BY J .
E. T K E \ . O R
Co/ollijoirci/fs. Tlie starting point of Gibbs’s tlierniodyiiaiiiic theor), of cheiiiical eqiiililiriiiiii is t h e cliffereiitial equation’
d€
/til/
-p&’
/l,dJ//,
+
f /.12d///2
,
. .
J.
/l,,dl//”
expressiiig the relation lioitliiig aiiioiig the differentials of tlie eiiergj’, eiitropj., ~.oluilie, anci iiiasses of the several coiiipoiieiits of a ( ( phase ) ) of its sulistances. T h e I’ coiiipoiieiits caii lie chosen in axil. \ w j . compatible with the coiitlitioii tliat the differentials of their masses shall be iiiclepentleiit and shall espress ei’ery possilde variation iii the coiiipositioii of tlie phase.’ Gilib;; s1ion.s further’ that wlien h relatioiis sulisist aiiioiig tlie units of these coiiipoiieiits, the nuiii1)er of ( ( iiitlepeiicleiitly \rarialile coiiiponeiits ) ) of the phase is 11
=
1’-
(1)
//$
and that the saiiie is true of an!. s1,steiii of phases taken as a whole, T h i s matter of tlie relation lietweeii the cu/n/wucnfs and the iicticj f j z u e u / ! i , w r i t r h / c /.on/joircuf.s of a sj,steiii has lieeii put verj. clearlj. aiicl conipletelj. 1)). Gililis.’ but iii so alistract a iiiaiiiier tliat its signification caii 1. well conil~relieiicletl0 1 1 1 ~1)).~ readers gifted \\.it11 iiiiaginatiun aiitl ti.aiiied into n tliol-ougliguiiig faiiiiliarit?. i v i t l i tlie pertineiit facts. Bailcroft lias accortliiigl!~ d u n e a i i iiiilportaiit service’ i i i liicitllj~statiiig the relatioil ant1 i i i giI,iiig it illustratioii I)!, all iiiaiiiier of applicatioii to material sj.stt.iiis uf \.arious t?.pes. It lias also lieen clearl\. aiid corrcctlj. eiiiplojwl iii special cases i i i tlie tlieriiic)tl).iiaiiiic studies of Du 11eiii .,-’ ‘ G i l i l x Trans. C n i i i i . .Icarl. 3 , I 16 i I S ; . ~ ) . -‘I. c. I r:. 7 1 . c. pages I 20 alii1 tlierexfter. 4Baticroft. The Phase Rule, 2 2 7 ( 1897). jDulietii. Dissol~itioiiset 1116laiiges.Tray. et
iiieni.
des fac. de Lille.
OslJ/ofic Sj~sfc77zs. T h e gain iii practical coniprehcnsioti so made led Bancroft one step further' to tlie very interesting conclusion that since the introduction of osmotic walls into a material system may increase the nuinber of pressures there, the 71 a variables in Gibbs's formulation for the iiuniber of degrees of freedom or the vn ri~z ?ict of the sy st em
+
iJ-72
+2-7
+ +
.
are thereby iiicreased to 72 a n , wlieii the addition of osmotic walls introduces n new pressures.' H e thus recognized that t h e variance of a system containiiig osmotic pressures is changed in a perfectlJ7 definite tiianiier by the appearance of these p r e s s ~ i r e s . ~ An advaiice i n this clirection is .very iiiiportamt, for it lightens tlie difficulties hitherto attendant upon treating the osniotic phetiometia i n their proper connection with the simpler types of equililiriuni with which they are allied-it is a step toward a iiiore comprehensive classification by variance than h a s hitherto been possible. But ~ipoiicloser exatniiiation of the matter i n its present state one readily sees that the subject requires treatment i i i a iiiore general w a y , for through the iiitroduction of osmotic walls into a material sj.stern the 71 a Gilibsian variables are, in the general case, increased by tlie
+
'Bailcroft. 1. c. 235. *Gihhs. 1. c. 153 (1S76). 3Gibbs. 1. c. 13% JThe terminology of I'ariance used here h a s heen etriployed by the writer for soiiie years in all matters relating to a classification of iiiaterial systems according to the number of tlieir degrees of tliermodynatilic freetlorii ( see this Journal, I , 167, a n d also Banci'oft, Phase Rule, pi). j a n d 4 ) . According to it all ( 1 1 7 2)-phase systems of the ortliiiary phase rule are iiotivariatit systems, all ( u + /)-phase sj.sterris are rnonovariaiit ones, all u-pliase systenis are divariants, ( 1 ) - /)-phase s).stems are trivariants, atid so 011 ; oiie may also refer to systems of variaiice greater tliun unity a s polyvariatits, ~ r h i c is l ~often convenient. A pure liquid with its saturated vapor coiistitutes therefore a orie-component nionovariant system, ati unsaturated solution with vapor is a two-cotnporient divariatit, a mixture of t\vo gases is a trivariarit, a n d so 011. It is shoa~nin this paper that in osmotic systems the variance is not necessarily 7'
= 1L 7 2
- I',
and it is readily seeti also that the same is true of niatly voltaic comhinatiotis.
