QY
The EQUATION of STATE of a PERFECT GAS R. ROSEMAN
AND
S. KATZOFF
The Johns Hopkins University, Baltimore, Maryland
I
N "A Defence of the Doctrine Touching the Spring and Weight of the Air, . . . . ," published in 1662, ( I ) , Robert Boyle showed that the volume of a gas varies inversely as the pressure (temperature remaining constant). Among Englishspeaking peoples this law is usually called "Boyle's Law," though on the continent of Europe i t is attributed to Edme Mariotte, who, however, did not publish it until 1676 (2). Credit for the discovery of the law of equal thermal expansion of gases is given t o J. L. GayLussac (1802)
.
E. Clapeyron (1834)* appears to have been the first person to amve a t a single formula connecting the
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~
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premiirc Crcndui dc aingtquax pauces, que k pidr enricr de I'&r rux twit quuu du m h c poidr. Jc fir hire encore quclquer autrcr cxp&ieucct fimblablcs, hitT?n~ pitm ou mains d'rir d m t mtmc ruym,ou dnnsd'surm plurou'moint p n d s i & i s rrovvri roujourc, qu'apr& I'~.cxp&nc~ fiitc, 12 lopor d; tlon d s P i r dihtf i Pircndne ds cdui qu'on woir llim nu mcrcurc wanr Pr~pi.liencc,fmith mCmc uc ceUe de vingrhuirpouccr de ~LICUIC. qui CR 15 poids cnrbr dc?'Arrnofphdre, h Pexca de vinphuit par d r i h la hauteur ob ildcmeuroitaprlrPexp61icnce: ce qui a. connoitre lui%~mmcnr,don p u t rendrc pour unc Aglc cerrrinc ou loi de la nature, quc ?,air le c o n d h 2 proporrim der poidr dont il dl ehrrg6
{aut
THE PAGEIIROM MARIOTTE'S ESSAY REPRODUCED FROM TKE 1717 EDITION OII HIS WORKS\ ON Wmca HE STATES TAAT"L'AIRSE CONDENSE A PROPORTION DES POIDS DONTIL EST CAARGO." THISESSAYFIRST APPEAREDIN 1676 I k E PAGE PROM B O n E ' S ARTICLE (REPRODUCED PROM BIRCH'S SECOND EDITION OF HIS WORKS) ON WHICH
pressure, volume, and temperature of a given mass of HYPO~ESIS "THATSUPPOSES THE EXPANSIONS TO BE W RECIPROCAL gas (5). By combining the Law of Boyle and Mariotte PROPORTION." ARTICLE FIRSTAPPEAREDI N 1662
IS ENUNCIATED
THE
PRESSUlLES AND
(3), although J.A. C. Charles, in some earlier unpublished work (about 1787), had found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent from 0' to 80' R (4).
~~~
le rn6molre Jut env& ie I1 jan& 1833. ~ e n d ud l'auteur, ilyii Pdlid par cduici, dans 2e Journal de L'Ecole Polytechnique, 23e cahier, 1834, Pnges 153190."Private communication from the perpetual secretaries of the Academie des sciences, Paris, De~
cember 18, 1933.
D K
'57
C e r n l t r
D E CItlMIB.. 175 sit6 et la qoantit6 d'eau qu'ile fiemeot en dissalution, et toutes lesvapeurr,se dilnbnt Pgalement par Ira memes degds de ehsleur. lo. Pour lco gaz permanens, I'eugmentation de volume que chacun d'eux reqoit depuia Ic degvd de la glaep fondame jurqe'i eelui de I'em buuillante est Pgale aur 5i:ndu volume primitif pour le thermom~tre&.is& en 80 parties, ou sux du m h e solunle Pour le tllermomktre eenligrade. 11 me m t e ,pour cdmpl6ter ee travail ,& dftcrminer la loi de la dilatation der knr dcs vapeurs , a h d'en eooclcre le codfficient dr 1s.dilalotion pour un rlegd queleonque de elmlrur dllrrminf et h m'nssurerde la rraic momlne rlu thermombtre. Jevais m'occuper de ces uouvelles recherche#; elquand dtcr seront termindes, j'nurni I'honorur de ler 0ommuniquGr h I'lnatil~t.
