J . Phys. Chem. 1988, 92, 1332-1337
1332
and experimental Gibbs free energies agree to within 1.5 kJ mol-' (1.1 kl mol-l is due to a different AGP298)up to 100 OC. At higher temperatures, AI3+(aq) becomes less stable relative to the hydrolyzed species, and values for AGf0AAl3+,aq)cannot be obtained accurately from any of the solubility data reported to date. Values for AGfOT(A13+,aq)derived from this calculation are also included in Table X. We think that the extrapolated Gibbs energies for A13+(aq) and Al(OH),-(aq) are the best values now available.
We have calculated the Gibbs free energies for Al(0H)Jaq) from 0 to 300 OC using these revised values and have listed these new values in Table IX. We also show the curve associated with our regression of the equation of type (44) to the direct solubility measurements on boehmite in Figure 8. We have used data from the same sources cited previously and Tables VI1 and VI11 (except for the revised Gibbs free energy and entropy for Al(OH) 0, c = @p,93’2p is the pressure, @ = (kT)-l, k is the Boltzman constant, T i s the temperature, p is the density ( N + l)/L, B(R-ma) is the Heaviside step function, and a is defined by the hard-core potential. The Laplace transforms in the integrands are Q / ( s ) = JmG(l,r) exp[-sr
Q(c) = JmG(r) exp[-cr
- &(r)] d r - Pd(r)] d r
(10) (1 1)
2
4 R
0
2
4
Figure 1. Space-dependent potentials @ (dashed lines) and spin-dependent potentials J (solid lines) studied in this work (a) the square well potentials, eq 17 and 18; (b) the hard-core plus Yukawa potentials, eq 19 and 20. The area under the rectangle is equal to the area under the exponential curves for both @ and J .
where I is the length per particle. We consider two particular classes of the interaction potential, displayed in Figure 1. The first potential (Figure la) is expressed in terms of square-well potentials d(R) =
G(1,r) and G(r) are defined by eq 8 with no explicit reference to their arguments. None of the infinite sums in eq 9 introduces any divergence; the sum over m is finite due to the step function, and the sum over I converges due to the limiting behavior of the Bessel functions in the integrand. The inverse Laplace transform in eq 9 may be written as a contour integral when the interactions correspond to a hard-core finite-ranged potential. Integration over the spin variables in eq 9 yields the spinless pair correlation function, g(2)(R)
a,
R< 1
= y2Ah, 1 IR I 2
=0, R > 2 J(R) =
a,
R
(17)
2
(18)
and the second potential (Figure lb) uses hard-core Yukawa potentials,” where
+(R) = -T-e-B(R-S) A R
L denotes the Laplace transform and
Equation 12 is a genralization of the result obtained9,l0for the problem with no spin degrees of freedom. We have also derived eq 12 by using the Chapman-Kolmogorov equation for a stationary Markov process96and thinking of p(r) given by eq 13 as the probability of a walker along a chain of stepping at r. The mapping of “a pair of particles on the chain” and “a step at position rn was used in the spinless problem.1° Due to the spin coupling, no such interpretation can be established for the pair distribution function given by ,eq 9. For nearest-neighbor particles, the pair distribution function g(2)(R;u.u’) given by eq 9 becomes
where R is the distance between two contiguous particles with spins u and 6’. Further integration over the spin variables yields 1 1 gnn(R) = ex~[-cR - Pd(R)IG(R) (15) Equation 15 is analogous to the result obtained by Gursey’* for spin-independent potentials.
4. Equation of State and Pair Correlation Function The pair correlation function for two contiguous particles in the chain is given by eq 15. The equation of state isI2 (16) (95) Abramowitz, M.;Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1970. (96) van Kampen, N. G. Stochastic Processes in Physics and Chemistry; North Holland: Amsterdam. 1981.
A J(R) = - -e-B(R-s) R
- TP--le-“(R-s)), R 2 s
(19)
R A’ - -e-B’(R-s) R
, R L s
(20)
and +(R) = m = J(R) for R < s. The constants A, A’, B, and B’can be suitably chosen to reproduce pair potentials for different spin states. The quantities T and T‘are equal to I/2S(S + 1) u(u l), and we consider several different values of the total spin S and u = 1/2.97 We use the value h = 0.596 347 to make the areas under the exponential curve and the square in Figure 1 equal to each other. We have computed the equation of state for several choices of the potentials, detailed below.
