J. Phys. Chem. 1993,97, 8993-9001
8993
Probability Distribution in the Cell Theory of an Interacting Lattice Gas: Application to 129Xe NMR of Xenon in Zeolites T. T. P. Cheung Phillips Research Center, Phillips Petroleum Company, Bartlesville, Oklahoma 74004 Received: February 25, 1993; In Final Form: May 28, I993
A cell theory of an interacting lattice gas is derived from the basic principle of statistical mechanics that the equilibrium distribution of Nparticles among Mcells, when both N a n d M a r e very large, is given by the most probable distribution, with the constraints that the total number of particles and the average energy remain constant. The theory provides a direct way to calculate the probability distribution of the gas particles among the cells. A gas-liquid transition is observed at low temperatures where the probability distribution develops a bimodal shape indicating the separation of the cells into two populations, with one population having higher particle densities than the other. The derivatives of the Helmholtz free energy per cell with respect to the average number of particles per cell are examined. When the interaction energy between particles within a cell is proportional to the square of the number of particles in the cell, the smallest possible second derivative of the free energy is 4kT/J2, where J is the maximum number of particles that can be accommodated by the cell and T i s the absolute temperature. The probability distribution of the interacting lattice gas is applied to the adsorption of xenon in zeolites. In order to describe lZ9Xenuclear magnetic resonance data reported in the literature for xenon trapped in the a-cages of the NaA zeolite, it is necessary that, in addition to the attractive interactions between the particles, the repulsive interactions should also be included when the xenon atoms begin to fill the a-cages of the zeolite.
I. Introduction In this paper, we investigate the probability distribution in the cell theory of an interacting lattice gas. The system consists of a number of particles distributed among a fixed number of cells. Each cell can accommodatea finite number of particles, and the particles interact only with those within the same cell. The goal is to determine the probability that a cell is occupied by a certain number of particles and the distribution of this probability as a function of theaveragenumberofparticlespercell,i.e.,theaverage occupation number of the cells. If one allows the maximum occupation number to become infinite, one recovers the grand canonical ensemble. However, our interest here is to limit the maximum occupation number to a finite number, which may be smaller than 10. The reason is that the cell theory in this form can be used to describe the adsorption of gases in zeolitic materials. Zeolites such as type Y and A comprise cells or cages of uniform sizes in which adsorbed gas molecules are trapped. A direct way to study the distribution of adsorbed molecules in zeolites is by lZ9Xe nuclear magnetic resonance (NMR) of the adsorbed The generalconclusion is that while the hypergeometric distribution6 derived from the noninteracting lattice gas provides a qualitative description of the xenon distribution in A zeolites at high temperatures,discrepanciesat thelows and high4J loadings of xenon are evident. We shall show that the discrepancies disappear when interactions between particles are included in the lattice gas. At low loadings of xenon, attractive interactions alone suffice. However, at high xenon loadings, one needs to include the repulsive part of the interaction as well. We have also reported7.*the 1z9XeNMR of xenon adsorbed in Y zeolites at temperatures below the freezing temperature of the bulk xenon gas. At such low temperatures, interactions between the xenon atoms play an important role in determining the distribution among the zeolite cages and must be included in any credible theory. Statistical analysis of a system of particles in terms of probability distributions of the particles has been described by Guemez et aL9 While it is straightforward to derive the distribution for a system of noninteracting lattice gas using elementary probability argument,6it is not evident how one extends
such a treatment to situations in which interactions between particles, and therefore the temperature effects on the particle distribution, are important. A case in point is theself-condensation of the gas molecules at low temperatures. Guemez et al. circumvent part of the difficultiesby using conditionalprobabilities whose analytical form is postulated “a prior” and the interaction between particles is approximated by a mean field. Our analysis here differs from theirs in that we derive the probability distribution directly from the basic postulate in statistical mechanics that the equilibrium distribution is given by the most probable distribution. The energy term and therefore the temperature dependence are incorporated naturally in the basic equations. No specific assumption is made for the interactions between particles other than that they are pairwise between two nearest-neighbor particles within the same cell. It should be emphasized that the statistical mechanics approach is completely general and not limited to the lattice gas. In section 11, we derive thegeneral equationsfor the probability distribution in the cell theory of gases. The derivation follows standard methods in statistical mechanics. Its main purpose is to establish the necessary equations and notations for the applications to the interacting lattice gas, which is treated in details in section IV. The cell theory predicts the Occurrence of a gas-liquid phase transition at low temperatures, as indicated by the bimodal distribution in the probability. Comparison of the results of the interacting lattice gas with the 129XeNMR of xenon in zeolites will be made. In section 111, we briefly review several models of gases in which the probability distribution can be solved analytically.
