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STATISTICAL MECHANICS OF CONDENSING SYSTEMS.
V1
Two-component Systems JOSEPH E. MAYER Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland Received October
1%,
1938
I. INTRODUCTION
An extension of the method of Ursell (8) for treating imperfect gases by the author, working with Ackermann and with This method has succeeded in predicting the conditions under which condensation occurs in a one-component system, and is in agreement with experiments showing that the usual classical conception of the critical point is not altogether correct (3). The theory was reported by Born (1) to the van der Waals Congress in Amsterdam. Since then Born has, with Fuchs (2), greatly improved its mathematical basis, as have also Kahn and Uhlenbeck (4). In this paper the method, with a mathematical approach modelled after that of Born and Fuchs, will be applied to two-component systems. In general, for any system, the work function A, or Helmholtz free energy, may be calculated as has been developed Harrison (5, 6, 7).
A
=
QÁNi,V_,T)'
kT
II-Y-.i
(1)
-
in which N¡ is the number of molecules of type i. QT, the volume-dependent part of the expression, is the configuration integral of the system, an integral over the configuration space of the molecules. For the usual classical mechanical case QT is given as Q7
=
jj j ...
e~u!kT
dn
·
dr,·
·
·
·
dr.v
(2)
in which U is the potential energy of the system as a function of the coordinates of the molecules, and dr, is the element of configuration space of Presented at the Symposium on Inter-molecular Action, held at Brown University, Providence, Rhode Island, December 27-29, 1938, under the auspices of the Division of Physical and Inorganic Chemistry of the American Chemical Society. 1
71
72
JOSEPH E. MAYER
the jth molecule in the system composed of N molecules. For the simple of monatomic molecules with no internal degrees of freedom
case
dr,·
=
dxjdy,-dzj
For the sake of simplicity of expression for which equation 3 is valid.
we
(3)
shall limit ourselves to
cases
The essential feature of the method followed is to split the integrand, e~ulkT, of equation 2 into a sum of terms, the first being unity throughout the configuration space, the others differing from zero only where certain numbers, m¡, of clusters of l specified molecules each are close to each other
In the case that the potential U of the system is a sum of 1) terms, each a function only of the one distance between one %N(N pair of molecules, this analysis into a sum of terms can be made readily from the mathematical form of the integrand. The integral of such a general term in this sum is a product of “cluster integrals,” Nvb¡, each raised to the power m¡, where v is the volume per molecule in the system. Kahn and Uhlenbeck have shown this to be a general result, independent of the use of the classical expression (equation 2) for QT, and independent of the assumption that the potential is the sum of that between molecule pairs. Those terms characterized by the same set of numbers mi, but with different specified molecules associated together in the clusters, lead to the same integral in a one-component system, and correspond all to one definite distribution of the molecules in the configuration space. This distribution is simply one in which there are m¡ clusters, each containing l molecules close to each other, in excess of the random expectation. Distributions described in this way are naturally significant only as long as the total volume is so large that some of the molecules, at least, are at distances large compared to the reach of the molecular forces from any neighbor. It is therefore not surprising that the method fails to give simple results, at least without forcing, for volumes lower than that of the condensed in the cluster. —
phase.
are two types of molecules, with a total of L molecules of type a, molecules of type a, in the system, with
If there and
L +
=
N
(4)
must distinguish the clusters by two numbers, l and , the numbers of molecules of types a and a, respectively, in the cluster. The number of numerically equal terms in the integral QT, with the same set of the numbers m¡,\ of clusters of l and X molecules each of types a and a, respectively, is just the number of ways in which the L and numbered we
73
STATISTICAL MECHANICS OF CONDENSING SYSTEMS
molecules of the two types can be distributed among the clusters. leads to a numerical coefficient for each product
This
(Nvh,x)m‘*
in Q„ such that *
Qr(L, A, vb) L! !
V
_
2
,
w
mLx\
¡iX
lm¡,\
—
L
=
A
The cluster integrals, b¡,x, are volume-independent at large volumes. The volume per molecule ty, below which the bi,x’s become significantly volume-dependent, is that of the condensed phase. We shall assume, in the subsequent discussion, that v is always greater than ty. It is only in the case that the potential U may be written as a sum of interactions um, uaa, and between pairs of types aa, aa and aa, L-l
h
Jj
O.au
n>m m-=l
0 mn).
