ON THE T R I P L E POINT BY PAUL SAUREL
Bakhuis Roozeboom has recently given' a complete classification of the various possible kinds of triple point. Although there is nothing to be added to his results, it may be of interest to indicate two other methods of obtaining t h e m One of these methods is due to Gibbs, but up to the present it does not seem to have received the attention it deserves. The fundamental equations of the triple point are well known. They are
I n these equations vi, videnote the entropy and the volume of the unit of mass of the i-th phase, and dIli/dTi denotes the slope of the pressure-temperature curve of the i-th univariant system, that is to say, of the system which is formed by suppressing the i-th phase of the invariant system. Let us suppose that the phases are so numbered that 1'
and let us write equation
I
> > 2'
(3)
'31
in the form
If we observe that, in virtue of 3, the coefficients (a, - vJ, (aI- a,) are positive, and also that '1
-
= (74 - V3>
t
(Vl
-4,
it follows from 4, by a well-known theorem in algebra, that Die heterogenen Gleichgewichte,
I,
189 (1901).
400
Paul Saurel
dIIJdTz is intermediate in value between drIIl/dTIand dIIJdT3. We thus obtain the following theorem of Duhetn : THEOREM I. --At a temperature slzghtly hzgher or slightly lower than that of the trz;ble point, the univariant curve which corresponds to the transformation that is accompanied by the greatest change i n volume lies between the other two curves. In like manner, let us imagine for a moment that the phases are so numbered that 9,
> % > %,
(5)
and let us write equation 2 in the form
From this equation we obtain at once the following theorem of Roozeboom : * THEOREM 11.- Under apressure slightlygreaterorslzghtly less than that of the trz;bZe point, the univariant curve which corresponds to the transformation that is accompanied by thegreatest change i n entropy lies between the other two curves. The univariant system can be grouped into two classes. T h e systems of one class can exist in stable equilibrium only at temperatures higher than that of the triple point ; the systems of the other class can exist in stable equilibrium only at temperatures lower than that of the triple point. We have shown3 that the coefficients (v2- v3),(a,- vI),(vI- v,) in equation I enable lis to form these two classes without, however, enabling us to say which class corresponds to the higher temperatures and which to the lower. T h e univariant systems which correspond to positive coefficients form one class while the univariant systems which correspond to negative coefficients form the other. If we suppose that the phases are so numbered that ~
Zeit. phys. Chem. 8, 367 (1891). Trait6 616mentaire de Mecanique chimique, 2, 98. Die heterogenen Gleichgewichte, I , 98 (1901). Jour. Phys. Chem. 6, 257 (1902).
On the Trz)le Point
40’
conditions 3 are satisfied, then the univariant systems I and 3 form one class while the univariant system 2 forms the other. We may accordingly state the following theorem : THEOREM 111.- The univariant system which corresponds to the transformation that involvps the greatest change in volume is in stable epuilibrium at temperatures which lie on one side of‘ the tr+lepoint while the two other univariant sysLems are in stable equilibrium at temperatures which lie on the other side of the trz)le point. We can also group the univariant systems into two classes by putting into one class the systems which can exist in stable equilibrium only under pressures greater than that of the triple point and into the other class the systems which can exist in stable equilibrium only under pressures less than that of the triple point. We have shown2 that the coefficients ( q , - q3), (v3- 7,))(7,- 7,)in equation 2 enable us to form these two classes. T h e systems corresponding to positive coefficients form one class, while the systems corresponding to negative coefficients form the other. If we suppose that the phases are so numbered that conditions 5 are satisfied, then the univariant systems I and 3 form one class while the univariant system 2 forms the other. We may accordingly state the following theorem : 3 THEOREM IV. - The univariant system which corresponds to the transformation thbt involves the greatest change in entropy i s in stable equilibrium under pressures which lie on one side o f the tv+le point while the two other univariant systems aye in stable equilibrium underpressures which lie on the other side of the tr$lepoint. By combining theorems I. and 111. or theorems 11. and IV. we get at once the following theorem of Gibbs : 4 THEOREM V. -rf we describe a smnll closed curve about Duheni. Zeit. phys. Chem. 8, 379 (1891). Trait6 elementaire de MCcanique chimique, 2, 123. Jour. Phys. Chem. 6, 257 (1902). Roozeboom. Die heterogenen Gleichgewichte, I, 98 (1901). * “On the Equilibrium of Heterogeneous Substances,” p. 174.