II. Cell Theory of Interacting Particles Weconsider a system of Nidentical particles distributed among Midentical cells. Both Nand Mare considered to be infinitively large. Each cell can hold up to a maximum of Jparticles. There are go’) ways by which j distinct particles can be accommodated within a cell. (The fact that the particles are actually indistinguishable and that permutations among them should not be counted will be dealt with explicitly. See the text before eq 2.4.)
0022-365419312097-8993$04.00/0 0 1993 American Chemical Society
8994
The Journal of Physical Chemistry, Vol. 97, No. 35, 1993
The particles interact only with those within the same cell. The system is in equilibrium with a heat reservoir with which it is in thermal contact, but there is no exchange of particles. Let r (no,nl,...,nJ) be the number of ways of distributing N identical particles in M cells with cells containing zero particle, nl cells containing one particle, and so on. The equilibrium distribution is then given by the set (no,nl,...,nJ), which leads to a maximum in r(no,nl,...,nJ), while at the same time satisfies the conditions that J
&nj = M
Cheung particle interaction is not modified by the particlewall interaction. Consider the situation of distributing two particles between two cells. The total particlewall interaction energy remains the same whether we have one particle in each cell or two particles in one cell while the other is empty. Therefore, the particle-wall interaction is responsible for the adsorption of the particles into the cells, but it does not influence the distribution of the particles after the adsorption. With the method of Lagrange multipliers and introducing the parameters cy, 8, and p, one obtains 6nj(ln[g(j)/j!] - ln(nj) and the equilibrium nj is
'I
J
E n j=N
(2.6)
+
i=o
J
c E j n j = constant
+ cy - j3E) + j p ) = 0
(2.3)
j*
Ej is the energy of a cell with j particles. I'(no,nl, ...,nJ) can be calculated by noting that there are M!ways to arrange M cells, but for each j, nJ permutations among the cells with j particles and j! permutations among j identical particles within the cell should not be counted. However, for each cell with j particles,
there are additionalgQ) degrees of freedom that must be included. Therefore, one has
(2.7) nj = [g(j)/j!] exp(cy j p - @E;) It should be pointed out that g o ) does not include contributions from the internal degrees of freedom of individual particle. If the internal degrees of freedom were included, they would give rise to a term proportional to bi, where b is the number of internal degrees of freedom of each particle. Since In(&) = j ln(b), the contribution of such a term is already taken into account by the parameter p in eq 2.6. This is precisely the same reason why we have E*j instead of Ej in eq 2.6. The parameter a is determined by condition 2.1. It can be eliminated by using a normalized probability PQ;(j)),which is defined as the fraction of the cells containing j particles given that the average number of particles per cell (j) is N / M ,
P ( j ; ( j ) )= nj/M
To find the maximum of r(no,nl,...,nJ) which satisfies conditions 2.1-2.3 is equivalent to have
[gci)/j!l exp(ip - BE*,) (2.8)
In other words, P ( j ; ( j ) )is the probability that a cell chosen in random will contain j particles if the average number of particles per cell (j)is N / M . According to condition 2.2, J
=O
j-0
(2.5)
with 6nj satisfying
(2.9) (2.1')
with Q defined as
J
G 6 n j=o
(2.2')
j=o
Equation 2.9 determines the parameter p. It is clear that both and P ( j ; ( j ) )are functions of (j).By defining
p
J
cE*,Gnj= 0
s * =~ (@T)-' ln[g(j)/j!]
