 1852), first Duke of Wellin oton and b ,

by letters to the Till1 es. Gurney's attempt to deve lop a steam-driven vehicle using ordi nary roads was short­li ved because o f th e strong oppos iti on of tra in support.ers , who organi zed sabotage of the vehicl es by putting piles of stones on the road and getting thi s mean of transportation to be subject to heavy turnpike tolls. Thi s part icular projec t encouraged Herapath to study the rapidly expanding rail ways. In 1835 he began to write articles on engi neering and commercial aspec ts of the new English rail way lines and in 1836 he became the editor of the Rail way Magazine and Annals (~f

ScieIJ Cl' . The name of the journ al was eventually changed to Heropafh \ Raih vav JOI/rl/a l. Manag ing thi s publi cation allowed Herapath an easy source for ex pounding his kinetic theory and its applications, for examp le, a calculati on of the veloc ity of sound in air, whi ch he had announced at a meetin a of the British Society fo r the Advance ment of Scienc; in 1832

12• This

was the first known calculation of the speed of a molecule from the kinetic theory of gases, which was later on ex tended to ca lcul at ing the wind resi stance encountered by a fast railway locomoti ve " .

Herapath was the first to show, more or less. that the kinetic theory can pro vide simple exp lanat ions for changes of state, diffu sion, and the propagati on of sound . Whil e hi s ideas were not entirely correc t. the hostility they aroused seems to have ari sen rather fro l11 the difference between the sce ne in which he found himself and that of a century earlier. By the 1800's. in England at leas t, ori ginal speculati on of any kind and mathemati ca l theory in parti cular were d istru ·ted. Mechanics had been es tablished and ex hausted by Newton . Heat was ex plained in gross by the caloric theory and on the molecul ar scale by the vibratory moti on of atoms within a xtherial substance, not a fit subj ect for mathematics

7

Scientific work Before proceeding to look into some parti cul ar

contributi ons of Herapa th , it is important to apprec iate hi s interpretation of the phys ica l world , as pre. en ted in

. II hi S book M(lfh ell/(lficol PIl\'sics .

According to He rapath matter as observed is composed of hard , so lid , inert atoms, indestructi ble. indivi sible, and of different shapes and sizes. The number of atoms or partic les distributed throughout a given space, unrelated to the density or homogeneity of the body, is termed Illll11emfOiIl.

Th roughout all space is diffused a very subt le and highly elastic medium, which ewton ca lled ~ether. composed of ex tremely small and act ive parti cles. This medium is of the sa me nature as air or any other gas , and can be ex panded or contrac ted by an increa 'e or diminution of heat or temperature. IE ther becomes very rarefied when approachin g the dense and hot bodies of the sun and planets and denser the farth er one recedes from them. It pervades the pores of all bod ies and hence acts upon the ir separate parti cles . The laws that resuit from the action of such a medium are precisely those that are manifested by grav ity.

Regarding the nature of heat Herapath adopts the positi on of Francis Bacon ( 156 1-1626) and Newton and assumes th at it consists in the " intestinal moti on" of the particles of bodies. The greater the heat the greater the motion and the greater the expansion of the bodies affected. The quantity of heat in a body is merely the quantity of motion in one of the parti cles multiplied by the number or particles in the body. At thi s point Herapath makes a turn in the wrong directioll and states th at if the particles are divi ded into a greater number, th en th e in d i vid u(ll q u{/ nfifV of moti on dimini shes, and the temperature sinks though the /0101

278 INDIAN 1. CHEM. TECHNOL., MARCH 2004

quantity of motion in the entire body remains unchanged. For him thi s ex pl ains the phenomena of latent heat, the rise and fall of temperature in chemical combinations, and oth er phenomena of similar nature. Now, the relllpemrllre of a body is measured not by the quantity of Illotion in the whole body but by the share fallin g on each particle . In ' other words, it is not the total heat or quantit y of motion th at determines th e body's temperature, it is the quotient of the whole motion by the number of particles in the body. Thus if we coul d mcasu re th e ratio of the temperature of two fixed points in the thermometric scale, as those of ice melting and water boiling. we would determine the point of

Ilh ({bsoillre cold (see be low) . Herapath would later show that the knowledge of the point of absolute cold allowed hi s kinetic theory to determine of the velocity of sou nd , the laws of pressure and temperature in the atmosphere, the quantity of hea t generated by a measured compress ion of a given quantity of air, etc ., etc. Accord ing to Brush ~ :; , Herapath 's ex planations of these phenomena were not always completely correct, but hi s ideas were always ingeni oll s.

From [his description it is clear that Herapath supported the vibrational theory of heat and not the calori c one. Alt hough he accepted that heat consisted in the vibratory moti on of the parti cles of the body he added the wrong assumpti on that temperature was measured not by the ve loc ity or the particles but by rheir individual I/lOlIIenrUI11 . Thus two bodies, to be or the same temperature, would have to have their partic les vibrating individually with the same force. It was immaterial how these vibrations were made and in wh ich direction, the resulting motions were propagated from particle to particle throughout the body. If the vibrations were greater in one part of the body than in anot her, the tendency would be to an equality of vibratory fo rce, that is, of intestine motion. To Herapath, temperature was a measure of the degree of in test ine motion.