appearance of new potentials as well as that of lien' pressures.' T h e iiiaiiiiei- iii wliich this circumstance must Le takeii into account shall. iiow be considered. 1 1 ~ 7
r . iVo Osiirofic TT/cz//s. To fiutl the iiuiiilier of degrees of freeof the ther~nocl~~tiamic l,ehai.ior of a systeiii. from the nuiiiber of its iiitlepetideiiily variable components aiid that of its phases, G-ibhs coiisitlers the differeiitial equation for the energy of any one phase tlotii
I,
(i?& = f d l /
4-z J ( d l ) l
-$(it'
I
(.
( 2 )
integratiiig this, oi.er a change i n the quantity of substance from zero to t h e xiven finite aiiiouiit, at constant state (of temperature, pressure aiicl coiiipositioii ,) ,
+ 27,Liill, IS
&
= /I/
i
-$?I
(3)
and theii differentiating for the likewise perfectly general expression Y
de = f d l / --kt
+ i/d&
i
( 2 ,)
WQ+ SJJL~,LL, b'
-
~ h i c l 1in coiiiliriation with
+ 2pdill (4)
gives Y
0
= ?/df -7Yiy
+ ZIIld,II, I
(5)
iiitlicatiiig that there exists for each phase a relatioil . . . (6) which is easily slionw to lie ( ( fuiitlaiiiental ) ) i n Gibl~s'ssense. T h e variables appenriiig- iii ( 6 ) iiiaiiitaiii tiiiiforiii Ixlues tlirougliout a systeiii of ' I * coesisteiit phases, a n d h relations olitaiii among the Y potentials, so [equation ( I )]
.Xf.P, /(,, / i 2
(7/-//)
/it,)
+_.=71+2
vai-ialdes remain independelit for each pliase and the). are subject to 1' conditions-to a; niaiiy coiiditioiis a s there are equations among
theni. T h e tiuiiiber of degrees of freetloiii or tlie variance of tlie systeiii is therefore
-7.-
" j = 7 1 I
J'.
(7) a . @ J L C ///n// mid NOs + o m t ~ o u . In the cases thus treated by Gibbs there are ho\ve\,er no osiiiotic pressures, the proof supposes tlie equilibriuiii pressures to lia\ve the same values at all coexisteiit phases. A simple case where this is 110 loiiger true is afforded by a two-coiiipoiient system of solveiit ant1 solute, separated into two portions 1)~. ari osmotic wall perinealile 0111). for the solrelit. These separated portions iiitist now lie regarded as iiitli~.itlualphases, which involves a slight ex.tensioii of Gilibs's tlefiiiitioii. T h e attraction between the solute aiicl tlie solvent will then establish ail osmotic pressure upoii oiie side of the wall, wliich will necessaril~~ differ from the equililxiuiii pressure iipoii the other ; the fuiidameiital equations for the I . phases of tlie s).steiii will then l,ecoiiie--tleiiotiiiS 11). I the phases 011 tlie osmotic side of t h e wall, by I1 those 011 the other and by subscript I and s the solvent arid solute respectively'-
It is liere supposed, for tlie sake of siniplicit!., tliat the h extra compoiieiits 1iai.e alreatlj. Iiecii eliiiiiiiatetl. There are i i i tliese cquatioiis 3 variables in 11 potentials. one teiiiperature and t\vo pressures, 11 all ant1 I* relntioiis lioldiiig aiiioiig- tlieiii, so tlie \.arialice of tlie sj,steiii is
+
;' : z
72
-,-
?,
- g.