(2.3')
I=O
To obtain the first equality in eq 2.5, we have used the Stirling's approximation for n) Since nis are very large numbers, the resulting error of the approximationis negligible. E*j in condition 2.3' represents the part of E, which is not proportional to j. The reason of replacing E, by E*) is that the part of E, which is proportional to j-such as the internal energies of the particles and energies due to interactions between the particles and the wall of the cell-already satisfies condition 2.2'. It is redundant to include it in condition 2.3'. The fact that the interactions between the particles and the cell wall do not affect the distribution of the particles is true in general so long as the particle-wall and particle-particle interactions are independent of each other. That is, the p a r t i c b
(2.1 1)
and
A*j = E*j - TS*,
(2.12)
eq 2.8 can be rewritten as
T is the absolute temperature. Although A*] and S*j have respectively the forms of the Helmholtzfree energy and the entropy of a cell with j particles, they are not the true free energy and entropy because they include only the energy due to interactions between particles (see thecomment after eq 2.3') and the degrees
129XeNMR of Xenon in Zeolites
The Journal of Physical Chemistry, Vol. 97, No. 35, 1993 8995
of freedom resulting from the arrangement of the particles within the cell. We may refer to them as the free energy of interactions and cellular entropy. Of course, terminology like this is meaningful only when J is large. When J is infinite, eq 2.13 should become that of the grand canonicalensemble. Therefore, onehasB= l/kT,where kis theBoltzmannconstant. Tocalculate P(j;(j)), one needs to specify g(j) and E*j, Le., the gas model one uses. Then one determines the parameter p from eq 2.9. Since the entropy per cell S of the system is given by
S=
words,
go')= j! One obtains from eq 2.10
(3.5)
+ d)'
(3.6)
Q = (1 and from eq 2.9 p = ln(j)
k In r(no,nl,...,nJ)
J! ( J - j ) ! j!
- ln(J- (j))
(3.7)
which yields
M
one finds, by substituting eq 2.7 into eq 2.4, the Helmholtz free energy per cell A as A/kT= (j)p-lnQ
When plotted as a function of j , eq 3.8 shows only a single peak with a maximum located near (j). The second moment of the distribution can be obtained by substituting eq 3.7 into eq 2.16
It follows from eq 2.9 that p=-- 1 aA k T a(j>
(2.14)
We define the ith moment of the probability distribution as I
Comparing eq 3.4 and eq 3.9, one sees that the width of the distribution of the noninteracting lattice gas is always smaller than that of the ideal pointlike gas. The same problem can also be solved by probability theory which yields a hypergeometric distribution6
The second moment w2 expresses the dispersion of the distribution and, according to eq 2.9, is related to p by the equation (2.16) In the following, we apply results derived above to several models of lattice gas. For some models, p can be evaluated explicitly, and therefore, analytic solutions for P(j;(j)) are available. Comparison with known results in the literature will be made. We shall also examine models in which p can only be determined numerically.
III. Exactly Solvable Gas Models i. Ideal Pointlike Gas: E*] According to eq 2.10, one finds
=:
0, J
-
w,
and go') = 1.
Equation 3.10 reduces to eq 3.8 in the limit of N >> J and M >> J. iii. Interacting Lattice Gas with J = 2. The treatment of an interacting lattice gas with J = 2 is similarto that of the adsorption of gas molecules on a surface where the surface sites can accommodate a pair of mo1ecules.l0 Therefore, our discussion here will be brief. Let the interaction energy between a pair of particles in a cell be -e. Since we are mainly interested in the attractive interaction, e 1 0. Then using the fact that go')/j! equals 1,2, and 1 respectively forj = 0, 1, and 2, one obtains from eq 2.10
Q=1
+ 2 d + eZr+@'
(3.1 1)
and from eq 2.9
= exp(d)
(3.1)
Therefore, from eq 2.9,
P =WU))
(3.2)
which is a quadratic equation of W. Since w L 0, only the positive root
from eq 2.8 PG;w) = (W'/j!) exp(-Ci)) and from eq 2.16 w2=
ci)
(3.3) (3.4)
Equation 3.3 is the well-known Poisson distribution. One can obtainthis distribution directly from elementary probability theory which gives the result in terms of a binomial distribution.6 The binomial distribution reduces to eq 3.3 in the limit of N - + a, M =, and N >> J. ii. Noninteracting Lattice Gas: E*j = 0, go') = J!/(J-j)! In the lattice gas model, the particles occupy mutually exclusive lattice sites in the cell and there are only J lattice sites available in each cell. Taking into account that permutations among unoccupied sites should not be counted, there are J!/(J-j)! ways of distributingj distinct particles among J lattice sites. In other
-
is acceptable. For (j) = 1, one has PO'=O;l) = P0'=2;1)
= {2[1
+ exp(-&/2)]]-'
(3.14)
exP(-W) (3.15) 1 + exp(+c/2) Equations 3.14and 3.15 show that,asthetemperatureislowered, P(j=O;l) and P(j=2;1) grow at the expense of P(j=l;l). The same is true for an arbitrary (j). In Figure 1, we depict the distributiortas a function of temperature for the case of (j) = 0.5. At tdperatures below c/[k2 ln(2)], eq 3.15 indicates that the probability of single occupancy is smaller than those of zero and double occupancy. That is, below this temperature, the particles are more likely to form dimers. The parameter p as a Po'=1;1) =
8996 The Journal of Physical Chemistry, Vol. 97, No. 35, 1993
I
1.0
J=2 (j)=0.5 0.6
v .-. _
I
I
--
J
0.2
P(2;0.5) ___,,,,,..,...........,”’ , ( , . . . . . . . . ,’ . , . . . . .
