O S THE GESER-\L PROHLEAI O F CHEMIC.\14 STAITICSx
INTRODUCTION
T h e present paper is a commentary 011 and a coiiipleiiieiit to the celebrated iiieiiioir of J. TTillard Gibbs : ' On the Equilibriuiii of Heterogeneous Substances." -411 cheiiiists iiow ktion-, lion-, in this memoir, the illustrious S e w Haven professor has deduced from the principles of tliermodyiiaiiiics the eqnatioiii wliicli represent the states of equilibrium of all chemical SJ bteiiis ; all kiiow, under the naiiie of the pliase rule,' the w-eiglity theorems u hich lie has found through comparing the nuiiiher of uiiknowii qnaiitities to be determined, with the iiuiiiher of iiidepeiideiit equations subsisting aiiioiig them. T h e labors of H. TT. Rakhnis Roozeboom and of his pupils h a l e illustrated the bearing of the phase rule, i n applj ing it to the discussion of a host of coiiiples clieiiiical reactions ; and the ipwial treatises of TT'. IIe~-erhoffer3a i d of n'ilder D. Rancroft. ha\-e asseiiibled under it tlie chief problems of clieiiiical iiieclianics. Kow, it seems to me that the deiiioiistratioiis given by certain authors, for the phase rule, leal-e something to be desired in oiit Translated. from tlie author's French manuscript. hy J . E. Tre\ or Trans. C o n n . .Iced., 3, 10s and j ~ (j1S7j to r S i S J . Die I'haseiiregel, Leipzig und IVien ( 1S9j 1. ' The Phase Rule, Ithaca. S . l-.,U. S. .I.( 1S97 I . I
P. Dz~hcm
2
point : they admit, at least implicitly, that the independently variable componeiits of each of the phases of a system are identical with the independently variable components of the system as a whole. But this is not i n general the case : it may be that one of the independently variable components of the system is excluded from one of the phases, or it may be that two of the independently variable components can appear i n one of the phases onl?. in the fixed ratio of a definite compound. Consider for example a case made classic by the work of Debray, the system made up of the two independently variable components, quickliine and carbon diosid : here none of the three coexistent phases, in which the system ordinarily appears, admits both of these components ; one phase admits no coinponeiit other than the quicklime, a second no other than the carbon dioxid, \vliile the third contains calcium carbonate alone. Gibbs, however, has S ~ O ~ T ’ I I , ~ in a few lines, how the demonstration of the phase rule can be made wholly general ; and it has seemed to me, hereupon, to be desirable to develop, with all possible rigor, this denionstration which he has indicated. I have sought, fmtherinore, to add to the propositions discovered by Gibbs certain new theorems having, likewise, wholly general character, These theorems are not deduced solely from the principles of thermodynamics ; they rest as well upon two very simple postulates, which are readily verifiable in any realizable case. T h e y are as follows : 1-P nlZd f h E ColJl$OSiti(l7Z O f I. 1 f 7 / l E ~ l t/l&?JZflSS, fhE tl?JJlp~J*fZflf fliiid nrr mninfni~2rdcoizstnrzt, the Z W Z I ~ J J ~ oC f the flziid alill dc( - I T ~ S C ziihrrz its ) r ~ s s z i ~ , c is iJzci~cns~4; nrid f h r wz’e?*sr; II. 1571ieii the houiog~rzeitj~ o f n m i x t z i w (fbodies is di.sfzil,bcd, it a f i Z Z teiid to be msforctd dzfzisiori. These postulates, whose character is seen to be essentiallj phj-sical, suffice to establish important propositions of chemical mechanics, of which in the first instance the following theorems may be cited :nlZd f h c Jl1NSSC.I O f I. l f i z P l 1 f / l C ifElJl@?7,nif211eC, the $l’CSSZlI’C, ~
1
J
Trans. Conti. .%cad., 3 , I j;
J