V e quoique j ' e ~ r ewconouuo grand nombrc de fair que lrr gaz oxigknc, azdte h y d r 6 @ne ct wide carbonique, el Pair atmosph6
,
rique se dilarnt (galemerit depuis o' jusqu'h Soo le cit. Charles araitrcrnamu6 deiruis
.
.
. .
<>
II avail ausi eherch6 B &termioer la dilatation des gaz rolublcr dam l'cau, rt il avait t r o u d & rhacun une dilalation particulikre ct diITPrenle de relle dm a u W g. A cct igard mrs experiences diffkmt beaucoup drr sicnner. 1.e cit. Charles s'6taitser.i pour apparcil
,
,
,
dunl: oulvtid b tirer de lout rcquc jr ricns de dire lrs concluric ns suivaotea. ~ U . T u u sIrs S ~ L qtr:Iqu~ , slriml bur den
PAGESFROM GAYLIJSSAC'S ARTICLE(1802) On page 157 first mention is m a d e of Charles' previous unpublished work. Pages 174 a n d 175 are the concluding pages o f t h e article. POISAHCB IIOTRICE
1 q
MBMOIRE SY.U
PUISSANCE MOTRICE DE LA CFIALEUR, P*n R. CLAPEYRON
.
mCblmn DS m.
s.&dnirom~~ p r w i de o In ~ qurntitd d'aetioo m r i m u m q u e p n r d ~ i ~ l ~ ~ ~ d ' ~ ~ c q ~ . . t de iG cbaleur, d o o d'uo n & mrpseotreleou i u m ~ p e n t n r ed d t m i n & i un c o r p emtreteou bi uoe t p m p h t b w moindre, et noma arrivemns 1d e nladoos muxeller eotrele volume. I. pio. ,la trmpkatuie et la quaotitd absolve de chalemr ou le enlorique lalent, d s corps solids, liqoida ou gazeux. deuz c a r p A et B, et s u p p ~ qoe s la tempemture 1 Repreoms du corps B soit inldrieore Zu'uoe quaoilte iotibiment petite dl i la lempiroture t du corps A. NOUSruppmerons d'abord que ce soit un gaz qui sene i la tnoamisioo du calarique d o mp A an carps B. Soil uo le volume dm gaz solu la prarion p. ct 1 la tempirature I.; wimt p et u lc volume et la pression du m4me poi& d~ grz i la lem+tam r d u carps A.LP Loi deMariotte, combink svrc celle de GayL o s w , &lit enhe an quaotitk divela relation
om porsnt, pour simplifier, I1 s t pem de p e l i o n s plm d i p d e b I'atuntion des g&m&lrer el d a physician,, que oelle qui sa nppmtent i I. constitution de g u a drs v a p u n ; le r61eqdils jooentdnnr la nature* le p r t i qu'eo tire I'industrie erp1iqme.t lea nombreus et i m p r u m travaur dont ilr oot 6td I'objet; mais cette wste qoation n t loin d'btre Cpui.(e. Ir loi de Mariotte et cclle de M. GajLusyc 6tablisreot l a relations qui existent enlre le volume, l a prrssioo el la temperatore d'une m h c quami6 de gar ; tout d e u oot obtenu depuis longtemp I'awntiment d e swans. Les expfrienfes nowella I d e s p r MM. A n g el Dulaog oe laissent plus aueun doute sur l'exaetitude de la premiere eotre d e l i m i t s t r k dtendua de la pression; mais ns rkulutr imp r u o a n'appredoent rieo rur la quantilddeehaleur que w e n t I n gar, r t qu'en ddgagent Is pression ou un pbaiwment de tempdlature, iir ne donoent par la loi d s c a l o ~ i p e s+cifiqua i presaion coos
,
X.Xl,P C ~ h l w .
20
eL
= R: pv = R(267 4 t ) .