(Tal
T h i s forniulatioii is the phase nile for s!.stenis of the t).pe coiisitlered. T h e result iiia!' lie illustrated 11). a s).steiii iiiatle up of a salt i n water on one side of tlie osiiiotic all and piire water w i t h its vapor on t h e other, each coiiipoiieiit ilia!. lie supposed Leiit at constant 'The /,anrl s iiiiglit relate to ' l i q u i d ' a n d 'salt'
Osniofir Prtssii rc
i r icil
I 'arin?lcc
353
voluiiie by a closely fitting piston. T h e attraction betweeti the salt and the water will draw water into tlie solution coiiipartinetit until the osniotic pressure so produced tliere i d 1 have attained a w l u e dependent upon tlie concentration. T h e concentration can lie varied at constant temperature and the pressures will vary \\it11 varying temperature, so the equililiriuiii of the s).steiii is a dirariant one although it has three, or ~ z ' I , phases'. T h e addition of a iieiv phase, as for example ice in the water conipartnient or solid salt in that of the solution', must diminish the variance by iiiiity, and we readily see from the arrangenient iiiclicatecl b y Fig. I that the coli-
+
I
I
I
Salt FIG. I .
centration of the solution, and coiisequently the osniotic pressure, can not be varied at coiistant teinperature,-such change can be due to the change of temperature only, the system is monovariant. This follows immediately from ( ' i n ) , for 72 = 2 and Y = p so that v=1z
f3
--Y
= I.
LVith the appearance of ice this systeni would become nonrariant ; with solution on one side a n d solivent on tlie other it woulcl lie trivariant, and this appears also froiii tlie eqiiation, wliicli for JI = z and Y = z gives 7)
==2 + 3
-2
-
- 3. 'Bailcroft. Phase Rule, 235. 'The appearance, under ordinary circunistances, of a vapor phase 011 the side containing solution is escluded, because tlie pressure there is greater than tlie vapor pressare of the pure liclnicl.
./. E . Trevor
354
3 . Oirc WnZl mid Oiie Scfini*ation. T h e deteriiiinatioii of variance in cases like those just cited is the matter elucidated by Baiicroft, but we pass to the iiitroductioii of new potentials a s well a s of new pressures ~7lienwe consider cases in which the sulistaiices for which t h e wall is iiiiperinealile are present i n phases separated by the ~ l l .Under such circiiiiistaiices the fiiiiclanieiital eq~iatiotisof the phases are
+
There are here n I potentials, one teiiiperatui-e and two pressures, a 4 variables iii all aiicl consequently a variance of
+
= 7-1
+4
(76) Suppose for example the four-phase system represetitecl in Fig. z %I
I
I
- Y.
____
FIG. 2 .
T h e two solutions \vi11 Iia1.e iii general different coinpositions aiid accordingl?. different poteiitials of their components, the concentration aiid pressure of the solution i n I1 oiil!, can be varied at coiistaiit temperature, so the equilibrium is a tlivariaiit one ; this follows from (78), which for 71 = z and 7'- g becomes 'i'=zz 2 + 4 - 4 = 2
as stated.
T h e transfer to a monovariaiit system may be made
through the appearance of ice in the unsaturated solution, tlie system then assuiiiing the nionorariaiit eciuililjrimi of a fiision curve. With the freezing of ice froin the compartment I the two sides of the s\.steiii would become identical and the influence of the ostiiotic all upon tlie equilibrium pheiioiiieiia would disappear. T h i s circuiiistaiice is illustrated iiiore siiiiply by the appearance of salt in 11, above, with iiicreasing coiiceiitratioii there. W e should then have, as in Fig. 3, the phases of solid salt identical and also
I
I1
FIG. 3.
those of solution, so that tlie two masses of salt for example form one phase as truly as do iiidividual crjrstals of the salt and tlie situation is in no wise altered if the \.oluiiie lie increased so as to permit the appearance of a second mass of the vapor phase. T h e solutions niust eventually collie to tlie satlie level tlirough the influence of gravity ; the pressures and potentials are the saiiie on each side of tlie nieiiilirane and the variance is simply n + z - Y or z z -3j a nionovariaiice, just as if the nieniljraiie were not present. Such identity of phases and consequent disappearance of osmotic pressures must not be overlooked in variance problems where osmotic walls are conceriied. A similar state of affairs appears when a solution with other possible phases is separated l q . like o m o t i c walls froni two different conipartiiients containing pure solvent, with or without l.apor,-the outside compartments contain but a single set of phases and so but one osmotic wall conies really into the consideration. T h i s may also be slionrii, if it be so desired, by writing out the fundamental equations for all the phases. 4 . Two Like Walls aiid Ozc Sejaration. A n increased complexity is reached by the iiitroductioil of two osmotic walls into a systeiii, we will first consider these walls to be impertiieable for the same components. T h e set of fuiidaniental equations for the phases
+
f. E. Trevor
3 56
of such a system, when it is supposed that the outside compartiiient I11 contains no solute, are
+
There appear here 71 - I 2 potentials, one temperature and three pressures, or n 5 variables iii all with tlie consequent variance of
+
v=n+5-r. ( 7c) T h e simple arrangement, under this head, which is represented in F i g 4, has one possibility of a variation of an equilibrium pressure
FIG.4.