The aggregate (tota l) mot ion divided by the aggrega te mot ion of some other standard body of the same weight or volume, and of the same temperature, is the spec(Jlc heat of the body. Therefore, the specific heat is really no more than the ratio of the number of particles in a given weight or volume, to the number of partic les in the same weight or volume of the standard body.

.:!.I-LI5 Kinetic th eory

The kinetic theory was first put into mathematical form by Dan iel Bernoui lli (1700-1 782) as a minor item

in hi s book on hydrodynam icsl6

. In hi s scheme he considered a cy linder of unit height, fitted with a movab le piston having a we ight equal to the fo re of the atmosphere and exerting a pressure PI!' To Bernoulli the paJticles enclosed in the holl ow cylinder were "I 'er\'

lIIinure corpuscles, which ore driven hilher olld rhilher wilh a very rapid marion ", and "pracliw lly in/illire ill ilL/mba" The particles constituted an elastic fluid . which expanded if the weight on the piston was removed or reduced , or it became denser if the weight was increased. He then considered what happened if the pressure was increased to a va lue P that red uced the vo lume to a fraction.\' of its orig in al va lue . The compress ion resulted in a larger force appli ed to the piston because the surface density of the particles was hi gher and because any gi ven parti cle made more frequent impacts aga inst the piston.

Be rn oulli assumed that the p ressu re was proportional to the surface density of the particles and inversely proportional to the mean di stance between them. After assuming th at the parti cles were spherical with a diameter d and with mean distance D between their centres, he derived the following equation for the ratio PiP,

J

J c; I P ~ Ds ·'-{. ......J!.. =s -' ---- . .. ( I ) P D -d

In order to determine the rati o Did, he considered the situation in which the pressure was increased indefinitely until the particles were in mutual contact , in these circumstances the vo lume had decreased to a

-//3 fraction /1l of the orig inal. Then Did = 111 and Eq. ( I) became,

. . . (2) I 2 Po ,

S - 111_" S -

According to Bernoulli the experi mental evidence available indicated that air could be compressed enough to make /11 = 0, so that Eq. (2) predicted th at the force of co mpress ion was approximately inverse ly proportional to the volume. The pressure could also be increased by a rise in temperature and addition of heat increased the internal motion of the particles.

Consequentl y, increasing the veloci ty required a "reater wei"ht and it was "not difficul t to see that the b b

weight should be proportional to the square of this velocity because, when the velocity u increases, not only the number of impacts but also the intensi ty of

EDUCATOR 279

eac h ~ f the m increases equally" . In other words, P =Ku where K is a dimen sionless constan{

The kinetic theory remained dormant until 1816 when Herapath proposed hi s theory, which essentially fol lowed that of Bernoulli. According to Brush

23

Herapath apparently was not aware of the work of the latter, otherwise, his own theory would have been little different but, in advancing it , he might have benefited by Bernoulli 's authority.

In Herapath 's original view of the physical world heat was the consequence of an elastic fluid and gases were composed of particles endued with the power of mutually repelling each other. Anyhow, he was unable to und erstand "ho w any intestine 1II0tion could augmellT or diminish this power" so th at "after I fwd revolved the subject a few times ill my mind, it st ruck me that if gases ill stead of havin g th e ir particles endued with repulsive forces , subject to so curiolls a /imitation as Newton proposed, were made up of particles or atoms, f11utually impinging on one a noth er, and th e s ides of th e vessel co ntainin g them ... Such hodi es I eas ilv saw possessed several of the properTies of gases; for illstance, they would expand, and, if the particles were injinitely small, contract aimost indejinitely; their elastic force would increase hy an increase of Inoti o ll or Tempe rature, alld diminish by a diminution ... lVou ld gen e rat e hea t by sudden cO f71pression. and destrov it by sudden rarefaction (llId anv TlVO, havin o ever so slllall a communication,

. '-") Ill'

would quickly and eq ually interm ix ." In other words , gases being in "a state of intestine mobility" had no distinctive properties, no active ones, but simply a mechanical existence of internal unrestrained motion. The elasticity of a gas was the force with which the gas tried to expand itse lf, or with which it resisted co mpress ion . It was calculated by " the amount of its ac ti on against similar and equal portions of the containing bodies" (in modern terms, pressure is the forc e applied per unit area, or the number of collisions per unit area and unit time).

In working out his theo ry of gases, Herapath met with a rather se rious obstacle: "!f gases are made of moms. Th ese IIntst be absolutelv hard. th ey mllst adf11it 11 0 hreaking , splirring, sharrering, or any il11pression whatever; and yet, if the gases are to l/1aintain their elastic properties, and this property br the /'es II It of the particles mutually impinging on ail e ({ lI other and the sides of the containing vessel.

th e particles o r atoms, must likel,vise be elastic; that is, they must be soft, for elasticitv, according to the ideas we ha ve, is nothing but {{ctive sojfn ess. Therefore, it appeared to me that the ultif11ate atoms ought to possess two properties ill direct contrariety. hardn ess and softn ess, which is manifest /v impossible ... by what means was it that if the parts of gases were abso lu tely hard. that th ey lI 'ere reflected back into the ,'J1edium .fi'Oln th e sides oj' th e containing vessel .? - ".