flir iizdc~eiiderit[~l ztnvin6le coiii$onc.nts of n sjisfeiii ai’e gizvii, tlic coqkisition nttniiicd by ~ t z c h(if f h c $Arises of the systcni, W I ~ C J ~ cqzi ilibriu vi cxsiies, is zi 12 iyzie(y dcfci-iiii m d ; 2. I f the z o l i i i i i c occii$it-’d 6 ~ 1the -ystcJiz 6c gizvii, iiisfmd @ thr $rcssiii-e ziiide’elyaihiclc it sfam’s, theit not O J L ( J ~ f h c co)ii$osifioii bi(f nlso the densify of each phnsc is ziJiiyiip(?l defciwiincd. Capillary action and terms due to friction will be neglected in establishing these propositions ; relating, further, only to t n i f cqiiilihrin, they y i l l fail to applJ- to real or apparent (falsi cqiiiliDJ-in;’ it may be said, in fact, that the falsity of these propositions characterizes the false equilibria. T o these propositions must be added the two following : 3. If f h e f ~ J l l ~ € I * ~ f Z lOl ~f P SJlStE111, nlld the $ITSSlii’C ilildC?* 7~lh ich it sfnnds, be iiin iiitniiicd corist nizt, f h e chcm icnl eqziili6i*iiiiii rf f h c sysfrni cnii be O J Z ( ? ~stnhlc or iizdifercrit ; 4. [f the tcnipcmiitrc (fthe s ~ ~ s t c norid i , the zvliinie zohicli if ocqbics, be nicriritni~zcd coiistn i i f , flic ~-cszi Zfiug I hemica1 eyziilibrizi 111 iii ii s t be, likea fisc, cifh CY stn Ole 07. ?;ltdifercJit. T h e relation of these propositions to the postulates upon which they rest leads to the remark, which seems to me to have its peculiar importance in natural philosophy, tliatiiftvibziti~i of ' the fluids I , 2, , . 72 3 n-e shall obtain 3 second mixture, like to the first hut with a Lfold mass : if we neglect the actions exerted by the different parts of a same illistiire upon one another, we can attribute to the second mixture a total thermodynamic potential A-fold that of the first inistiire, a i d shall then have.(
and applying the theorem of Enler to the fiiiction
x we find
F,--l f ,F, -- - 7-13 ,F ( 2 ) T h e functions FI, F2, . . ., F are evidently homogeneous functions of the zero degree of the variables AI1, AI2( . . . , ; applyX
lf
,
ing the theorem of Euler to each of these fiiictions 11-e find the identities
... The definition of the functions tions ( I 1, furnishes the identities
I FI, F2, . . . , F,!$ given by
(3)
equa-
through which the equations (3) can be repIaced by the equations-
... 2.
I
sfnbilif)) nf thr p h j i s i c n l Eqziih'briiiiii
qf n
~?)JJZO~~CIZ?O~LS
ZI 11 dCi* [-[I72 S f 0 1 l t 7 ' 1 suri..--n'e are to consider a mixture of the IZ fluids I , 2, . . ., ? I , at the temperature T, and subject to the action of no exterior force other than a uniform pressure II ; it is clear that at equilibrium this misture will be homogeneotis. ,4dniitting that its equilibrium is a stable one, and that if tlie homogeneity of the mixture be disturbed it will he restored of necessity 11)- diffusion, we will seek the properties which these circumstances assign to the functions F1,F2, . . . , F;,. Supposing tlie initially homogeneous mixture to be coiiiposed of tlie masses 2111,2112( . . ., 231,,, of the flnids I , 2. . , . 71, its tlieniiod-)~nainic potential, for the constant pressure II,is
&fl:rf21 I %
~
(
2
. . ., ~ L I , ,n, . T)
z S~r 2 , 1
~
~
or, since tlie function FK is homogeneous and of the first degree n-it.h respect to the masses of the iiiised fluids,
..
2 ~ ( 1 Sr?. 1 ~ ~ .,
if,',
IT, T I .
Ry means of a surface S, let the mixture be divided into tn-o parts -4and B>each having the saiiie total mass : each \vi11 then contain the masses >II, AI,, . . . , l I J ,of the fluids I , 2 , . . . 12. Caiise the infinitesiiiial inasses SIII, 61I2,. . . AM,, of these fluicls to pass from the part to the part R,the pressure and teniperature reiiiaiiiiiig constant. If ~
~
m1_ _
.~
811-
--.-
...