at constant temperature, iiaiiiely by changing, directly or by additional pressure, tlie compositioii of the solutioii in I. T h e system is therefore divariant,-and this result follows from equation (7c), for with n = 2 and Y =5 v=2+5-5 =2
a divariance. Changes of variance through the introduction or disappearance of phases, and simplification of tlie system by tlie appearance of identical phases upon both sides of a wall, are readily
Osiiiofic Prcssiire and Variame
357
foreseen. Further, addition of solute to the compartment 111 is seen at once to introduce a new potential into the system and therefore to raise its variance by unitj. for a n y same number of phases. 5 . Two U7ilike l+'al/s aiid N o Scpnmtiou. It should not be omitted to indicate at this point that tlie considerations employed in the foregoing hold in unchanged wise when the osmotic walls which appear in a given system are permeable not for the same but for cl i fferen t co tii poi1en t s. Kaoult has shown' that walls of vulcanized rubber are pernieable for ethyl ether but not for inethyl alcohol, while such of swine's bladder are permeable for the alcohol but not fur the ether. It is therefore possible, with the aid of these membranes, to construct osmotic s y s t e m in which a mixture of tlie two substances named shall be in equilibrium with each of the pure components,-as is indicated in Fig. 5 . W e h a r e here to do with two potetitials, one Stops
stops alcohol
ether
FIG.j.
temperature and three different. equililiriuiil pressures, six variables in all with five phases, i. c. fi1.e relations holdiiig aiiioiig these variables ; the variance is therefore 6 - 5 = I , a n ~ o n o v a r i a ~ ~ cT e .h e equililirium pressures of the system can obviously not be varied isothermally, so changes with chaiiging temperature alone arise for consideration, the ecluilibriuni is actually a iiionovariant one, 6. 7 k o Unlike W a l l s aiid 0 7 i c SEparafioiL. T h e formation of a solution in one of the outer compartments of the above systetn must increase the variance by unity, of course ; the process would produce for example tlie arrangement of Fig. 6 . Here are two alcohol poteiitials, one for ether, one teiiiperature and three pressures to be considered; seven variables \vith five relations among them-a -~
'Raoult, Zeit. pliys Cherri. 17, 797 ( 1 S g j ) .
J. E. Trcvov
358
Stops alcoliol
Stops ether
,
I
~~
Vapor (Ethcr)
K7fiOV
~
I
(Efher)
FIG. 6 .
Vajor
Vafior
~
- -
~~~
(Ether)
ElhM
i
(Akohd)
,
~~
(Ether) (Alc~hoz)
~
A ~ c o ~ o ~
Osiiio f ic Pmss w e n nd
VaYiajice
359
alcohol iii 111, 011 tlie other liaiicl, will coiiie to equilibriuiii with tlie alcohol of I V , tlie resultiiig pressure in I11 delieiidiiig upon the voliiiiie aiid tlie aniount of ether there. Wlien I1 aiid 111 are, finally, set in communication they iiiust coiiie to a n equilibrium between theniselves which caii oiilj. he wheii tlie composition of the latter hecoiiies equal to that of tlie foriiier-for ether can pass froiii I to 111 (or the reverse) aiid tlie displaced alcohol in I11 caii flow out' into IY. Tlie t\vo solutioiis will ),et stand under different pressures, because, altliougli the), seeiii to 1iai.e tlie saiiie coiiiposition, oiie of them is in eciuilibriuiii with ether aiid the other is so with alcohol. Tlie resultiiig eqiiilibriuiii of the entire s).steiii is governed by the relative iiiass of the alcohol in 11, wliicli caii be varied at constant teiiiperature, tlie etitire s).steiii of pressures will T.ary of course also with the teiiiperature so the s).steiii is a di\.ariant one. T h a t such is tlie result of the theory is sliowii b y counting tlie variables,-one poteiitial for ether, two for alcohol, one teiiiperature and four different pressures,-eight 1.ariablt.s aiid six phases, with a resultiiig variance of two as sliowii by tlie experiiiiental exaiiiinatioii. In coiiiiectioii with sucli cases it is instructive to consider the siiiiilar arraiigeiiieiit iii \vliicli tlie separatiiig osiiiotic wails are all of tlie saiiie kiiitl ant1 iiiipertiiea1)le for oil; coinponent, saj. tlie solute. Such a s).steiii is preseiited i i i Fig. 8 . Here tlie coiicentra-
FIG.