Herapath 's initial answer to the questi on was that e lasticity must spring from a different source to what was commonly be lieved , and that it might be a property of hardness, s ince, in genera l, the harder a body the more elastic it is. Eventually he changed hi s interpre tation to one in which collis ions had to be instantaneous: a perfectly e lastic body had its shape changed by force, but recovered it again with ene rgy equal to the force by which it was changed. Thi s recovery mechanism meant that as much motion was generated in the recovery as was destroyed in the loss of the shape so that the result of the collis ion cou ld be dete rmined by using the principle of conservation of momentum , even if it was not possible to describe what actually happen ed durin g the col li sio n . In Herapath's words : "It lIIay be asked bv ",,/hat means is it, if the parts of the gases are absolutelv hard. that they are re.flected from th e s ides of The containing vessel? This question is easily answered. !f' we allovlI heat to consist in WI intestin e motion of' the parts of the body. for then the parts of' a solid of equal temperature H'ith a gas must have the sOll1e quantity, though they have not the same .fi·eedom of motion as the parts of' a gas have; a11.d hence the parts of the gas will have the same rejlection from th e sides of a vessel which they ha ve ji'Oll1 Oll e

lIt . allother ."

Herapath then proceeded to develop his genera l theory of matter, based on the following postulates: (a) matter is composed of hard atoms , (b) all so lid and fluid bodies have their smaller parts composed of these atoms, which may be of diffe rent s izes (:\ml shapes, (c) all gaseous bodies consist of atoms or particles moving about in perfect freedom , (d) heat ari ses from an intestine motion of the atoms and is proportional to the ir individual momentum. (e) a gaseous body of very great tenuousness in all its parts fill all of space, and extends to its utmost I imits, and (f) the motion of the particles is independent of the shape of the containi ng vesse l and hence it must depend entirely upon the temperature of the body.

180 INDIA N .I . C1-IEM . TECHNOL, MARCH 2004

Herapath went onto derive the first part of the ideal gas law by considering the pressure exerted by two equal portions of the same gas in containers of different sizes, at the same temperature (Note I): "No w, because the onlv change Ihat is supposed to take place is in lire space 1111ich the gas occupies. the /7/ ot ions ({nd collisions 0/ a parlicle in rhe one IFill /Jp silllilur 10 Ihose 0/ {{ corresponding particle ill Ihe olhCl: (l lId th e lelllpemlure. thaI is in rhis case Ih e I'etocit\', beillg Ihe salli e ill each. Ihe numbers 0/ revo luriolls (g oill g alld cOlllill g back ./i'Oll1 a collisioll ) that IIW correspondillg particles in the 111'0 Ill edia lIIake ill a given lime lIlusl be in verse/v pmpUrliollal /() the pOlh Ihe pOrlicles describe; that is. rhese pmhs heillg {{like, {f lld described lI 'ith equal I'e/ocilies. in Ihe ill verse sub-triplicale mlio 0/ the spaces occllp ied hr rhe equal pOrlions of gases (a I IV 113) . Bul bec{{us e th e elas lieitv of a gas is pmporliollul /() Ih e aelioll (~/ irs particles (/gaillsl a givell porl ioll of Ih e sl.IIface of lhe contailling bodv, Ih e mlio of Ih e e l{{ sl ic fo rces. arisin g ./i'om Ih e repeated aeliolls of equ(/ I lIumbers of correspollding pOrlieles ill Ih e rH'O l11 edi{{ , Ivill lik ewise, th eir ,'plocil ies beillg the same ill bOlh lII edia, be in versely os th e spaces oCC/lpied ... Bul the number of parlicles of allY oll e II/ edium Ih ar strik e aga ill st a given pOrlioll (~f lhe cOlltaining Sillfoce of one medium is 10 th e IILIll/h er of pa rlicles of Ih e corresponding slralllill Ihal strike agaillst an equal and sill/ilar portioll of Ih e o th e r medium; ill th e dup lisubtripl icate (a I/V w ) of th e flum eratoms (lIull1h er of particles ill ullit vo lum e} ... Therelore ... that ratio musl be equal.) ro the simple ill verse ra lio of the spaces occupied .Ik." Since the

_ . ID

pressure ot . the gas was proportI onal to both I IV II.,

and to IIV then it had to be in verse ly proportional to the vo lume.