"4"
^
U(rmn)
=
fUr^)
=
=
/.«(V)
^
µ=1 m=l
that the integral definition of the
6¡,x
fmn
-1
h
hi.
=
'.
. ^ µ)
^
"4*
Uaa(rµ·.)
(6)
»->µ µ—1
cqn be readily given.
=
-
/„*
=
e-u“(r^)lkT
/„„
-
e^')lkT
-
-
Defining
1
(7a)
1
(7b)
1
(7c)
the equation for 6/,x is
j j
=
Ih.X
‘
|
(sum
^| over
’
^LrfIT/ / µ/µ díl
all products consistent with
a
·
·
d/;
'
dri
1
drx
‘
'
(8)
single cluster)
in which di is used for the element of configuration space of molecules of type a, and dr for molecules of type a. The sum goes over all possible products of the f’s, such that all molecules are at least singly connected with functions / to every other. The integrals 6¡,o and bo.z are identical in form with that written as b¡ in previous papers on one-component systems.
For the classical case, equation 2, and when expression 6 is assumed for U, irreducible integrals Bk,K may be defined as Bk,K
(sum
=
over
JJ
*
*
‘
J
fmnfmµfµv
dtl
*
*
*
dífcdri
·
·
·
di",
all products with all molecules more than singly connected)
(9)
74
JOSEPH E. MAYER
The cluster integrals bj> can be shown to be sums of products of the Bk,K raised to certain powers. These integrals, Bt,„, differ from the irreducible integrals jS* of the earlier papers for one-component systems in their normalization. The two notations are so related that Bk,0 and So,*.· are the same
as
The numerical coefficient with which a certain product of irreducible integrals Bk,K occurs in the cluster integral b*,x is not easy to evaluate or simple to express in closed analytical form when evaluated. In appendix B these coefficients are calculated, and b*,x is derived in terms of the Bk,K as a Cauchy integral. This form (equation B-23) is sufficiently explicit for our
purposes. Of course it would be formally possible to define the quantities Bk,K in terms of the integrals b*,x by the inverse of equation B-23. By this artifice the irreducible integrals are definable even when equation 6 is not valid or even in the quantum-mechanical case that Qr is not given by equation 2. The subsequent equations of this article are therefore formally valid under all conditions, but, as pointed out by Kahn and Uhlenbeck, this seems extremely artificial. Certain sums, related to the g, sums of previous articles, occur repeatedly in the equations. These sums occur both as functions of the b¡,x with certain variables z and f, and also as functions of Bk,K with other variables y and . In order to shorten the notation the two cases will be distinguished by using lower case and capital letters, respectively. The sums are defined as gm.„
g„,„
=
=
4-0
*-0
1**0 X-0
rvb*.; :'fx
r/B^/v
(bo.o
(B„,„
=
0
=
0)
if k +
(10a)
«
< 2)
(10b)
The relationships
(lla)
=
=(ub) are
II.
seen
to hold from the definitions
THE METHOD
OF USING
la and 10b of the sums.
THE LOGARITHM
OF THE LARGEST
TERM IN Q,
Expression 5 for Q, consists of a sum of terms, each characterized by a certain set of the numbers w*> defining a distribution of the molecules in space.
The logarithm of
QT,
for large values of L and , may be equated to that
STATISTICAL
MECHANICS
OF CONDENSING SYSTEMS
IO
The largest term, subject to the conditions of the largest term in the sum. L and SXm¡,x that , , is found by the method of undetermined multipliers, using the symbols In z and In £ for the two parameters. The details of the manipulation do not differ appreciably from those used previously for one-component systems (5), and will not be repeated. One obtains for the quantity a(x), defined as =
=
limit at constant x
L/N,
=
as
IV
—>
x
(12)
A/hV
=
=o
the equation
a(x) (x
+
1), in which
=
z
=
and
x
In
£ are
z
+
In £
-
(13)
vg0.o
determined by the two equations, x =
01,0
(14a)
-
v
(14b)
From equation 1 the quantity a{x) is seen to be related to the work function A, per mole of material, by the equation (1-5) so
that A
=
RT
x
In
The pressure,
, kT
/
„
is found to be,
with the
use
¡7o,o
+i (x
, —
vgi.o)
3
In
—
z
--f-
dv
.