+
5. Results and Discussion We have calculated an expression for the pair correlation function as a function of the separation and internal (spin) degrees of freedom of two particles in a chain. When all the spin degrees of freedom have been integrated out, the pair correlation function is given in terms of the correlation function for nearest-neighbor particles. This result generalizes the result which Sells, Harris, and Guthlo proved for potentials with only space dependence. A mapping between a Markov chain and the one-dimensional arrangement of particles has been used. An analogous result has not been obtained for the spin-dependent pair correlation due to the spin-spin coupling. The pair correlation function calculated by Salsburg, Kirkwood, and Zwanzig9 can be obtained by integrating out all the spin degrees of freedom in our expression and considering space-dependent interactions only. We obtain Gursey’s expression12 when we further reduce the pair correlation to nearest-neighbor particles. Silver5 calculated the equation of state by taking the pressure derivative of the chemical potential obtained from the solution of the integral equation analogous to the transfer matrix eigenvalue problem for king spins.4 Such equation of state is in agreement (97) Pathria, R. K. Sfatistical Mechanics; Pergamon Oxford, U.K., 1972.
Juands i Timoneda and Haymet
1336 The Journal of Physical Chemistry, Vol. 92, No. 5, 1988
-
56 7 0
;I------0 l 40
u -_
- - - - _ _ _ _- - - - _ _ .
05 15 25 35 45 5 5 1 / P
Figure 4. Equation of state P / k T for the same potentials as Figure 3, for the temperature T" = 0.25. 70
1
i
1 4 -
!..;=
,
-
00
05
1
---__
15 25 35 45 5 5 1/ P
Figure 5. Equation of state P / k T for the same potentials as Figure 3, for the temperature T" = 0.1.
Figure 3. Equation of state P / k T for T" = 1, for the same spin-dependent potential as Figure 2 (solid line), for the spin-independent hard-core attractive (dash) and repulsive (dot dash) potentials, and hard particles (dotted). The parameters are the same as those in Figure 2. (Note that the dotted and solid lines are almost superposed.) with the equation of state that we calculate in the preceding section. Tanaka, Morita, and Hiroike14 studied the one-dimensional lattice gas equivalent to an Ising chain with nearest-neighbor and next-nearest-neighbor interactions. Katsura and Tago13 studied a chain of particles with space-dependent interactions9 as square-well potentials. W e have computed the equation of state in the preceding section for several potentials (see Figures 1-6). The equations of state computed in previous ~ o r k ' ~and , ' ~our equations of state for attractive and space- and spin-dependent potentials show the common trend of a sharp change in the slope as the reduced temperature, P,decreases. This behavior was noted by Gtirsey12 who suggested its interpretation as the onedimensional analogue of a change of phase, even though the partition function is analytic. We have investigated in detail several spin- and spacedependent potentials. The first is (A) the spin-dependent Yukawa potential defined by eq 19 and 20. The equation of state for this potential is calculated from eq 16 and shown in Figure 2, which displays the teduced pressure p / k T as a function of the volume per particle p-'. From top to bottom the curves in Figure 2 range from high to low temperatures. As expected, at high temperature the spin dependence of the potential has no effect on the equation of state, since the properties of the system are dominated by the repulsive, 'hard-core" forces. In Figures 3, 4, and 5 we display the equations of state for the spin-dependent Yukawa potential (potential A) at the reduced temperatures P = 1,0.25, and 0.1, respectively. In these figures we compare the spin-dependent potential A with three other potentials which are purely space-depeddent: (B) the usual hard-particle potential (dotted line); (C) the usual hard-core potential plus attractive Yukawa tail (dashed line); (D) the hard-core potential plus a purely repulsive Yukawa tail (dot-dash line). Outside the hard-core region, the last two potentials may be written 4(R) = f(A/R) exp[-B(R-s)], r2s where the plus sign holds for the repulsive tail (D), and the minus sign for attractive tail (C). These are, of course, the limits of the
05
j
00
05
15 25
3 5 45 5 5
1 / P
Figure 6. Equation of state P / k T as a function of the inverse density p-l from eq 16 for the spin-dependent square-well potential (eq 17 and 18). From left to right the curves correspond to the temperatures T" = 0.02, 0.1, 0.25, 0.5, and 1.0. spin-dependent potential A; for all pairs of particles, potential C is more attractive than the spin-dependent potential A, and potential D is more repulsive. Figures 3, 4, and 5 show that as the temperature is lowered the spin dependence of the potential plays an increasingly important role. The attractive forces between particles with antiparallel spins lower the pressure of the system (but of course not as much as the spin-independent potential C which has attractive forces between all particles)?* The sharp changes in slope which are found a t low temperatures (Figures 4 and 5) for attractive potentials (A and C) occur a t higher densities when space dependence is neglected and the spin coupling is pair-independent,14 and a t lower densities when the potential is spin-independent." We have also investigated a second type of spin-dependent potential, based on square well potentials. This potential is defined by eq 17 and 18, and plotted in Figure la. The equation of state for this spin-dependent potential is shown in Figure 6,for the reduced temperatures 1.0,0.5,0.25,0.1, and 0.02. This potential has a similar equation of state as the spin-dependent hard-core Yukawa potential, but has the advantage that the nearest-neighbor correlation function g,(R,y), where R = Ixl - x21 and cos y = ul-u2,can be calculated in closed form from eq 14. In Figures 7 and 8 we display the nearest-neighbor correlation function as ( 9 8 ) A s a check, we calculated the equation of state by thermodynamic perturbahon theory. The results agree with those shown by the curves for attractive and repulsive potentials at high temperature (Figure 3). Discrepancies arise at lower temperature due to the breakdown of thermodynamic perturbation theory in this region, as should be expected.