Leco.psAestmir enco~ucl~veclegaz. Soit me", ae=p(tig.3). Si I'on dilate le gar d'une qoaotitd iofinimeot petite du = eg, la tempirature resten cmrunte h cause de lr pkreoce de la source de chrleur A ; la preuioo dimin.cn et devieudn i p l e 1 l'ordoon& hg. Maintennot, om iearto lecorp A et I'm dilate le gaz dam une eweloppe imprm(lhle 1la chaleor, d'une qunntile ioti~imrotpetite gk, jusqu'i ee que 1. chaleur dnenue lateote &ise la temphture du gar d'uoequaotil6ihtioiment pelite dt, et I'amhe dnsi i la temperrlure I dl du Nrpr B. Par suite de eel ahisremen1 de temperature , In presioo dinxinuera plus rapidemcot que dam la premiere partie dr I'opiration, et devieodra eh. Nous approehons maintenant le c a r p B e l .our dduisons le volumsmh d9uoequaotiG infinimeot petite f h , el caluulPc de laeon ice que phdsot n l t e mmprrssion, la gaz d c

with that of Charles and GayLussac, he obtained the equation, P V = R(267 1). The figure 267 was obtained from the accepted value of the coefficient of expansion, 0.00375, due to GayLussac. If for it we put 273.1, the figure resulting from the later and more accurate experiments of Regnault and others, we have the familiar general gas equation, PV = R(273.1 f t ) , or PV = RT. This is occasionally referred to as Clapeyron's equation of stale of a perfect gas. Today, one hundred years after the first appearance
+
equations and leave the student with three formulas to remember. I have found it advantageous to combine them into one formula which applies in all cases. But there are always some students who object to having the formula, PVT' = P'V'T, thrust upon them bodily: they want to know why it is so. Some of them are satisfied by seeing it developed in the following manner: since T varies jointly with P and V, it varies as their product, i. e., T : T1::PV:P'V'; whence PVT' = P'V'T.
Those to whom joint variations remain a mystery we usually placated in another way. Write the three [ormulas thus, and apply the axiom of multiplication:
of Clapeyron's equation of state, it "is by no means nncommon to find this simple formula deduced by a vicious combination of Boyle's and GayLussac's laws." Thus, (a) in a recent wellknown textbook of physiological chemistry, the "derivation" is given as follows:
,V
=
pRI (Boyle's Law);
Boyle's Law, PV = P'V'
V = knT (GayLussac'slaw); P = ksT
(The last expression is a corollary of the first two laws; it states that a t constant volume, the pressure of a gas is proportional to its absolute temperature.) Multiplying these together, T2
P P = klklks ;
P
VP2
=
klk2k8P; or
PV = 4klknkrF = k ~ . "
(b) In an article published in THIS JOURNAL several years ago, we read the following: "Most highschool texts will develop the gas law
Charles' Law
{ PT' VT'
= =
P'T V'T
(PVT')2 = (P'V'T)2; whence, by applying another axiom, we have PVT' = P'V'T."
I t is hardly necessary to point out that the equations of Boyle's and GayLussac's (or Charles') Laws are not simultaneous, and cannot, therefore, be combined as is done above. Furthermore, the present authors have a feeling that

"The pictures of Charles and Regnault (p. 353) were very kindly furnished us by the perpetual secretaries of the Acadhie des sciences, Paris.
same mass of gas in still another state, it can be shown that
In general, then, for a given mass of Eas, the value of (pressure).(volume) the expression, is invariable, (absolute temperature) ' Z. e.,
(B) In algebra it is proved that "if x, y , z are variable magnitudes such that x a y , when z is constant, and x = z, when y is constant, then, x a yz, when y and z vary together; and . . . if x varies as y , when z is constant, and x varies inversely as e, when y is constant, then, x y / z when y and z both vary" (6). Applying this theorem to the present problem, we have that, since
. . .
V ir T, when P is constant (Law of Charles and GayLussac), and 1 V = when T is constant (Law of Boyle and Mariotte), then 1
V = 7 when T and P both vary
a good many students of chemistry are not familiar with the correct deduction of Clapeyron's equation of state of a perfed gas, starting with the Law of Boyle and Mariotte and that of Charles and GayLussac. Accordingly, the following acceptable methods for the derivation of this equation, one of the most important in the whole range of physical chemistry, are presented, with the hope that they may be of some interest to the readers of THISJOURNAL. ( A ) Let PO,Va, and TObe the pressure, volume, and absolute temperature, respectively, of a given mass of gas in an initial state (I), and let PI,V,, and T I be the pressure, volume, and absolute temperature, respectively, of the same mass of gas in a final state (111). Suppose this latter state to be achieved as follows: (I) keeping POfixed, (n), keeping TI fixed, Pa, Vo, To Po, V TL change PO to PI change TOto Tt
.