In more simple words, Herapath postulated that heat was moti on and that the indi vidual momentum of the particles measured its intensity (temperature). A :ystem wil l then be in thermal equilibrium when all its ato ms have the same momentum. Now, the pressure or a gas at constant volume is proporti ona l to the intensity of the colli sions aga inst the walls of the contai ner and to their frequency. Since the intens ity of th~ coll isions (temperature) is proporti onal to the mo mentum , and th e ir frequ ency to the ve loc ity, then the pressure is proporti onal to the squ are of the momen tum , that is, to th e square of th e

~ - -L1 1;1

,c illperarure . III other words, Herapath correctl y deduced that pressure is proportional to the number of

particles per unit vo lu me, ? thei r mass M, and the square of their ve loc ity, but failed in identify ing the square of the velocity with the square of the temperature:

pT 2 p oe M . .. (3)

Another of Herapath 's conclusions was that the spaces occupi ed by equal masses of the sa me gas at constant e lastic ity were direc tl y proporti onal to the squares of the temperatures . This conc lusion came from the fact that at constant elas tic ity the numeratom is inve rse ly proportional to th e square of rh e temperatu re. For equal amount of gases the vo lume is in verse ly proportional to the numeratom so that the volume is proporti onal to the square of rhe temperature. In equati on form,

E= BoT 2 V

. .. (4)

,. (5)

. . . (6)

where E is the elasticity under the conditions considered, II

N the numeratom in the unit vo lume at 32 F and elastic ity B ) = N ), D the density of the unit vo lume at

II I I 0

32 F and elastic ity Bil = G", V is the volume at 32 F and elastic ity B ) = I, and T is the true temperature,

I II

having a value of I when the temperature is 32 F The value of Eo is 30 inches in the Briti sh system and 0.760 meter in the French one. •

Herapath used the va lue reponed by Pierre-Loui s Dulong ( 1785- 1838) and Joseph-Louis Gay-Lussac's ( I 778-1 850) for the thermal coe ffi cient of ex pansion

I) I) · 1

of gases at 0 C ( 1/266.7 at m, C , iha t is, a decrease in pressure of 0,00375 atm for every degree of cooling)

I)

to deduce th at the true zero was 480 F below the me lting point of ice. The improved measurements or

I) - I 0 Regnault ( 1/277.7 at m. C at 0 C) led to the better

I)

value of 49 I F (Note 2). At the true zero all corpuscular mot ion would cease and the temperature woul d be

Ilh nothing, or it would be "abso lu tely co ld" ,

Th e re lati on tha t co nn ec ts He ra pat h 's tru e temperature and the temperature in degrees Fahl·enheit

F is then

EDUCATOR 28 1

T = 1000 448+F

480 . .. (7)

The factors 1000 and 448 enter in this relati on beca use He rap at h dec id ed to make th e tru e temperature or the ice point (32°F) equa l to 1000. The square root ari ses from hi s incorrect formation of the . I~

Idea l gas law . The observant reader wi" notice that the two main

differences between Herapath 's kinetic theory and the one accepted today (after James Clerk Maxwell , 1831-1879) is that Herapath stressed the conservati on of average momentum (11111 ) in th e co lli sion between particles. and that he defined an abso lute temperature, whi c!!, is not related lin early to the temperature. in of (o r C). Co nsequently, if tw o fluid s are mix ed ad iaba ti ca ll y Herapath 's th eo ry predicts a final temperature different from the simple average of the ori ginal temperatures (see below).

In 18 16 Herapath publi shed hi s first findings in the Annols of Philosoph\' as a short memoir to tes t the reaction of the sc ienti fic community. , He then elaborated hi s ideas in more detail and submitte'd a detailed account to the Roya l Soc iety in 1820. As exp lained above, Davy rejected the paper because it was too speculati ve and complicated and did not have sufficient experimental justifica ti on. In partic1Jiar, he rejected the hypothesis of an a~so lute temperature impl ying an absolute zero of co ld .

Herapath 's theory went into obli vion until thirty­five years later, when Joule, August Karl Kronig (1822-1879), Rudolf Julius Emanuel Clausius ( 1822-1888), and Maxwell revived the kinetic theor/

l.

Maxwell recognized Herapath as a precursor of his own research in kinetic theory and oave the followino

b b

assess ment of Herapath 's work: "His theory ol the collision olpelfec tly hard bodies, such as he supposed the molecules to be, is laultv ... This authol; however has applied his theO/~' to 'th e numerical results oj· experiment in mal1\1 cases, and his speculations are alwavs ingenious. and ojfen throw IIllIch real light in the quest ions treated. In partieulm; the theorv of the temper({/ure and pressure ond ~ases and diFfitsion are

IX I JJ

clea rlv pointed out ' ''.

Velocity of SOl/lid

Tn order to comprehend Herapath 's arguments in calculating the ve locity of sound using hi s kinetic theory it is necessary to understand the concepts that were prevalent then on the st ructure of a mixture of gases .