s.
tioiis iii I1 aiid in I11 are in geiieral different, so these two coiiipartiiieiits must stand Ltiicler different pressures to be each in equilibriurii with the solveiit aiid tlie vapor. This too is coiiipatilile with equilibi-iuiii betweeii I1 and 111. For suppose p,,, > p,, ; then, for given voluiiies of each of these two phases, tlie attractioii of the solute in TI, S,,, for the pure solveiit, plus the difference of tlie attractions
which the solute in 111, S,,, has , for the solvent in I1 and S,, has for that in 111, are exactly couiiterbalanced hjr tlie attraction which S,,, has for tlie solvent in IV. T h i s is iiidicated by the nieditim, light and heavy arrows in tlie figure. T h e coiiipartment I1 with its two membraiies may in fact be regarded a s constitutiiig a single ineiiibrane iiiipernieable for tlie solute, the attraction of this latter for the pure solvent is then tlie same-from 111-in either direction and it is immaterial how the pressure changes in the interior of tlie meinbraue. In such treatment tlie voliiiiie and mass of solute in I1 need not be supposed coiistaiit, that is to saj' tlie osiiiotic pressure is independent of tlie nature of the membrane.' Tlie total result would be altered in 110 wise if the two end coinpartiiients should be connected directly, thus foriiiing iii appearance as they do in fact but one phase each of pure liquid and of vapor. I t will be remarked that only conibinations of two components h a r e been discussed in the above illustratioiis, the application of theory to systeiiis of more than two coiiiponents would seem to offer no esperial difficulty. 8. The Geiternliwd Phase Rille. I t is iiow desiralile, after tlie general canvass which has been made of the field, to condense the above considerations into a rule for deterniining tlie variance of any given osmotic system froiii its osmotic peculiarities and the nutiiher of its compoi~ents-into a logical extension of the phase rule. To do so we h a r e merely to note that tlie variance is always the total number of varialiles wliicli a s>.steni exhibits, cliniinislied bj. tlie iiuinber, Y , of its phases. Tlie variables are ?L potentials, being at least one for each iiitlepenclently variable coinponent, one for the temperature and one for at least one pressure, with a- for tlie added pressures introduced by tlie appearance of .z' osiiiotic walls' aiicl J] for each of tlie 11 srfltri-citioirs of a coniponent by sucli a wall. Tlie total number of variables is therefore 1/
f
2
4-J'
.1'
and tlie variance is 'Compare, in coniiectioii with this, van 't Hoff, Zeit. phys. Chein. 9,477 (IS9.2).
'Save when a set of phases is simply reproduced and two pressures become equal.
Osmotic Presswe a d Variaiice 2
a=11+
+x +y
361
-v.
This is the generalized phase rule, as applicable to systems containing osmotic pressures. For tlie liniiting case in which all osmotic walls are absent we have both x = o a i d J' = o and consequently tlie Gibbsian variance of '-- n 2 - ??.
+
CI
111. Test of tht Rille
T o test the correctness of this foniiulation and at the saiiie time illustrate the convenience, the readiness, of its application let US apply it successively to tlie p s e s adduced in the foregoing section. Following the previous order arid numbering we have, first, for single walls and 110 separations, SOllt f ;oil
Vajor
Salt
Solveizt
one extra pressure and no separations, 71 = 2 ,
A-
= I ,-11 = 0 ,
" J = = z + z +r + o - - g =r,
a monovariant : then, when the solution is unsaturated,
,
I
'v= 2
+ + r + o -3 2
=2,
a divariant ; next, when the solvent freezes,
Vafio,v Solii/ioii
SaIf
v=z
Zce
Solveiif I + + r + o -5 2
=0,
a nonvariant ; and finally wlieii the solution and solvent are present alone.