(In)
PI,
v,,TI.
For the change from state (I) to state (11) the Law of Charles and GayLussac applies, i. e.,
v*  To V ,  Z' For the change from state (11) to state (111) the Law of Boyle and Mariotte applies, i. e.,
v,
V'  =  PI or V' = P, . v Po' Po
Substituting this value for V' in the preceding equation, we obtain P.lr0
To = 
P,V,
Tz'
., =a
To
=
P31, TI
Similarly, if Pz,Vz,and Te represent the pressure, volume, and absolute temperature, respectively, of the
* See footnote. p. 352.
.:
or^^
V = 
=
KT.
(C) The volume of a gas is a function of the pressure and temperature, i. e.,
By the fundamental equation of partial dBerentiation,
Substituting these values for
in
equation (I), we have, Now from the Law of Boyle and Mariotte, which states that V =
dV += d P . d T V P T
K, (T constant). P
This, upon integration, gives
we obtain, by differentiation,
InV+lnP =InT+InK,or P V = KT.
and from the Law of Charles and GayLussac, which states that V
= K P T ( P constant),
we obtain, by differentiation,
In conclusion, it is to be noted that the methods presented here are not strictly independent of cach other. Thus, the general theorem on variation stated in (B), which has frequent application in the physical sciences, is usually proved by a process similar to that employed in ( A ) , but can be derived more elegantly by method
0. LITERATURE CITED
J. B., "Trait6 de physique," 1816, Vol. I, Chap. IX. (I) B o n ~R., , "New physicomechanical experiments," London, 4) Ref. (3a), p. 157. 1662, Part 11,Chap. V; see also, T . B m c ~"The , works of E., " M h o i r e sur la puissance motrice de la 15) CLAPEYRON, the honourable Robert Boyle. In six volumes. To which chaleur," I.dcole polyteck., 14, No. 23, 164 (1834). is prefixed the life of the author." 2nd ed., London, 1772, Vol. I, pp. 15663. (6) MELLOR.I. W., "Higher mathematics for students of chemistry and physics," 4th ed., Longmans, Green & Co., New (2) See "Oeuvres de M. Mariotte, . . ," P. Vander Aa, Levden, 1717. Vol. I. "Avis au lecteur" and "Discours de York City. 1931, p. 24; H. S. HLL AND S. R. KNIGHT. "Higher &bra,"4th ed., Macmillan & Co., Ltd., Lonla hature de i'air," 6. 152. don, 1929, pp. 235. (3) (a) GAYLussac,J. L., "ReSherches sur la dilatation des gaz et des vapeurs," Ann. chrm., 43, 13775 (1802); (b) B ~ o r ,
....
GENERAL BIBLIOGRAPHY
J. S. AMES,"The free expansion of gases. Memoirs by GayLussae. Joule, and Joule and Thomson," Harper & Bro., New York City, 1898, 106 pp. C. BARUS,"The laws of gases. Memoirs by Robert Boyle and E. H. Amagat," Harper & Bro., New York City. 1899,110 pp. awes," H. Halt & Co., New York City, R. M. CAVEN."Gas and . 1927, 256 PP. L. DARMsrAEDTER, "The life of Edme Mariotte," J. CHEM. Enuc.. 4,3202 (1927).
W. S. JAMES, "The discowry of the gas laws. I. Boyle's law," Sci. Progress, 23, 26372 (1928). W. S. JAMES,"The discovery of the gas laws. I . GayLussac's law," ibid., 24, 5771 (1929). W. S. JAMES,"The discovery of the gas laws. 111. The theory of gases and van der Waals' equation," ibid., 25, 2329 (1930). W. W. RANDALL, "The expansion of gases by heat. Memoirs by Dalton, GayLussac, Regnault, and Chappuis," American Book Co.,New York City, 1902, 166 pp.