402
Paul Saurel
the trz$Ze point, this cuyve aZternnteZy cuts stnbZe and unstabde branches of the three univariant curves. Roozeboom has shown1 that the five theorems which we have just given suffice, when taken in connection with the wellknown equations
Vl,
719
1
a,,
721
1
v,,
73,
1
,
(8)
On the Trz)Ze Point
719
VI,
1
721
nz,
1
73,
a81
1
Jour. Phys. Chem. 6, 261 ( 1 9 2 ) .
,
403
(9)
404
Pau l Saurel
If we turn the v 7 plane through a right angle so that the r ] axis becomes parallel to the T axis, and the v axis parallel to the negative direction of the II axis, we can restate theorems VI. and VII. as follows : THEOREM VIII. - If the v r ] plane is so placed that the 7 axis is Parallel to the T axis and the v axis parallel to the negative direction of the II axis and from apoint v.ithin the triangle whose vertices are (vj,vi) we drop perpendiculars to the three sides of the triangle, these three lines are resjectively parallel to the directions at the tn$e point o f the stable portions of the three univariant curves. This theorem is due to Tanirnann,I who has stated it without adequate demonstration. By means of the last theorem we can at once enumerate the various types of triple point. T h e results are given in the accompanying diagram (Fig. I). T h e eight figures correspond -respectively to the following eight sets of conditions :
I
7, VI!
'ul9
1
rlL!
'up9
1
> > 78. rl2
I
1 < 0.
It now remains to show that the above classification can be very easily obtained by carrying out in detail a course of reasoning outlined by Gibbs." Consider a system consisting of three phases formed by means of a single component. If we denote by CP, the total thermodynamic potential of the unit of mass of the i-th phase, the well-known conditions of equilibrium of the invariant system are Drude's Ann. 6, 65 (1901). "On the Equilibrium of Heterogeneous Substances," p. 174.
On the Tr;Ple Poiizt
40.5
cp, = Q?* = as. (11; Por the three univariant systems that can be obtained by taking the phases in pairs the well-known conditions of equilibrium are
Fig.
I
Let us take in space a system of rectangular axes along, which we shall measure the temperature T, the pressure IT, and
406
Paul Saurel
the thermodynamic potential @. At a given temperature ‘I’ and under a given pressure ll we shall have three values @I, a2,Q3 for the thermodynamic potential, corresponding to the three phases. We thus obtain three surfaces or rather three sheets of the same surface. T h e three sheets of our potential surface taken in pairs intersect in three lines which, by equations 12,represent the states of equilibrium of the three univariant systems ; and the point in which these three lines intersect represents the state of equilibrium of the invariant system. T h e projections, upon the temperature-pressure plane, of these lines and this point are respectively the pressure-temperature curves of the univariant systems and the triple point of the invariant system. At a given temperature and under a given pressure the most stable state of equilibrium of a system is that for which the thermodynamic potential has the smallest value. Accordingly, at a given temperature and under a given pressure, the lowest of the three sheets in our diagram will represent the most stable state of equilibrium of our one-component system. T h e three univariant curves divide the space about the triple point into six regions. By considering a small closed curve surrounding the triple point we shall be able to determine, in each of the six regions, which sheet of our thermodynamic surface is the lowest and consequently which phase is the stable phase. For this purpose we shall need the equation which connects the change in the thermodynamic potential with the changes of temperature and pressure, viz : d@, = -rlr),dT v,dIII. (13) Through the triple point M (Fig. 2) draw two lines MT’, MII’ parallel respectively to the positive directions of the temperature and pressure axes. If we move from the triple point a short distance along the line MT’ to a point MI, we shall have, from equation 13, for the corresponding changes in the thermodynamic potential d@, = - v,dT,
+
d@., = - 17 dT d@, == - V,dT. 2
j
(14)
On the Tn&e Point
407
T h e relative positions of the three sheets above the point Mr depend upon the relative values of qI, T, 7,. Thus, for example, if
> >
711
we shall have
172
713,
< d@2 < d@3,
and the three sheets, beginning with the lowest, are arranged in the order I, 2, 3. At the temperature and under the pressure corresponding to the point MI, the phase I is the stable phase. In like manner, if we move from the triple point a short distance along the line MII' to a point &Iz, we shall have, from equation 13, for the corresponding changes in the thermodynamic potential d@, = v,dII, d@, = v z d n , d@, = v3dn.