Spin-Dependent Force Model in One Dimension
benchmark for more approximate calculations on spin- and space-dependent potentials in three dimensions.
0.03
d
Acknowledgment. This research was supported, in part, by N I H Grant GM34668. A.D.J.H. thanks Marc Kramer for many helpful discussions in the early stages of this work.
,0.02 h M aLl.
9 0.01 0.00
The Journal of Physical Chemistry, Vol. 92, No. 5, 1988 1337
u
0.0
2.0
0.0
2.0
4.0
R Figure 7. Nearest-neighbor correlation function pg,(R,y), where cos y = u1w2as a function of distance and spin projection y given by eq 14 with the potentials (eq 17 and 18) at the reduced temperatures (a) Tc = 1 and (b) 7' = 0.5. The curves in the region 1 I R I2 correspond from , 0. top to bottom to y = H, 4nJ5, 3 ~ 1 5 2, ~ 1 5 HIS,
Appendix The derivation of eq 9 for the space- and spin-dependent pair correlation function proceeds as follows. Let us assume that the two selected spins in the chain are labeled k and k + p, p > 1. (If p = 1, the spin integration is straightforward because the spins are nearest neighbors.) Spin integration for all the particles i C k and i > k p in the chain yields factors
+
Gl J + l e-86,1+1
(A. 1)
The spin integration wHich remains is ldak+l
... dak+p-l
exd8[Jk,k+lak'ak+l + ... + Jk+p-l,k+p'Jk+p-l*~k+pll
(A.2)
Each one of the integrals Jd.,
exP(8[J/-IJu/-PJ/ + J/J+l~/.~/+lll
('4.3)
in eq A2 can be performed by using the expansionsg5 00
20
00
20
40
R
Figure 8. Same as Figure 7, for the reduced temperatures (a) P = 0.25 and (b) T* = 0.1. The curves in the region 1 IR I2 correspond from top to bottom to y = H, 4 ~ 1 5 2, ~ 1 5and , HIS.
a function of distance R for several fixed spin orientations y and the reduced temperatures TI = 1.0 (Figure 7a), TI = 0.5 (Figure 7b), TI = 0.25 (Figure 8a), and TI = 0.10 (Figure 8b). These nearest-neighbor correlation functions are of course independent of the spin orientation for distances larger than the range of the spin dependent potential (that is, for R > 2). For all temperatures the antiparallel configuration is favored, and this fact dominates the structure of the system at moderate and low temperatures. The perfectly parallel configuration becomes improbable rapidly as the temperature is lowered. At high temperatures, the probability of finding a nearest-neighbor pair separated by a distance larger than the range of the potential is greater than the probability of being bound, but this is reversed in a narrow range of lower temperatures. These analytic studies are intended as an extension of much earlier results on spin-independent potentials and also as a
m
exp(z cos 6') = Zo(z)
+ 2cZn(2) cos (ne) n= 1
(A.4)
for XY and Heisenberg spins, respectively. The orthogonality relationships that affect the angular parts allow us to cast the integrals (A.2) in terms of the angle y between b k and bk+p as given in eq 6. Equation 9 can now be derived by using the result that when the functions f and h satisfy the convolution product
their Laplace transforms satisfy
('4.7) If we identifyfwith Ge-@@ and h with G(l)e-@@ given in eq 8 and we proceed in an analogous manner as in the work by Salsburg, Zwanzig, and Kirkwood? eq 9 is obtained. [~y>lq+'[m)l"