1'1 John Dalton ( 1766-1 844) bel ieved that gases mi xed by diffu sion and that once equilibrium was achieved the particles of one gas were not elastic or repul jve in regard to the partic les of another gas, but only to the particles of their own kind. Consequently in a vesse l co ntainin g a mi xt ure of two gases , each act ed independentl y upon the vessel, with its proper pressure. just as if the others were absent. The lowest part ic le of one gas in the mixture susta ined the weight of all the particles of the same gas above it and the wei ght of no other. There was no movement of the pa rti cles, the particl es of each gas be ing ordered as lattices of poi nts regularly arranged, eac h gas havi ng its own lattice, and th e patterns be in g sup e rimp osed. 111 thi s arrangement sound was propagated independentl y by each lattice, that is, if the atmosphere cons isted ch iefly of two elastic media, then distant sound s ough t to be heard doubl e. The same sound would be heard twice, according if one or the other of the gases carri ed it. Dalton calculated that if sOllnd trave lled at the rate of 1000 feet per second in nitrogen then it would move in oxygen at 930 feet per second and at 1175 feel per second in water vapour. Accordingl y " if ({ strong ({ li d

loud sOllnd was prodllced 13 miles off the first \wllld he a weak impressioll of it brought by the atl110sphere of water vapOLII; in 59 seconds. the second would be (he strongest alaI/ brought hv nitrogen ill 68.5 seconds, and the third would be l11uch inlerior to the secolld. brought hv oxvgen in 74 seconds. It is known that the report of a canon fired at {{ distance of 13 miles ./i'Ol17 the observer did not strike the eor as a single soulld, b

. I~ ut was repeated 5 or 6 tllnes close to each other .,.

Herapath used the same argument to reach the opposite conclusion . He could not dec ide whether air was a compound or a mi xture. He proposed some carefu l experiments on gas diffusi on to decide the matter, but later concluded that , after all, they co uld

b . . III

never e sensiti ve enough . To Herapath , the best proof that air was a chemical compound and not a mechanical mixture was the fac t that there is no duplicati on of sound . To emphasize the point he discussed the propagation of sound through a med ium whi ch manifestl y is a mixture, namely fog: " In foggy weather it is weI/-kno wn that there o.liell ore ta'o sOllllds. one transmitted 1'110re rapidly by the aqueous vapour and the second by the atl11osphere; {[nd so it wOllld be pretty generally if air was (I mechanical mix tllre of two airs. Th e most IIsual sound would cOlli e .limn nitrogen. A second feebler sOllnd, hOll'evel; \·wuld he transmitted by the oxygen at a velocity oFf) . " flo slower

282 INDIAN J. CHEM. TECH OL., MARCH 2004

flial llie principal sound. But no such thing ever takes place. fl' it did, it would be ill1possible af a dis(({nce to distinguish notes, as we now can, in a

fin e pie ce 01' music. There would be comple te conji.lsion and des/ruction of a ll harmon y with music heard (/ long W(/v off" Therefore air must be a

II I - ..

compound . In a fo ll ow ing paper he reported his calculation

of the veloc ity of sound, first presented to the Briti sh I '

Assoc iation meeting at Oxford in 1832 -. In this paper Herapath reported the correct first approximation for the speed of sound and gave the first explic it value for the speed of a molecule.

Herapa th assumed that the ve locity of sound in a gas was simply the mean rectilinear veloc ity of the particles of the gas . To calculate this va lue he reasoned that since a part icle may strike the side of a vessel in

• () 11. • anx angle trom 0 to 90 , the II1termedtate a ng l ~ was 4S so that the average momentum transferred IS not 2mu but 2l17u/~Since the length of the path traveled between a given revolution is 21N I13 the number of revolu! ions per second is VNI 1312 and, consequently, the number of particles striking the unit area (pressure) is V20/2 where D is the density of the gas . Now, if E is the elasti city of the gas (the weight it will support) then the product Eg (where g is the grav ity constant) also represents the total number of co llisions per unit area. Equating both equations , V20/2 = Eg, leads to the final express ion V = JEg.J2 ; 0 for the velocity of the particle. This formula pred icts, for example, a ve locity of 1089.65 ft/s in a ir at the freezing point of water. Comparison with various experimental results shows that it is accurate to better than one per cent at

Il Il I I k f"iT)'/J temperatures between - 30 F and +60 F . u= v 3Pld

In 1848 Joule published a pape/ in which he used Herapath ' theory to calculate the velocity of hydrogen molecules from experimental densities. Joule's method of ca lcul ati on can be summed up by the expression for the mean square velocity where P is pressure and d is density in mass per unit vo lume. Even though he did not explicitly use this equa ti on" t,l e arri ved at the equi va lent result in his

_ .. l

argument .