-
631 ,,
(5) XI 31> ' the two parts will retain their original composition. the mixture will remain liomogeneous, and its tlieriiiocl-).namic potential for constant pressure will suffer no variation. If, on tlie other hand. ~
the equations ( j ) are not satisfied, the mixture will become heterogeneous ; this change, however, will be impossible, while the opposite change will be possible but irreversible. T h e virtual variation in question would accordingly correspond to an increase of the therinodynaniic potential for constant pressure. At the close of this virtual transfer of matter, X will contain the inasses (MI-8AII), (X2-8J12), . . . , (JIt2- 811,,), its thermodynamic potential for the constant pressure II becoming ~ ( h l , - 8 M ~ ,M,-811z,
0
,
Mz2-SllJ,
II,T ) .
Developing this expression, with consideration of infinitesimals of the second order, and comparing with the equations (I), we find
..
X ( ? Y ILL, ~, ., Ai,,, II,T) -Fl(Ml, M,, AI,,,II, T)8M1
-F,(M,,AI,, -
- e ,
* , M,,, IT, T)8M,
where the sign 2’ denotes a summation over all the possible 3
.
combinations ij of the indices I , 2, . ., 12, taken two at a time. T h e part B contains the masses (AII+ 8&TI), (MD-8h12), . . , (Mtt+ 8hI,,) ; its thermodynamic potential, developed to inclnde infinitesiinals of the second order, is
.
-
X(RI,, M,,* * , RI,,,II,T)
--F,(?YI~, hi,,
-F,(M,,
hf,,
- .. . -rF,(AIl,hl,,
..., M,,, n,T)M, - LI,,, n,T)8Mz *
-
9
,
., hl,,, 11, T)8M7‘
T h e final thennodynamic potential of the whole iiiixtiire t 111is becomes
.
G H - ( M , ~AI?.. ,, )I,,,
rr, T )
its increase during the virtual transfer reducing to
whereby we reach the two propositions : TYlieii the quantities 6111,. . ., 611>dsatisfy equations is), we haye
svhile, on the other hand, this siiiii is positive m-lien the quantities 81II,. . .( 1311~~ do not satisfy equations (j). T h e first proposition brings nothing new ; it is easy to see, in fact. that it fol1or;s directly from equations (3). These two propositions can be cast into a slightly different X2$ . . ., X,, be any I Z finite quantiform-as follows: Let XI, ties ; one can then always put-
nr1
2 tlie masses of the seyeral coniponents of tlie mixture ; tlie inner tlieriiiocl~-iiamicpotential of tlie system may then be represented by $( >IITAI2, . . ., J I , , T‘, T). From one of the fundamental relations of the theory of the ther~nodynamicpotential we have
a
-av c ~ ( l I , ,AI,,
, hl,,, T‘, T)= - II,
(9)
II being the equilibrium pressure of the system at the volume IT. This equation can be solved for IT,T‘( AI,? AI,.
V:
. . ., hI,,, IT,T ) ,
( 10)
the \-olume occupied b~ the mixture at equilibrium under the pressure IT at the temperature T. If in the expression
$(M,? AI-%
a ,
T ) -- I I V ,
XI),
n-e replace T‘ b j its 1-alue in ( I O ) , we sliall obtain the function &c(JI1, AI2, . . JI , II,Ti. Equation ( I O ) thus transforms tlie equation . 1
Il, the identit\- in the row i, we ha\-e \T
a?x ax,alr
~-
32:Pf
a\-
- 323 .-
~~~
a m \ - all,.
aJI'ax,
Eq-nation (91,on the other hand, becomes an identity \\-lien its is replaced bj- the expressioll ( I O ) ; diffei-entiatiiig it with respect to we find 323 ~~
a\-aJr.
32s ay
~ _ -
av:
~~
.~~
=0
a31.)
. 1
and the last two e q u a t i o ~ ~yield s the relation a?.iiig lietiveeii c) sild I sucli that
Ikcxiiw of ecjitatioiii
1 I 2)
and
[ 2 5 j>
thi5 eqtiatioii caii lie writteii
variation of these masses that is defined b j eqnations ( 2 2 ) ; to this virtual variation J\ ill correspond a secoiid \-ariation S2f of the inlier tlierinodj-naiiiic potential, n-hich secoiid 1 ariatioii is the product of eZ 2 and the first iiieiiiber of equation (26). Let 61I denote the iiifiiiitesiiiial increiiient, due to the virtual variation considered, of the pressure of the surrouncliiig walls upon the mixture. -kcording to tlie proposition proved at the close of tlie Pi-eliiiiiiiarj Chapter, for eciiiation (26) to liold it is necessary and sufficient that the equation
8rIz
0
holds good, as also the equations id,
- ?If,
-
h(6)
711'9 - 1/29 ~~
~
P*.p(0)
...
