OS THE GES;ERSLIZ.\TIOS O F C L X P E T R O N ’ S E QV X TI 0S
BY P.iUL SAUREL
1-arious simple tlemonstrations of Clapeyron’s equation for uni variant systems have been given, but as yet no simple demonstration of the extension of this equation to the indifferent points of bivariant and inidtivariant systems has appeared in an easily accessible publication. T h e object of the present note is to offer such a demonstration. Before giving the demonstration it is necessary to recall the characteristic properties of univariant systems and also those of the indifferent points of bivariant and multivariant systems. T h e temperature and the pressure of a univariant system in equilibrium are functions one of the other. At a given temperature and under the corresponding equilibrium pressure, the equilibriuin concentrations of the phases are determinate, bnt the volume and the entropy of the system inay be varied. T h e indifferent points of a hivariant or inidtivariant system form a continuous series analogous to the series of states of equilibrium of a univariaiit system. At a given temperature, the pressure of the corresponding indifferent point and the concentrations of the phases at that point are determinate. T h e ratios of the masses of the independent components are also determinate, but the volume and the entropy of the system may he varied. In passing from one indifferent point to an adjacent indifferent point, the temperature, the pressure, the concentrations of the phases and the masses of the independent components of the system vary.‘ Let us denote the temperature by T, the pressure by IT, the 1
Jour. Phgs. Chem. 5, 47 (1901).
volume by \', the entropy by H, and the energy bj. E. l y e shall also iiiake use of the functions F, C, and @ defined by the equations F=E-TH, (1) G =E + Ill7, ( 2 ) @ = E - T H 7 Hl-. (3)
F is the free energ?. and @ the total tbermoclj naiiiic potential. I t can be showiil that for the i-th phase of the sj.steiii, the following equation holds II
2
Y , d n ~-
J
--
F,,dl\I,.
(4)
1
H,, ITl and G Lare respectivel). the entropj-, the volume arid the theriiiod\-namic potential of the phase, >ITJis the mass of the j-tli component which is present in the i-th phase, F,, is what what may be called the cheinical potential of the /-th component in the i-th phase, and n is the number of independent coniponents. ITlieii the systeni of 7' phases is in equilibriuin, we Fr,:
F, =
...
-~
F,--= F,,
1'-
I, 2,
. . ., 72.
If then we add the varioiis equations of the form 4 and if we recollect that , I i
1 - 1
1 = 1
2 = 1
I = I
9 1 T l being the total mass of the /-th component, we get ,I
d G = - H ~ 4T\'dl3
+
2
F,dW,.
(5)
J'I
From this equation and equations difficulty I
''
Jour. Phys. Chetn. 5, 49 (1901). Jour. Phys. Chem. 5 , 49 (1901).
I, 2
and 3, we obtain without
J'= I
Each of these equations will enable 11s to establish Clape!,ron's equation for the indifferent points of bil-ariant or inultivariant systems as well as for univariant systems. Consider a univariant system, or a bivariant or inulti\-ariatit system at a n indifferelit point. Keeping the teniperatare, tlie pressure and the masses of the independent coiiipoiieiits constant n-e may vary the eiitrop?. rind the x-oluine of the s).steni. If we denote tlie initial and the terminal states of eqiiilibriuin b!. the subscripts I and 2 , the four equations just writteii give i i i QL-@,=O,
E, - E, F,-~ F,
< C] I
Ti H, - H I 1 ~
IIf Y1- I-, 1
--
n(\->- Trl
~
I.
1101 i
G1- G, = T t H , - H, ) .