0.oc 0
1
2
---_-----__ 1
4
3
5
E/kT Figure 1. P ( j ; ( i ) ) as a function of c/kTfor an interactinglattice gas with a maximum occupancy of two particles per cell and the average number of particles per cell (i)= 0.5. 4 is the interaction energy between two particles.
J=2
E/kT = 100
-481
-531 0.0
/t
0.5
1.0
1.5
1
2.0
p( (i)) as a function of the average number of particles per cell an interacting lattice gas with a maximum occupancy of two particlesper cell and r / k T = 100. The dashed line representsthe smallest slope possible for p ( ( i ) ) according to eq 4.13.
Figure 2. (i)for
functionof (i)isplottedin Figure 2 for e/kT= 100. Thesmallest slope possible for p is represented by the dashed line. It occurs when (i)= 1 and T+ 0, and has a value of unity. The discussion of the slope of p will be given later.
IV. Interacting Lattice Cas with J > 2 To describe phenomena like phase transitions and adsorption of gases into porous materials, one needs to include interactions between particles. For an interacting lattice gas, one no longer has analytic solutionsfor eq 2.9 and P ( j ; ( i ) )except for the simple case of J = 2 described earlier. However, solutions for eq 2.9 can be obtained easily by numerical methods. We shall divide our discussion into two parts: one with attractive interactions only and the other including also repulsive interactions when the cells are nearly filled. i. Luttice Gas with Attractive Interaction Only. Since we consider the interaction between the particles to be pairwise and attractive, the interaction energy E*] can be written as with { 2 0. The energy parameter t is determined from the approximation that only the nearest-neighbor interaction is important and otherscan be neglected. When a cell is completely full,
E*, = -[Jz(J)/~]E where-e, with E 10,is the interaction energy between twonearestneighbor particles, and z(J) is the average number of nearestneighbor sites around a latticesite within a cell. z(J) is a function of J but becomes independent of J when J is large. Comparing these two expressions of E*J, one finds
Equation 4.1 with {given by the above expression is the same as
Cheung that of the Bragg-Williams approximationllJ2 except that z(J) is not fixed but is a function of J. Therefore, short-range correlations among particles are implicitly neglected.’2 This also implies that the degrees of freedom g0)of the interacting lattice gas will remain the same as that of the noninteracting one given by eq 3.5. While the Bragg-Williams approximation yields only qualitative results for an interacting lattice gas when J QJ, it should provide a more accurate description when J is small and z(J) is of the order of J because now each particle in the cell can interact with almost all the other particles within the same cell. Consequently,short-range correlation becomes irrelevant. Notice that there is a term in eq 4.1 proportional t o j . As mentioned earlier, such a term can be neglected in the determination of P ( j ; ( j ) ) ,and a simpler expression
-
E*j =
-u2
(4.1’)