112 A - ?/2A - -
~ ( 0 )'
But it has been s1ion.n in tlie preceding section that these two equations are incompatible. Our propositiol; is therefore demonstrated. lye shall next show that : If the triri$rrntzir*r nizd iolzcnic -of, fhr systrm ni'r mnintniizcn! constniit, its stntc' of cyziiliDi~zz~m vzll 6c n stable stnfr. Let = ??La,
p3
= ?/!e.
*
,
PA
ZZZ
?/?A
(27)
be the values of pa! p 3 , . . ., p,, correspoiiclitig to equilibriuiii. The proposition in question is equivalent to the assertion that, of all the values that the function J(pn. p3! . . . , p,,,T', T)can take for given values of AIr, 112,. . . , 11 , \-, T, the value f'(uz,, i z B , . . . , mA,T-, T)is a ininimum. Ti-e kiio~valready that the equations ( 2 7 ) are such that equation [ 19) is satisfied whenever equations ( 3 ) are satisfied ; so there reinailis merely to demonstrate that
whenever 8pa, 8p3,
,
. ., 6pA satisfJ- equations
( 3 ) , 6V aiid 6T
being zero. S o n . , because of chapter, we can write
-1deiiioiistratioii siiiiilar to that which teriiiiiiates the preceding section shows that n-e caiiiiot have
On tlie other haiid, equation ( 18) of the Preliininar!. Cliapter ~i1;es
ay --> a\--
0.
These two inequalities, taken together \Tit11 equation ( 29)> cleiiioiistrate the inequalit!- ( 2 8 1 , and, therefore, tlie p r o p i tioii that v a s to be established. CHAPTER I1
I.
1 izi-lbzis Dc)~i~z’tz‘o~i.r.-lTeshall consider
R SJ
steiii, at tlie
temperature T, subject to the action of no exterior force other than a iioriiial arid uiiiforni pressure rI ; we sliall, further, neglect all capillary actioni, and suppose the SJ steiii to be composed eiitii-el!
26
of homogeneous masses. I t may be that two of these inasses have the same composition and state, that an element of one is identical with an equal element of the other ; in such cases i t shall be said that the two masses belong to the same phase. JYe will suppose the system to comprise several phases in contact with one another, each being made n p of one or more homogeiieoiis masses. If none of the various virtual changes that the system can undergo can effect a variation in the nature or mass of the bodies constituting some given phase, we shall then have to do with an i ~ z o a ~ * i n h ~ ~ p ~ h ahsicc, h\vi11 play no part in the chemical statics of the system concerned, and which shall therefore be neglected in the folloming developments. T h e system to be examined shall be supposed to comprise variable phases, which shall be distinguished by the indices a , p, . . ., ; f h p rrrimhcr (if 11/16 z!ni-in6lcphnst.s into ailiich the sysfcm s@nrntes is oiic of the ~zziiiihe~=~ flint chnvacferizc the S J / S ~ E I I Zfi’oi12 , thc,poirzt of iicsc! o f clicnzicnl mcchnizics. Let us consider the different simple bodies that enter into the composition of the whole sj-stem. T h e definition of the s! stem involves, in general, certain conditions, which make it impossible to choose arbitrarily the masses of each of these simple bodies : if, for example, two of them, -A and B, appear only in the forin of a certain definite compound containing both, then knowing the mass of A in the system is sufficient to make known the mass of R. I t shall be admitted that c simple or compound bodies can be found, such that : I , X system of the kind studied can be formed from the arbitrary masses 3111,31i2. . . 31tc; 2 , -1 determinate system of the kind studied corresponds to a set of determinate values of the masses 31tI, 91T2,. . ., 91i, of the bodies T h e bodies I , 2, . . c shall be termed the zkdcI , 2 , . . . , c. pcndcnt coirzpnizcizts of the system. I n certain cases it is possible to choose in different ways the independent components of systems of a gi\-en kind ; it may be that these systems can be regarded as formed from the independent components I , 2 , . . .. c, or from the independent components a , p, . . ,, y . This lack of determinateness, however, can not affect the number of the independent components
+
+
+
~
.