11'
112'
Siippose now that the pressure and temperature are cliaiigetl. T h e states of equilibrimn I and 2 will change into tivo n e v states of equilibrium. T h e equations last written gil-e 11' d(.@, - @,) = 0, (13) I d ( E, - E, 1 Td H L-- H, - nil' I' V2---(~?--~~~rz~~-(\~~--\-,)dn, d ( F, - FI ) = n d,I-,, 1-1 ')) - - (I-,, irfn, 15 , d ( G , GI ) = T d ( H, -- H, ) - ( H1-- HI 1 d T . 1161 I
\-)
~
~~
~~
If we are considering the change from one iticiifferent point to another, tlie changes in the temperatiire aiid tlie pressure must, i n general, he accoinpaiiiecl bj- changes in the masses of tlie i i i dependent components. If, on the contrary, the sJ.stein coil-
sidered is a iinivariant system, the masses of the independent coinponen ts need not change. In both cases, however, equations 5>6, j and S give 11s
If we substitute these 7-allies in equations 13, 14, I j and 16, we get i n each case the equation
T h i s equation may of course be written in the familiar Clapeyron form
i f we denote b y Q the heat absorbed bj. tlie sj-stem cluring the rei-ersible change at coilstant temperature and under constant pressure. If the system uiider consideration be a iinivariant systeiii it is not necessary to consider changes in tlie masses of the independent coiiiponents. Equations 5, 6, ;and S then take the more familiar forms
Froin these eqiiations we can derive simple proofs of Clapejxou's eqaatioii for univariant systems. T o do this we slid1 iiiake use of the fact that for a sj-steni of constant inass we may choose the eiitropj. and the volunie as independent variables ivliich fix the state of the system, or we may take as independent variables either of the above variables and the temperature or the pressure.
Equation 9 shows that for a anivariant system @ is a function of T alone. If then we write equation 2 1 in the fonii
it follows that the coefficient of d T is a function of T alone. If we take, as our second independent variable, tlie volume, we must have a H t \ - d drI T)-o.
a \-
(-
T h i s reduces to 1111 I
Consider next equation 2 2 and take, as independent variables, the volume and the entrap:-. Expressing the fact tliat the right-hand ineinber of tlie equation is a perfect differential, n-e get
In virtue of the well-known relation
this be comes
I n deriving this formula we have not made use of tlie fact tliat the system iinder consideration is a univariant systein. Consider next equation 23 and take as independent variables the temperature and the volume. Expressing the fact that the right-hand ineinber (of the equation is a perfect differential, n e get
398
PauZ Sazircl
Similarly, if we take as independent variables the pressure and the entropy, equation ,?4 will give 11s
I n deriving equations 1-and \’I we have not made use of the fact that the system under consideration is univariant. For a univariant system the pressure is a function of the temperatiire alone ; we may therefore omit the subscript which appears in the left-hand inembers of equations II’, T and VI. Moreover. for a univariatit system, we may write
(
E 1, ( ;TH > n =
m. H, - H,
=
Equations 111, IT-,\- and TI thns reduce to the form dII __ H1-HH, dT V2-V, ’ ~
T h e demonstration by which we obtained equation T7 seems to be due to van der 1Taals.I T h e demonstrations just given can be modified so as to apply to the indifferent points of bivariant or multivariant systems, but the demonstrations lose their simplicity. For the sake of completeness, however, we shall indicate the more important steps i n the work. K e shall denote the total inass of the system by 9Kl and we shall suppose this total mass to remain constant. Moreover, we shall write 9K, = m,9li,
so that IU, may he called the mean concentration of t h e j - t h coinponent in the system. Altliough 91i is to remain constant, and mi may vary. Equations 5, 6, 7 and 8 now become Bakhuis Roozehoom. Sur les conditions d’Cquilibre de deux corps dans les troiq i.tats, solide, liquide et gazeux d’apres 11. v. d. IVaals. Recueil des travans chiniiques des Pays-Bas, 5 , 336 (18S6).
C
ti
/j 1'J
n
TdH
dG=
\'d
- 9K
3
F,dm,.
(28)
,=I
I t is to be remembered that for univariant systems every dm, is equal to zero, and that for the indifferent points of bivariant and multivariant systems F, and ~ I L , are functions of the temperature alone or, if we please, of the pressure alone. We may also say that F, is a function of ?E,. T h e four deinoiistrations given above now take the following forms: Equation 2 j yields
j=I
From this we get, as before
Equation 26 yields
,=
,'=:
I
Since F, is a function of wz,, aF,
av
~VZ, -
aH
aF, aH
am, ai'
__ --OJ
and the preceding equation reduces, as before, to
This may again be reduced to the form
(;3”=(;3r Equation 27 yields
J
I
T h i s gives
( ;: > , = -( ;: )T Finally, equation
28
yields
By the same reasoning as before, these equations may be reduced to the form of equation I. _\i.w
Jbrk, .llnidz 5, zgcrt.