Laten! heal Accordin g to Herapath the phenomena of latent

heat co uld be explained without difficu lty assuming lilal the speci fic heal depends on the number of particles. If pa rti cles of bod ies unite, it must be when they have lhe leas t d isposition to fl y off, that is, when they come

together with moti ons towards the same parts, or when they are moving in the same direc tion. When these particles unite their moti ons are added together, and they form a smaller number of particles, but eac h has a greater motion or temperatu re. When a so li d is heated the vibrations of the atoms increase, until fi nally the cohesion holding together th e so lid begins to be overcome. However, as soon as some particles break free, there is a lowering of the temperature since the same motion (i .e. , momentum ) must be distributed amon g a larger number of parti cles. Thus when gases condense into liquids and liquids are converted in to so lids , generall y th e particles aggregate and the temperature ri ses. On the contrary, when sol ids become fluids, and fluid s become gases, the parti cles split into

~

more, and the temperature .~ ink s . In the condensa ti on of vapour into liquid water

the particles form ed are larger and lesser in number, leading to an increase in the temperature. A sim il ar phenomenon takes pl ace ill the so lidificat ion of li qui d water into ice. From these facts Herapath inferred that per unit weight , the spec ifi c heat of steam had to be greater than that of liquid water, and that of liquid water greater than the specific heat of ice. Today we know that thi s is incorrect: the speci fi c heat of steam is lower than that of liquid water. Herapath also assumed implicitly that the spec ific heat of a liquid remClined constant between its melting and boiling points.

Temperature of mixillg We have seen that Herapath derived an ideal ga~

law of the form PV ex i instead of the usual PV ex T The difference is more than just a matter of definiti on since Herapath believed that his true temperature was the one that had the properties usuall y attributed to the ordinary temperature. Thus, if one mixed two equal portions of the same substance at different temperatures TI and T

2, the temperature of the mixture should be

(T + T, )12. If one mixed, fo r example, equal portions I - (J II . J-I h' of water at 32 and 2 12 F, then according to erapat 0 s

theory the temperature of the mixture should be 11 8.4 F (Note 3) . Herapath menti oned an experiment by J ea~~1 Andre Oe Luc ( 1727- I 8 17), who, according to Ure­obtained the result 11 9

I1F 2.1. Herapath urged the Royal

Society to do further experiments to determine the true law for mixtures, using for example some substances like mercury whi ch is fluid over a greater range of temperatures, so that the difference between the results predicted by the two theories would be greater. USin g experimental data for the mixing of water and mercury

EDUCATOR 283

Herapath deduced from this formula that a particle of mcrcury mu st contain 27 times as much matter as one of wate/.

In general , of course, the temperature of a mixture will depend on the specific heats of the components, which , according to Herapath , depend on the number of separate particles per unit volume: "that body whose particles are the smaller and more numerous, being that which has been supposed to have the greater capacity for caloric and vice versa". If one mixes a vol ume VI of one substance at a true temperature T, with another of volume V

2 and temperature T

2, the

temperature of the mixture would be (TINI VI + T2N2 V)/ (NI VI + N2 V2) where NI and N2 stand for the number of particles per unit volume (numeratoms) in the two

III substances .

The concept of mixing could also be extended to the case of gases by observing if the final volume was larger or smaller than the sum of the original ones. A volume contraction represents a decrease in the number of particles and hence it should be accompanied by an increase of temperature, and vice versa.

Diffusioll Herapath dedi cated over 80 pages of the second

volume of hi s book to the anal ys is of the phenomena of diffusion and the results obtained by Thomas Graham

lie ( 1805-1 869) ' . The apparatus employed by Graham was a graduated tube, closed or open at one end and closed at the other by a plug of plaster of Pari s, or some other porous substance (dijjitsio/l tube). When a tube of this sort was fill ed with any gas li ghter than air, hydrogen for in stance, and the open end immersed in water, the upper plaster of Paris end being dry, the hydrogen would rap idly escape, and the water, following the vacuum occasioned by the departing hydrogen, would ri se up into the tube. After some time the air, which fl owed in slower than the hydrogen flowed out, would begin to fill the tube and the water leve l descended until it reached its former level. Graham found that the more eas il y the gas was liquefiable. the more readil y i f would permeate through substances such as thin films of water, rubber, and humid membrancs.

To expl ai n this phenomenon, Herapath considered that the mi xture of gases was mechanical , and that no chemical action took place between the components. First. he stated that the number of particles of gas striking a given portion of the containing surface was proporti onal to the elasti city and inversel y proportional to the temperature. that is , as EIT. This result is a

consequence of the fact that E ex TC, or C ex EI T, where C is the number of collisions.

He then used his kinetic theory to demonstrate that the time of percolation into vacuum of a gi ven volume of gas through small apertures, kept at constant temperature and elasticity, was proportional to the volume and to the square root of the spec ific gravity. and inversely proportional to the temperature, that is. as VJ CIT. This result was a consequence of the fact th at the number of particl es flowin g out was proportional to the number of co lli sions in the same time, that is , proportional to EIT. Since the time of flow of a given volume was proportional to the number of molecules it contained and in versely proportional the number of molecules flowing out in a given time. it resulted that the time of percolation r IS

VEJGT vJG (8) T ex: --:;:-- - ex: --

T2 E T

Eq. (8) led to two inte res tin g and practical consequences: (I) for equal volumes and temperature. the times of perco lation of two gases are simply as the square root of their spec i fic gravity, without any relati on to their elas ticities, and (2) if two gases are confi ned in equal volumes at the same temperature, and the lighter gas is compressed as to be of the same density as the heav ier, then the two gases will ha ve different percolation times, but the ratio of the times will always be the same (as the rati o of the square gra vity), independent of the quantity of the gases confined in the

IIh same volumes .