~
for it may be asserted t h a t : ~ V ~ , C ihcJ Ziurkpcrzdeut Lnmpom i l t s of systems o f a giiv2n k i d caiz (le chosc~z iu ticlo dzJerejzt iiviys, the ~ u m 6 r of r thcsr conipoiicnts ,vnzniizs f h c stamc in both ca.ies. For, suppose indeed that the sjsteiiis of a certain kind can be regarded as formed either from the independent components I , 2, , ., c, or from the components a, p, . ., y. From the masses 91iI, :YC2, . . . , 9li( of the bodies I , 2 , . . . , c we can then make up a determinate sj-stem of the kind considered, and. on tlirt other hand, the system can be regarded as made up of the determinate masses pa, p?, , . ., py of the bodies a, ,B, . . .( y : to a determinate set of \-allies of z’llI, 9112,. . ., 9K there corresponds therefore a determinate set of values of pa, pB, . . ., PLv. and it can be s1io~v-nlikewise that a determinate set of values of 2TcI.91i2,. . . , 91cc corresponds to a determinate set of values of pa, pa, . . .. py. Let 11s examine how this reciprocal relation is to be established. T h e mass blCl must reappear, in one form or another, in the masses pa! p?, . . ., p.,. Aliiiongthe masses serving to form pa there occiirs a mass Plop, supplied bj- the body I ; the quantity P,,being a purely niunerical coefficient, depending upon the chemical coi~ipositionof the bodies I and a. ,Iniong the masses
.
.
T1, 9K*,. . ., 91i, and the masses
there obtain the relations
... P ,pa - P p
p@
+ . . + P,,p, = 9K-be slion-ii likewise that c can iiot be greater tliaii y : so it iiinst be that 6
y
:~--
n-hich proI.es the proposition stated. S o : f h r iiz/iizhrr (f7'7ido$riidrizt coiiipoizciifs of y ~ s f c i i i (f s n gi:icn kiizd is di;fiiiifi~(?lfi-i-c'R! This iizriiibei- c, nrid f h c 1ii/iiihci* $ (ffhr p/rnsc>s,chni-ncfci-isr flic .sii.sfciii9 j v i i i thc PoiIif o f ~'irzt' of chc'iizicnI sfnfirs. For a definite system foriiiecl from the iiiasses 41iI, 911, . , ., I:'Tc of the iiidepeiideiit coiiipoiieiits I , 2 , . . . , c. a i d separating into the phases a , ,B, . +,-the pliase a being iiiacle up of the iiiasses MI,, M,,,. . lira of the iiidepeiicleiit coiiipoiieiits of tlie sJ.steiii,--n.e must have
.
.)
. 1
... ..
AI,, - 31,: -- AI'(,,= 41i', RI,, -!- 1 1--~. ~ + i r 2 b ,,
...
'I
I
li
(1)
I
--
31(.8 -;-
..
I
xr,> = 91ir. I1
7-
From tlie circumstance that tlie iiiasses ?liI, :?1t2, . . , , I:'T, are arbitrary, it does iiot follow that tlie iiiasses JI,,, JI,,, . . .. lIcr5are all arbitrarj-. It iiiay be that certain of the iiidepencleiit components do not appear in one of the phases, in u for esaiiiple : the defiiiitioii of the s!-steiii imposes iiideecl, in general, a certain uuniber of conditions, such as ?\I . i.
.
- - L0,
...
. . ., I. 2, . . .
:-7 - 0.