can be used. For a given J, eq 2.9 is a Jth-order polynomial of @, and p is given by the real positive root of the polynomial. For large J, solutions to the polynomial can only be obtained by numerical methods. Before we proceed with the calculations, we should point out one special property of this model of interacting lattice gas. Namely,
=--- z(J)J e (4.3) (J- 1) kT when E*] is given by eq 4.1’ and g(j)/j! is an even function of (j-J/2), Le., go’)/j! = g(J--j)/(J-A! (4.4) The condition in eq 4.4 is clearly satisfied by the lattice gas (see eq 3.5). To prove eq 4.3, we first realize that p is a function of (i)and express (i)and J - (i)in terms of p ( ( i ) ) and p(J-(i)) using eq 2.9
The denominator on the right side of eq 4.6 can be casted into the form J
J
where C= exp[Jcc(J-u))] exp( {>/kO. The first equalityfollows from the fact that the summation of j from 0 to J is the same as that from J t o 0. Equation 4.4 has been used to obtain the second equality. Similarly, one can write the numerator on the right side of eq 4.6 as
lZ9XeNMR of Xenon in Zeolites
The Journal of Physical Chemistry, Vol. 97, No. 35, 1993 8997
$
=C
(J - j)[g(j)/j!] exp[-jp(J-(j))]
exp(-2fJj/kT
+
(4.9) Comparing eqs 4.9 and 4.5, one obtains eq 4.3. Three corollaries follow from eq 4.3. First, when (j) = J/2, p((j)=J/Z) = -{J/kT
=--- z(4J e (4.10) 2(J- 1) k T Therefore, for the special case of average half-filled cells, one has an analytic solution for P ( j ; ( j ) ) . For a noninteracting lattice gas, c = 0 and p((j)=J/2) = 0, which is in agreement with eq 3.7. The second corollary is P ( j ; ( j ) ) = P(J-j;J-(j)) (4.1 1) One obtains eq 4.1 1 by first writing the expression for P(J-j;J(j)) using eq 2.8, and then substituting in eq 4.1’ for E*,, eq 4.3 and eq 4.4. Equation 4.11 indicates that one needs only to determine P ( j ; ( j ) )for (j) IJ/2; the probabilities of the higher (j)’s can be obtained from the lower ones by reflection a t j = J/2. The third corollary, which concerns the moments of the distribution of P ( j ; ( j ) )as a function of (j), states that w,(ci)) = ( - 1 h V -
W)
(4.12)
It can be proven by first writing the expression for wi(J - (j)) using eq 2.15 and then making use of eq 4.1 1. Equation 4.12 indicates that, when expressed as a function of (j),the moments have a center of symmetry at (j) = J/2. w z ( ( j ) ) , which is a measure of the width of the distribution of P ( j ; ( j ) ) ,is an even function of ((j) - J/2), while w3((j)), which describes the asymmetry of the distribution, is an odd function. Since wp((j)=J/2) = 0, P(j;(j)=J/2) must be an even function of (j - J/2). The same conclusion can be arrived from eq 4.1 1 when (j) = J/2. According to eq 4.12, w z ( ( j ) ) has an extremum at (j) = J/2. One can show that the extremum is a maximum by the following argument. From the definition of the moments in eq 2.15, it follows that
--
awZ(Ci)) --w3((i))
dCi)
‘/*(O - J/2)2
+
+ {j2/kT) =
exp [(N k T )0’ + r k T/ 2 33 exp [-(rk T/ 2 0 The first exponential term on the right side of the above equation is a reciprocal of a Gaussian. It has a minimum at j = -bkT/2f and diverges symmetricallyon both sides away from the minimum.
