O S THE THEOKEIIS O F ROKIS -1SD O F 1 I O V T I E K €31-P A U L S.IURE1.
IZ
The total thermodynaiiiic potential @ of a system containing coinponents i n 7' phases is given b y tlie equation 71
Y
/
,
]
I
in which 11,,denotes the iiia5s of tlie 1-tli coinpoiient coiitaiiiecl in the z-tli phase, and F,, is the partial derivative of the potential with respect to that mass. If tlie SJ stem is in equilibrium at the temperatiire T and under the pressure TI, the \oliiiiie V and the entropy H are giveii 11~7the eqiiations
?
I
l
l
aiicl for a n y yirtiial change in the iiiasses
7
1
1
1
hccordingl\., for an! virtiial change in the s~stem n e ha\ e t
,L
1
7,
S@=8IIy, lI I
=\6II
l
-
J
'
i
,i
T \
-
,,
S T 7
i
Z
I
J
I
aF
a,r 11
-
,I
7'7 F,8 M 2 1 1 1 1
H6T
Let lis apply this equation to a uiiiJariaiit s!~teiii. -It a given temperature atid under the corresponding piessure such a system admits of a continuous series of states of equilibrium for
wliich tlie potential has the saiiie d u e , but for which the veliiiiie and tlie entropT- are different. Denoting aiiy two of these states by t h e siibscripts I aiid 2, we s l i d have Q >I-- Q ? ?
6Q,
a@.,
--
--
~
\-,GI1 - H,6T, T - < 6 r I H,6T. -~~
If Y qis greater than and if we coiisideI a yirtual cliaiig-e in \vhich T remains constant, we sliall have
> a@,. T h a t is to say, if we consider a t the temperature 'I' and under t h e pressure II -~6II two states of the system n-hicli m e states of equilihriuni at the teniperatiire T and under the pressiire n,then tlie state with tlie greater \-oliime lias the greater poteiitial. Eiit if a systeni is not iii q u i l i b r i u i i i a t a giI-eii teiiiperature :ind niider a gix-en pressure it uiidergoes a traiisforiiiatioii which causes the potential to decrease. A l c c o r d i n g l ~if. , we coiisider at the temperature T ant1 tiiider tlie pressiire n 611 a univariant system in a state which is a state of eqnilihriiiiii for tlie teiiiperatiire T anti the pressure 11, the sy,steiii \vi11 iindergo a change which, if it occurred at tlie teiiiperature 'I' ant1 under the eqiiilibriiiiii pressure TI, would be accoiiipanietl b!- a decrease i n 1-oluiiie. Similarly? nt a pressure slig-litly less than the ec1uililxiiiiii pressure, the transforiiiatioii is such that at the ecliiilibriuni pre,siire it \voiild be accoiiipaiiiecl 1-: aii increase in \,oluiiie. T h i s is Kohiii's T1ieoreni.I S e s t consider a 1-irtiial change iii which I1 reiiiaiiis coilstaiit, aiid let 11s suppose that €Il is greater thaii kI., T h e n , tlie equations iiiider disciissioii gil-e -
>:
6Q1.
T h a t is to sa)-, if we coiisider under the pressure 11 atid a t tlie temperature T 6T tn-o states of the s\-steiii n-liicli are states of eyiiilibriuiii under the pressiire TI arid at the teniperature T,
then the itate with the snialler entropy has tlie greater poteiitial. A\ccorcliiigly, if vie consider iincler tlie pressure I 3 and a t the teiiiperature T 6T '1 uiii\ ariant 5 ) stem in a state n liicli is a state of equilibriiiiii for the pressure IT and the temperature T, the iysteiii ill undergo a change which, if it occurred under tlie pres5ui-e II a i d at tlie equilibriuiii temperature T, u ould he accompanied 1 1 ~an increase it] entrop:, or, in other words, by an absorption of heat. Similarl! , at a teiiiperatiire sliglitl! lower tliaii the equilihriiuii temperatiire, the transforiiiatioii iz such that at the eciiiilibriiiin temperature it wonld give out heat. T h i i i i AIoiitier'C Tlieoi-eiii.l Let iii suppose firiall! tliat 6n and 6Tare equal to tlie actual change5 d I and tJT of pressure and temperature ab n e pais froiii oiie poiiit of the tranifoiiiiatioii curve to the nest. Then the two states of equililx-ium coiisidered will pass into two acljacetit states of eyuilihriiun for n hich the poteiitials will again be equal. ;Iccordinglj n e ha\ e ~
m2 (fa, 9
from
1 1liicli
\.e obtaiii
~ - , ~ i n - ~ ,vr, im~
--
H,~T,
or finall!
iii which Q represents tlle heat absorbed b!. tlie s!-steiii iii passi n g froin tlie state I to tlie state z at tlie teniperature T and tiiider the pressure IT. T h i s is the well-k1ion.n Clapej-roilClaiisiiis fortiiiila. / : O / d f ' t r 21.1'. *?/