( 2 )
being a1iioiig the indices (', and A, p> . , , p, . . $,-let $ he the iiuniber of these conditiolis ( 2 ) . Furtheriiiore, e\-eii when certain coiiiponeiits do appear in the phase a , for example< it iiiay be that their iiiasses are iiot wlioll!- ai-hitrarj- there ; it iiiaj. lie that certain of them can appear in tlie pliase a only in tlie foriii of certain definite coiiiponiids : i n other words, if 71tn~1hc. f h n f fhosr iirdqbrizd( ' l i t c.oiir)oizrrrts of the sy8sfc'iii f h n f o@$,*ni, i ~ zf h r phnsr a n r c 17 o f i i i d q k i i CJc 11f co i u j o ii cii f s f h ik $h n sc. Let n,, h,, . . . , X., lie the inclepeiicleiit coiiipoiieiits of tlie phase a. aiicl Iiztl,, iitb,, . . .~i i i k a tlie iiiasses, of these components, .I,
aiiiotig the indices a ,
.(
~
of which the phase a is iiiade up. A h i o i i gthose of the iiiasses N,,,, SI,,, . . Mc, that are not iiecessarilj. zero, and the masses moa,mb,, . . . , / u k a 3there obtain the relations .)
the coefficients -4being 1)itrelj. numerical coefficients, kiio\m \\-lien the chemical formulas of the bodies I , 2 , . . . , $ and of tllr liotlies oat ha\ . . , k, are kiion-11. T h e iiiiiiiber of these eqiiatioiis ( 3 )is equal to the number of the masses l I , a that are iiot coilstaiitl\. zero. hToreo1-er, when one of these iiiasscs is coiistaiitl?. zero, the eqnatioii
.
0
= JI,e,
n-liich then appears aiiioiig the eynations ( 2 has in fact the form of equations ( 3 ): it can therefore be suppressed in the series of equations ( 2 ) and be n-ritten ainoiig the eqiiatioiis ( 3 1T h e j equations ( 2 j caii thus lie suppressed, aiid 4 eqiiatioiis like (3'1lie written for each phase : to the coiitlitioiis ( I i \vi11 thus be added q b conditions oi the tj.pe ( 3 1. T h e coefficients .I of equations ( 3 )iiiiist therefore be such tlir,t these equatioiis iiii.ol1.e as coiisequeiices-
JI?= -;- 71Ltz0 - _ J~i l h , ?if l a
..-' ~
. . . . . --
JIca
*
= L91ia.
(4) :Net denoting the total iiiaijs of the phase a. In a \.irtual chanpe of state of the s).steiii, the iiiasses ~~
l f , = , K,. 7l1aal
712
jCi,
..
-
* )
* *
712%
M,,,* . . , 71t~,,.
*
*
,
& ,
i12kr!,.
\\-hose numher i. ccp
+ R, -+ /is -;- . . .
J -
k,,
(5'
\vi11 suffer the \-ariations 811,,,
8\12,z. . *
h J ? l t 2 , x ,6 ? J l [ , a .
..
. ( mrn3 . * .. 8\1:,8>, a
~
8711ta,
*
,
6112j:,4J~
)
\
(6)
30
P. Dzihem
which are the same in number, and are subject to the
4+
1)
7
relations-
S>qa
18MI@I
8 M J 0 T 8M$ I
.
T
. . . - SM,* . . &I2, =o, -0,
*
.
+ 8Mc, + . . . - 8Mc, =o,
SM,,
. e .
X,a,8nza+
T A,b4S?%b4
+
*
Ack,87?ZkO=
f
1
8hl~,$.
obtained by differentiating the conditions ( I ) and (3), whereby, in the former, the masses 911~) 31t2,. . ., 9Kc are supposed invariable. 2 . Oiz the I ~ ~ z ~ State P I Yof v n~Heterogeiz~ozis ~ Sjistcm. Let lis now examine whether i t be possible to produce in a heterogeneous system a yirtual change which shall not vary the composition of an)- phase. For such to be the case it is necessary and sufficient that8??laa - 81126,
__
-
1iIaa
nth,
-
S112ha ... -= Pa. k -
711 a
T h e question then becomes: whether it be possible to find quantities Pa, P,. . .! P, snch that the equations
.
lima,= Pama,,
8l?Zb,
... 8?it,, = Pm?h?,,,
.
= Pamb,,
,
8mka
+
= Pamk,,
(8) 8i?Z.b+
= P$,)?tbo,
.