J/2)2 = (J/2)2
Therefore, the smallest slope possible for p ( (j))is 4/52 or
wz(Ci))
Sincew3((j)=J/2) = O,theextremumisamaximumifw~((j)=J/ 2-K) > 0, where K is a small, positive number. To determine the sign of w3( (j)= J / ~ - K ) we , consider the exponential term in the numerator of eq 2.8, which can be written as expup
At (j) = J/2, one finds, by using eq 4.10, that the minimum of the inverted Gaussian is located at j = J/2. According to eq 2.1 6, p is an increasing function of (j). Therefore, p( (j)=J/ 2-K) is a negative number smallerthan p( (j)=J/2), which implies that the location of the minimum of the inverted Gaussian for (j) = J / 2 - K is larger than J/2. Since g(j)/j!is symmetric about J/2, one sees that P(j;(j)= J / ~ - K is ) asymmetric and skews toward largerj’s. Then it follows from eq 2.15 that w3( (j)= J / ~ - Kmust ) be larger than zero. Since at (j) = 0 and (j) = J, w2 = 0, the above discussion suggeststhat as (j)increasesfrom zero, w2 becomes increasingly larger. Then it goes through a maximum at (j) = J/2 before it decreases to zero as (j)increases to J. Equation 4.3,4.11, and 4.12 can be used to check theconsistency of the numerical solutions for r ( (j))and P ( j ; ( j ) ) . In Figures 3 and 4, we depict the results of P ( j ; ( j ) )as a function of j for variousvaluesof (j)ande/kTfor J = lOand J =100,respectively. We have set z(10) = 5 and z(100) = 10. Only results for (j) less than or equal to J / 2 are shown because of eq 4.11, For small c/kT, the probability P ( j ; ( j ) )as a function of j has only a single maximum. However, as c/kT increases, the distribution becomes broad and bimodal. This signifies the beginning of a gas-liquid phase transition as the system is dominated by two populations of cells, one with higher particle densities than the other. This transition can be followed closely by examining P(j;(j)=J/2), the probability distribution at the half-filled condition, since the analytic solution is available. At (j) = J/2, two maxima begin to appear in the distribution when c/kT is larger than 0.67 and 0.40 respectively for J = 10 and J = 100. The two maxima have the same height and are located symmetricallyon the opposite sides of the minimum at j = J/2, as expected from the fact that P(j;(j)=J/2) is an even function of (j - J/2). The occurrence of the bimodal distribution may have been anticipated from the exact solution of the case of J = 2, where we have shown that at low temperatures the probability of a cell with a single particle is smaller than those with zero and two particles. In Figures 5 and 6, we show p( (j))as a function of (j) for J = 10 and J = 100, respectively. In both cases, the curvature at the center portion of p ( ( j ) ) becomes flatter as c/kT increases. In Figure 6B, we expand the scale of the ordinate in order to show thecurvatureofp((j)) at c/kT= 0.8. Thesmallestslopespossible for p( (j))are represented by the dashed lines in the figures. They are determined as follows: According to eq 2.16, d p ( ( j ) ) / d ( j ) is the smallest when the second moment is the largest. We have shown that, at a given temperature, the second moment w2 has a maximumat (j)= J/2. SinceP(j;(j)=J/2) isaneven function of (j - J/2), the largest w2 is achieved when P(O;(j)=J/2) = P(J;(j)=J/2) = 1/2 and the probabilities of other j ’ s are zero. This occurs as T- 0. In other words, the largest second moment is simply
(4.13) It follows from eq 2.14 that (4.14) and the system is always stable. Let us compare the results of the cell theory with those of the canonicalensemble. In the latter representation,we have a lattice gas consisting of a single cell in thermal equilibriumwith the heat reservoir. J is infinitively large, and j can be treated as a
8998
The Journal of Physical Chemistry, Vol. 97, No. 35, 1993
e/kT = 0.67
E/kT = 0.18 I,
2
I
4 . 6
J
0
T
4.8
2
J
2
4 . 6
3
J
T
(1) =
(j) = 1
8 10
0
2
T
8 10
4.6
8 10
I
0
T
0
(1) = 3
0
2
4 . 6
J
2
8 10
4,8
0
J
P -
8 10
0
2
3
8 10
0
2
4 . 6
3
2
J
8 10
I
8 10
4 . 6
8 10
nr -
T
(1) = 5
=5
4 . 6
4.6
(j) = 3
I
(1)
2
1
IL L I
(1) = 3
0
I
I I,
(j) = 1
8 10
€/kT = 0.90
a/kT = 0.79 I I,
(j) = 1
0 1
Cheung
0
J
(1)-5
2
4 . 6
3
8 10
J=10 Z r 5 Figure 3. Probability P ( i ; ( j ) )as a function o f j for various values of c/kTand (j)for a lattice gas with attractive interactions. The maximum occupancy J is 10 particles per cell and z(J) is 5. e/kT = 0.40
€/kT = 0.10 I I,
'1
€/M= 0.80
e/kT = 0.44 I I,
1
I I,
(j) = 10
0 20 40.60 80100 0 20 40.60 80100 0 20 40.60 80100 0 20 40.60 80100
J
I
(1)
= 30
v W *T.)