Miscellalleolls A fascinating aspect of Herapath 's personality is

the fact that in every activity of hi s life he tri ed to explain the pel1inent phenomena using hi s kinetic theory. Some striking examples come after his sw itch into the railway industry and its problems . Herapath showed. for example , that according to hi s th eory. th e

o atmospheric temperature decreased nearl y I F r or every one hundred yard s increase in altitude. Since a "single degree )'1"i// he sufficient !p change a dump roil (I I Ih e temperature of about 32 F. into one gla-:.ed II 'ilh ie I.' . 0 11 which the train coult! not JJl ove, ir will evidelltlv he a matter of some cOllsequencc ill the 100'illg d ()\ \'Il ()f" rhese expellsi l'e lilies to (II'oid el'C/T c/wnce ofohs((l c/c or injun l ro rhefi'ce lI 'od ing oj" ttWIII . bv kecping rhcir summits os lOll ' (IS possible. so (I S to bc rhe ICOSI affected bv tel71pCralUre

CI". The pertinent caiculations.

described in de tail in Se ction V!. Boo k III of

2R4 INDIAN 1. CHEM. TECHNOL, MARCH 2004

Mothe/lwt icul Phvsics" are actually wrong because they are based on the false assumption that the earth 's atmosphere is in thermal equilibrium.

Herapa th examined the ex isting data on wind res istance to tra ins travelling at vari ous speeds and noted that the pressure was rather great at high speeds. but littl e re li ab le information was available. This fact suggested hi m an additional appl ica ti on of hi s kineti c theory to railways. According to the theory the forc or pressure that air exerts on a so lid surface is proportional to the squa re of the ve loci ty of its molecules, thus , a train travelling at the speed of sound , i. e. , at the average speed of air molecules, should encounter a res istance equal , to atmospheri c pressure , that is , about 14.7 poundslin - at a speed of 1090 feet/

13 s . At any lower speed th e res istance would be proporti onally small er. By this simple method Herapath was ab le to compute theoretica l va lues that were in good agreement with the handful of experimental resu lts ava ilable. From hi s ca lculati ons Herapath drew two irnport<lnt conclusions: first, the idea that trains could operate at 60 mph or more is absurd ( !!) ; second , every effort shou ld be made to design the shape of the locomoti ve so as to minimize air res istance . Another way to reduce air res istance, he suggested, would be to lay the railroad line in a moderately deep cu tting so that the air would strike the train at the front were the exposed surface is the leas t.

The above results are essentia lly deductions form the idea l gas theory, the particles are assumed to have no magnitude and to move freely th roughout the space in which they are enclosed. Herapath also attempted to deduce what variation from the idea l gas laws would

I k .d be ex pected if the particles had a finite size . He co mpared two airs, A and a, contained in cubes of vo lumes Vand v, respective ly, where A is an ideal gas whose particles are mere points, and a is a gas of part icles of finite size and after some mathematical steps showed the rati o between the volumes of the two airs, at the same temperature and elastic ity, was,

~ = r I + '63JN ]5 V . . . . (9)

In Eq 9 } is the diameter of a particle of the air a and lie

N the number in unit space of the ideal gas A These resu lts were first obtained in 184 1, at a

when time it was generally believed that all gas deviated from Mariotte's law (increase of elast icity proportional to the incrcase in density) by having a smaller increase of elasticity with density than expected . Herapath's

theory predicted that the deviation should be in the opposite direction. He was at first inclined to attribute the discrepancy to the cohes ion of the part ic les in th e

lid gas . However, in 1846 Regnaul t measured the compress ibility of air, N], CO2, and H2 at differen t temperatures and pressures and fou nd that air, N] and CO

2 presented a similar compressib ilit y, wh ich not on ly

was larger than the one predicted by Ma ri otle-Boyle 's law, but also increased with increased pressure. The results for H] were surpri sing in that they presenteci the oppos ite behaviour. From these results Regnault conc luded that the compressibility of a gas depe nded not only on the pressure and the temperature. but also on the nature of the gas

22.

On the basis of Regnault 's results fo r hyd rogen Herapath decided that hi s theory was correct after a ll : "My inl'estigation.l' in the eor/" pOri of / 844 led me to a directly opposite cOllclusion . Assllming th e gases 10 be pelfecl gases with particles of a sensiNe size, / fOllnd that their e lasticitv shollld in crease fas te r not slOlver th ot th e in verse ra tio of Ihe

/h.d compressio/l. . Regnuult fOll nd hvdrogen to von'

to th e opposite side of Mariott e ~\. iOlv" .