(15)
T h e relative positions of the three sheets above the point &I2 depend upon the relative values of ar, 3,) a,. Thus, for example, if 2'1 > v, > v3, we shall have d@,
> d@, > d@,,
and the three sheets, beginning with the lowest, are arranged in the order 3, 2, I. At the temperature and under the pressure corresponding to the point MD,the phase 3 is the stable phase. In our discussion there are the following four cases to be considered corresponding to four different arrangements of the univariant curves at the triple point : Case I. The three slopes dlTJdT, are positive ; Case 11. Two of the slopes are positive and one negative ; Case 111. One of the slopes is positive and two negative ; Case IV. The three slopes are negative. Throughout the discussion we shall suppose that the phases are so numbered that 711
> > 712
73.
(16)
Paul Saurel
408
Let us suppose that, at the triple point, the slopes of the three univariant curves are positive. Thus
Conditions 7, 16 and 17 yield at once vi
> vz > va*
(18)
From 14and 16 we find that, above the point MI (Fig. 2), the sheets beginning with the lowest, are arranged in the order I , 2, 3. It follows that, as we move along the closed circuit from M, towards M2,the first univariant curve encountered is either the curve that corresponds to the intersection of the sheets I , 2 or the curve that corresponds to the intersection of the sheets 2, 3. In like manner, from 15 and 18 we find that, above the point MI, the sheets, beginning with the lowest, are arranged in the order 3, 2, I. I t follows that, as we move along the circuit 7
n
r Fig.
2
-k Fig. 3
T
from illz towards MI, the first univariant curve encountered is either the curve that corresponds to the intersection of the sheets 3, z or the ciirve that corresponds to the intersection of the sheets 2, I. There are thus two sub-cases to be considered corresponding respectively to Figs. 2 and 3. If, in the first of these cases, we describe, from MI, a small circuit counter-clockwise about M, we shall find that the rela-
Oiz the Trz.Ze Point
409
tive positions of the three sheets in each of the six regions of the plane are given by the following table : 3
3
1
1
2
2
2
1
3
2
1
3
I
2
2
3
3
1
In this table the numbers in the same vertical column refer to the same region and indicate by their relative positions the relative positions of the corresponding sheets. If we erase the portions of the univariant curves which correspond to unstable states of equilibrium we obtain the first of the eight diagrams given in Fig. I. If, in Fig. 3, we describe a small circuit counter-clockwise about M, we shall find that the relative positions of the three sheets are given by the table : 3
2
2
1
1
3
2
3
1
2
3
1
I
1
3
3
2
2
If we erase the portions of the univariant curves which correspond to unstable states of equilibrium we obtain the fifth diagram in Fig. I. An analogous discussion of the remaining cases yields the remaining diagrams in Fig. I. Case 11. corresponds to the diagrams 2 and 6, Case 111. to 3 and 7, and Case IV. to 4and 8. T h e method yields not only the stable portions of the univariant curves, but also the regions of the plane in which the different phases are stable. T h e numbers in Fig. I have been placed so as to indicate these regions. New Yovk, May -15,rgoz.