pc
j
0 20 40:60 80 100 J
T
:
K=6;
k r o , lk 1: I n 1 IM j 1
0 20 40.60 80 100 0 20 40.60 80 100 0 20 40.60 : : j l =80 ; i 100
z(j):50t
(j) = 50 0 20 40.60 80100 0 20 40.60 80100 0 20 40.60 80100 0 20 40.60 60100
3
3
J=lOO
J
3
2=10
Figure 4. Probability PO;(j))as a function o f j for various values of s/kTand (j)for a lattice gas with attractive interactions. The maximum occupancy J is 100 particles per cell and z(J) is 10.
continuous variable. Then it is known that A*j as a function of
respectively to the liquid and gas phases. The equation
j will develop two local minima when the temperature is below
a critical temperature.*2J3 Between the two minima, a2AS!aj2 becomes negative and the system is unstable. It separates into two subsystems, each of which has a particle density and free energy corresponding to those of one of the minima. In other words, the system would go through a phase separation; the subsystems with higher and lower particle densities correspond
a2A*j/aj2= Q
(4.15)
has two solutions j* = ( J / 2 ) [ 1f ( 1 - 4 k T / c ~ ) ' / ~ ] When rlkT
(4.16)
< 412, dZA*j/aJ2> 0 for all j and there is no real
The Journal of Physical Chemistry, Vol. 97, No. 35, 1993 8999
lz9XeNMR of Xenon in Zeolites
J=lO
z=5
J=lO
1.6
1 l2 10
4
t
W
z = 5
5
t
0 2 6
/.-.-.-
4
A-.-./
2
n 0
Figure 5. p ( ( j ) ) as a function of the average number of particles per cell (j) for a lattice gaswithattractiveinteractions. The maximumoccupancy Jis 10 particles per cell and z(J) is 5 . Open triangles, c / k T = 0.18; solid triangles, c/kT = 0.67; open circles, c / k T = 0.79; solid circles, c / k T = 0.90. The dashed line represents the smallest slope possible for p( (j)) according to eq 4.13.
40
60
0
10
Figure7. Second moment w2( (j))of the probability distributionP(j;(j)) as a function of the average number of particles per cell (j) for a lattice gas with attractive interactions. ThemaximumoccupancyJis 10particles per cell and r(J)is 5. Inverted open triangles,c/ kT = 0,thenoninteracting 1atticegas;opentriangles, a / k T = 0.18;solid triangles, c / k T = 0.67;solid circles, c / k T = 0.79; open circles, kT = 0.90.
equation. One important difference between the canonical ensemble and the cell theory of the interacting lattice gas is that in the cell theory, dZA/a(j)z is always greater than or equal to zero (see eq 4.14) even whenP(j;(j)) hasa bimodaldistribution which signifies the gas-liquid transition. This is not true for the canonical ensemble; the gas-liquid transition occurs only when a2A*j/aj2 becomes negative. The analysis of Guemez et al., which is based on conditional probabilities and a mean field approximation for the interaction between particles, leads to P ( j ; ( j ) ) of the form
n
20
4
and z(J) = 10 is in agreement with eq 4.17, e/kT, = 0.67, for J = 10 and z(J) = 5 , is smaller than that expected from the same
J = 100 z = 10
-5.04 0
2
80
100
(j)
PU;Ci)) = PoU;Ci)) exp[U- C i ) ) z ( a / k r ) l (4.18)
-3.96
1
wherePo(j; (j)) denotes the probabilityof the noninteractinglattice gas given by eq 3.8 or eq 3.10, and a is the mean value of the interaction energy. While eq 4.18 yields particle distributions with features similar to those of Figures 3 and 4, it has a major difficulty that it is not self-consistent. One can observe from Figure 2 of ref 9 that, in general,
n h
*-
v W
3.
J
G P 0 U ; C i ) )exp[U- Ci>)2(a/w1# ci) -4.041
0
20
40
60
80
j=O
c
100
(i\ \.I I
Figure 6. (A) p( (j))as a function of the average number of particles per cell (j) for a lattice gas with attractive interactions. The maximum occupancy~is 100 and z ( is~10. open triangles,€ / k T
= 0.10; solid triangles, t / k T = 0.40; open circles, t / k T = 0.44; solid circles,c/kT=0.80. (B)p(u))asafunctionof (j)plottedinanexpanded scale in the ordinate for c / k T 0.80. The dashed line represents the smallest slope possible for p( (j))according to eq 4.1 3.
solution for eq 4.16. When c/kT > 412, both j + and j - are real, and azA*,/aj2 is negative for j in the range j -