J . E. Trevor
362
1
Solzifion
Sohleizt
ZJ=2+2+1+0-2
=3, the system is a trivariant. For single walls with one separation there were considered
11
x=r
=2
z=z+ 2+ I =2, the system is divariant ; hut in
y=r
+ 1-4 Vapor
Ice
Soliltion
SOZI~tiO1L
it is monovariant. T h e system (3) introduced no new pi-essures aiid no separations, so we pass to (4) with two walls a i d oiie separation,
;C=z
12 = 2
v =2
+ + a 3-I -5
y-r
2
=2,
a divariant ; then the same with two separations,
1
I Sozlcti011
1
Soliltion 11
,
n=z
'
1
Salt
Solufion 1ZI ,
x=2
3' = 2
~ = 2 + z + z + 2 - 5
=3 ,
1 VflPOY
(.Fa)
Osmotic P~.essiirc nizd Vn/nin7zre a trivariant. With two uiilike walls aticl
110
Etlrsi*
ieparatiotl we considered
I
I
Vu;hOY
Solutioii
~
n=z
,c '
363
vafioi, Alcohol
~
=2
(5)
~
0
+ + + 0-5
z'=2
2
2
= f,
a iiionovariant ; and with one separation,
I
vafior
Vafior.
Solu/im 1
~
I1
Solitlioii
~
=2 ~
For the
I
vnfiov
1
ELhci* 71
((
2+
(6)
~
r
I!=
L'=2 i'=2 + 2 +
a divariant.
'4/Coho(
1Z 1-5
2,
coinplex cases ) )
\ye
haye I VLZjOY
~
Soliifion
~
So/citioi/ 111
x=3
=2
Ilrohol
~
J'
=z
1
(7)
f
+I-6
L'=Z+Z+J
=2,
the s ~ s t e i i iis a tlii.ariant ; and, filially, with like membranes we h a r e the case ~
Vajor
I
1
sohcflt
~
I
I s o i f t f i o ) l lz
~
P
soi;lcfrt.
vnfio)-
sollttio)L ~
The two end coinpartniet1ts here have been shown (a5 is iiideecl tiow apparelit from inspection) to constitute h i t a iiligk pair of phases, so that they iiiiglit it1 fact be connectecl" b y a canal without disturbing the eqLiilibriuiii ; accordiiigly ?I
a=z
=2
a=z+2+2Tr-# -
-3,
1'
=f
364
J . E. Tmvor
the system is trivariant. This is obviously true, for the equilibriuni pressures can be varied in two ways isothermally, by changing the concentrations of either of the inside compartments. T h e variance as thus deterinined accords in every case with that worked out in the preceding section. Any number of further combinations might be added, but. it is believed that those given are sufficient for the present purpose. I t may be well nevertheless to add, in closing, that two-component systems can be arranged with 17apor alone or with vapor and ice i i i one of their compartments : Imagine for example the system depicted in Fig. g ; the solution, supposed
Solzitioit
FIG. 9.
sufficieiitly concentrated, may be allowed to expand until it shall have absorbed entirely the relatirely siiiall amount of solvent in 11, and it may then be brought into equilibrium with tlie unsaturated vapor of I1 by suitably altering the pressure. T h e consequent variance will be that conditioned by one extra pressure aiid no separatioii, namely v=P+P+T+o--P
=3 ) a trivariance', -and it would he reduced to a divarialice by cooling the systeiii to the temperature at which ice appears in the vapor compartment. It is well worth while to reflect upon such cases when studying osmotic plienoiiiena. 'It will be noted t h a t tlie pressure of the vapor will not ill general lie that of the saturated vapor of the solutioti a s it stands.
T h e peculiar ad\.aiitage accruing from definite determination of the variance of syteiiis coiitaiiiiiig osmotic pressures is that tlie relations of the equililxia of such systems to those of tlie simpler types of therniodyiiaiiiic equilibrium where osmotic pressures do not appear become thereby apparent. One may classify the eqiiilibria with which phj.sical clieiiiistry is concerned in two ways, which offer, each of tlieni, decided advantages ; one can classify either primarily according to the variance and secondarily according to the nuiiiber of iiitlepeiidentl~~ variable conipoiieiits-thus treating all monovariant systems together, then all dirariants and so 011-or classify primarily according to the nuiiiber of independently variable components atid secondarily according to the variance. When following the latter plan one completes the discussion of one-component systenis before attacking that of two-component systems ; and under the head of two-component systems one can begin the treatnieiit of ostnotic phenoinena, and discuss them in the order of the variance of the equilibria in hand. T h e whole scheme of presentation becomes thereby verj. orderl). aiid coherent, aiid it has for these reasons much to reconinieiid it for lucidity and conseqiience. But withoat due attention to the \.arialice of osmotic systems it is a difficult and an arbitrary matter to introduce the important phenomena, which they present, into a satisfactory treatment of thermodynamic equilibria in particular or of physical chemistry in general.