Epilogue Herapath was convinced that research Illu st be

mission oriented, an idea that he expressed as follow s: " Wherever it has been possible my res ults arc reduced to practical IItilit". FOI: a/ieroll, science is valuable so far as it is usefi fl. M ere speculative truths, I,vhich ha ve no beoring on the affa irs 0/ life, are of Little value. Th ey are not, certainlv, IU

be despised because th e ir oppl ica tion is not immediately seen, but thev lIlust afl,va)'s vield place to truth s more rea (Ii Iv ava ilab le . Lik e Ivinfe r garments, in sUl11mer th ev should noT be thrOl\'ll away, but laid ([side until th e season for th e ir

. . serVice arrives.

References I Anonymous. Railll'ol' M(/g({; ille QI/arfo Snies. 30 ( 1l'68)

234, 275 , 309.

2 Brush S G, AIIII Sci. 1:\ ( 1957) 188.

3 Brush S G, Notes lIlId Records Ro." Soc, 17 (1963 ) 161 .

4 Herapath J. AIIII Phil [2J, I (1821) 273 , :\40. 40 I.

5 Herapath J, AIIII Phil. 8 (I g J 6) 56. 6 Davy H, Elemellts of Chem ical PhiloSIJliitr. (J John~on

London). 1812, 53. 7 Truesdel l C, Essays ill titl' Hi.lturr of Mechallics (Springcr­

Verl ag, New York) , 1968.

8 Joule J P, Mem Proc Mall cit Lit Phil SOC [2J. 9 ( 1851 ) 107.

9 Herapath J, AlI lI Phil [2J, 2 (182 1) 50. 89. 20 1,256, 353. 4., 4.

EDUCATOR 285

10 Hcr:qJath J Ann Phil [2], 3 ( 1822) 16.

II Herapath J. Malhl'lIIalical Phrsics (Whittaker & Co., L ondon),

1847 . Facs imilc edition publi shed by Johnson Rcprint

Corporati on. London, 1<)72: (a) I , 242, (b) I. 249. (c) I, 267-

26<). (d)1. 27 1. 276. (e) I. 278-279, (I) I , 343, (g) II, 3-57,

34<)-368. (h) II. 6. ( i ) II. 8. 37. (j) II. 37 (k) II. 60-73 , (j) II ,

23<).

12 Herapath J. Rel,ort of Ihl' Second Meeling rd' Ih e Brilisll

Associal ioll , Ahst rac t. 557. 1832: /?aih m r Maga:ine, I ( 1836)

n 11)36.

13 Herapath J. Railwar Maga:in l', I ( 1836) 89.

14 Wh ittaker R D, j ChI'lli Edllc, 56 ( 1979) 325.

15 M cndoLa E, Bril j Hisl Sci , 8 ( 1975) 155.

16 Bernoulli D, Hr drodvlI{/fnica, sh 'e de viriiJlls 1'1 1II0lilms

.f/Ilidonllnconllilell tarii. 01' /(.1' acadell/icliln aiJ a/(clorl', dllln

Pelmlw/i {{gael, ('ullgeslllln: J. R. Dulseckcri , 1738. Translatcd

and l'Cpri ntcd as Hydrauli cs. Do ver Publications, New York ,

196i'l.

17 WI,niak J , Dcvclopmcnt or thc Concept or Absolutc Zc ro

Temperaturc. Illdiall .I Chelll Techllo l, acccptcd (2003).

18 M ax wcll J C, Phil Trails. 157 ( 1857) 49.

19 Dalton, J. A Nell' S\'slelll of Chelllical Philosuph.". R. Bickerstaff.

M anch estc r. 1808: in two vo lulll cs. Rcpubli shcd by

Phil osophi ca l Library. Ncw York. I <)64: Pan I. secti on 2. pp

15 1-1 93.

20 Ure A . Phil Trans, 2 ( 18 11» 208.

21 Hcrapath J. /?aihl'ar Mag(/jlle. I ( 1836) 1<).

22 Regnault V. MellI Acad Sci IlI sl F, : 21 ( 1847) 329.

23 Desormcs C B & Cl cmcnt N . .I Pln·s. 89 (1815 ) .+2 8.

24 Wi sniak J, The Nature of Heal. Illdian .I Chon Techll ol.

acceptcd (2003).

No tes I . Thc follow ing long quote is a good eX:lInple of Herapath's

twistcd way to explain si mplc idcas.

2. The expansion coe fficicnt was considered independent of the

tcmperaturc, thu s, every decrcasc in the tempcrature by one

degrcc led to a decrease in th e pressure or 0.00375 at m unti l

ze ro pressure was reached. In this ca lcul ation Herapath was

not showi ng much originality. i t had alrcady been used by

Charl es-Bernard Desormes ( 1777-1 ::;(2) and ico l:ls Clcment

( 177<)- 1842(.

The development of tile concept of zero abso lute tempcratul\: ,~

has bccn discussed elsewhcre- .

3. According to Eq. 7, thc va lues o f the tnle temperature ror :>2" o (I \I

and 212 F are 1000 and 11 72.6 rcspcc ti vc ly. Hence the

tcmperature of the mixture wi ll bc (1 000 + 11 72.6)/2 =

1086.3. Us ing Eq. 7) aga in th is is equi valent to I I ,' A2"F