*
~
8?i2k0
= P+i'?zkO,
correspond to a \-irtual change in the system. For this to be possible, it is necessary and sufficient that the quantities 8MIa, 8M2z,, . , , 8 N c b ,determined by ( 8 ) and (3n), satisfy the equations
Gcizrrvl P i ~ h l c i ~ofz C ~ P ~ J ZSi'nfics Z'CO~
31
i ~ r r ;) now, equations (3) show that the equations (8) and ( 3 0 ) give8 M r a = P,M,,,
...
8hl,, = PahIpa,
8 1 1 z b = P+lIr,, 8M,+= P+hI,,,
.. . ..
*
~
~
6hl,, ==PahIra,
611c+ = Pchf,+.
and it is necessary and sufficient, therefore, that these quantities satisfy the equations ( I O ) ; so, if is rzrccss07~~~ niid ~zifficiztf h n f srrfiLfithr c 4 qzrnntiz'ics Pa,Po, . . . , Pb cniz bc fozrizd, z~~t5ich I i i Z P O Y 072d h ~ i J Z O ~ ~ ~ l l C O c$JlL0fiOlZS2lS
ITlien these $ quantities can be found, one of them, at least, not being zero, it shall be said that 1/26 sysfriiz is Z'it n z iizdzJJcrciif sfnfc. I t appears, in the first place, that if the number 4 of the phases into which the system separates be greater than the n i m ber c of its independent components, the system is, in general, in an indifferent state. If, on the other hand, 4 is not greater than c, the sj.stem will be 0111~-exceptionall!. in an indifferent state. T o express that the state is indifferent, we must express that equations ( 9 ) are compatible in P,, P,, . . . P,, whereby we are snpplied with one or more relations homogeneous in AIl,, X?,, . . . Mea, Recause of equations ( 3 ) , these relations become Iioiiiogeiioiis in i i ~ , ' ~7iih,, , . . . uzk, : so they remain satisfied when all these masses are iiiultiplied by any same number, or. i n other words, \yl-hen the total mass of the phase a is changed without altering the coinposition of the phase. I t can be shown, likewise, that these relations remain satisfied when the mass of the phase /3 is changed I\ ithout altering the coiiiposition of this phase ; and so 011. It can be said, therefore, that~
)
11jiriz taio s y s f t w s h i ~ thc ~ z snmr im'cjcm'cut
coiizjorzcizis
gild fhf’SOiiZC J2161J(hPi~ (f$/ZflScS, fh(’~ 0 7 7 ’ ~ . ~ ~ i Z d i i ~ ~ nfp(’fffh /lffsc~ hffZ’iJ)NzLg,
I
J
4
Moreover, that the second state of the \!stem, like the first, can be formed from the masses 9K19 gl12, . . . 9Kc,it is necessarj- and sufficient that the qiiaiitities AIrI,, AI’2a, . . ., X’ca, . . . obtained through replacing i J z f l a , ? J i b a l . . mk,, . . . , 7 7 i L b , by . . ., 7 i z r L m in equations ( 3 ) satisf! the i d f l a , i d b a , . . , 7i/’1,, eqriatiotis--analogorrs to equations ( I )~
e
.
wIa=( 1 - p a ) 9
I
\I,,,
w2a==( 1 -pa) xa, . . . , w(a
(I- L P ~ ) A .)I ~ ~ ,
:
I
.
31'z, * * , AI','$ =- (I-+P,$)&'$,
.
wlience,-since l I r a l lIz0, . . , lI,,, . . . lIc4satisfy equatioiis ( I 1, bj- hypothesis,-in order that equations ( I b ) be satisfied, i t is necessarJ- atid snfficient that Pa, P,, . . ! Pmsatisfy equations (9). S o n-e ma\- say:~
.
1 1 1 ordei- ihnf the g i w J i 7JiffSSL'S :31iI, ?IT2, . . ., 9Kc (fthr iJZdCpL'lldt1Zt COllZ~oIl~'J1ts (f ff SJ1LfcJll [nil j i ) l P i 1 1 f i c ' 0 dl&i-PJlt S t ' t S (!,f- $ phrrsc~s,in siiis'z wise that 2'12 these two stntcs the $/insr.s shrill h n w thc sniiic r ~ ~ ~ i i ~ o s i tbzit i o i zdif(>rrizt iiinssc's, it is ~icc-cssniy~ and Si