,
(
—
\3ln£
t’0o.i)
—r—
dv
of equations 14a and 14b,
P
=
kTgo.o
The equation for the free energy, F, per mole is
(17)
76 so
JOSEPH E. MAYER
that the chemical potentials µ„
=
µ
=
µ„ and µ of the two molecular species are
r.
*
kT In |f
(19a)
\$TrmakTJ
kT In
\112
h2
(
(19b)
|
\2icmakT/
which shows the physical significance of z and The two sums ¡71,0 and ¡70,1 (equations 14a and 14b) will be convergent for sufficiently large values of v. Under these conditions the system will be one phase gaseous. The condensation volume v„ the volume per molecule of the saturated vapor, will be determined by that value of v for which 0i,o + 0o,i becomes divergent. For v > v, the quantities z and f may be expressed as power series in l/v, and the first few coefficients determined from equations 14a and 14b by algebraic methods. If these coefficients are expressed in the B*,„’s instead of the hi.x’s (which can be done without great difficulty, since the expressions for the lower 6¡,x’s in terms of the Bi,«’s are not difficult to evaluate), one finds that the lower terms agree with those of the equations z
=
f
=
(20a)
ye~°l'alv
(20b)
in which X
(21a)
V
X
(21b)
V
Inserting these values of z and f in 0o,o, and expressing this function as an inverse power series of v, one obtains, with equation 17,
P
=
kT(y +
—
(?i,o
—
6?o,i
+
(? , )
(22)
which is the virial expression for the pressure as an inverse power series in the volume. This method of using the logarithm of the largest term in QT for In Q, has certain advantages. It expresses the thermodynamic properties of the system in terms of the configuration integral over the equilibrium distribution of the molecules in space, and determines this equilibrium distribution. The numbers m¡,x of clusters of l and molecules of a and a, respectively, in the equilibrium distribution are given by TO;,X
=
JVvb|,xZ'fx
(23)
STATISTICAL
MECHANICS
OF CONDENSING SYSTEMS
77
that the individual terms Ivbi^z1^ and \vbi,\zof vg,t0 and vgo.i are the numbers of molecules of types a and a, respectively, in clusters of types l and divided by the total number of molecules in the system. The method, however, is scarcely convincing mathematically. In the first place, the author is not quite certain that the total number of terms in equation 5 is sufficiently small to permit proof that the method is justifiable even if all the terms are positive, as can be done for a one-component system. Secondly, the method is certainly unjustifiable if some of the fei.x’s are negative, which will be the case at sufficiently high temperatures. Thirdly, equations 20 to 22 have not been generally proved, but it has only been shown that as many terms in the power series development of them are correct as have been specifically calculated. Since the difficulty of calculation increases rapidly with the order of the terms, this method is not practical for terms even as low as the order v~*. Fourthly, the values of v, z, and f for which g¡ + g0,i diverges, and for which the gas starts to condense, are not readily calculable from the equations. We shall therefore resort in the next section to a more abstract mathematical method, based on that of Bom and Fuchs (2), by which the equations of this section are rigorously proved, and which shows the conditions for condensation. so
c¡
III.
THE EXACT MATHEMATICAL
TREATMENT
A rigorous proof of the equations for the system may be developed with use of four purely mathematical theorems which are proved in appendices A, B, C, and D. These theorems are as follows: Theorem A: The function defined as the
H(vb,r,P)
=
t±Q^\A’Vb)rY L·i ! -o
,-
(24)
is equal to
H(vb,r, where the quantities
z and:
f
1
P)
=
1
are now r
=
p
=
—
0(0i,o
+
0o,l)
(25)
defined by ze-”50'0
(26a)
(26b)
Theorem B: The solution to the combinatory problem of finding the coefficients of the powers of irreducible integrals B*,„ in b¡,\ is such that 61,x is given by equation B-23. Theorem C: From equation B-23 the sums gm,^(6¡,x, z, f) may be expressed in terms of the sums Gm,„ of the parameters B*,« and variables y and
JOSEPH E. MAYER
defined as functions of z and f by equations 20a and 20b, with the equations C-10, C-12, C-13a, and C-13b. The sums gm,, and 0„,µ are defined by the equations 10a and 10b. The functions defined as
W
=
y
fi
=
V
-
—
w
