J . Phys. Chem. 1987, 91, 414-417
414
Heat Capacltles of Rare Gases Adsorbed on Graphlte Franco Battagtia, Dipartimento di Chimica, 11 Universita-Tor
Vergata, 001 73 Roma, Italy
Young Sik Kim,+ Department of Chemistry, University of Rochester, Rochester, New York 14627
and Thomas F. George* Departments of Chemistry and Physics di Astronomy, State University of New York at Buffalo, Buffalo, New York 14260 (Received: August 1 1 , 1986)
The McQuistan-Hock model is used to investigate adsorption for realistic systems. After computation of the partition function in the most convenient ensemble for the problem under consideration, Le., the isothermal-isobaric ensemble, the heat capacity for submonolayer films of Ne, Ar, and Xe on graphite is computed for several coverage values. Heat capacity signatures exhibit maxima at temperature values that are in good agreement with experimental data.
I. Introduction The recent discovery that submonolayer films of adsorbates on some substrates have two-dimensional phases has opened a new field: the study of two-dimensional matter, its phases, and the transitions between them. A common way to realize two-dimensional systems is to physisorb gaseous particles (atoms or simple molecules) onto solid surfaces (substrates). Dependent on particle and substrates properties, the adsorbate condenses as a homogeneous film which might reveal two-dimensional properties. Often the only task the substrate is expected to accomplish is to force the adsorbed particles into a plane, so that the substrate disturbs the adsorbed system as little as possible. In order to investigate experimentally the physical properties of two-dimensional systems, one must have substrates of high specific surface areas such as graphite (and its modifications). The essential task in a thermodynamical investigation of a physisorbed two-dimensional system is the construction of a complete group of adsorption isotherms. In practice, for an accurate determination of the entropy, calorimetric experiments to determine the specific heat signatures are performed. Such experiments offer the possibility of studying low-dimensional order-disorder phenomena such as structural and melting phase transitions. With respect to the order of a melting transition, the situation depends strongly on whether one considers three- or two-dimensional melting. In the former case, there is no doubt that melting is always a discontinuous event at which the system absorbs latent heat from the surroundings. In the two-dimensional case, the question as to whether melting is a discontinuous or continuous process is still open. In the present paper we use the McQuistan-Hock model' to compute heat capacity signatures for two-dimensional systems of rare gases adsorbed on graphite. Experimental studies on these systems have demonstrated the existence of ordering and phase transitionsZand have identified regions of two-dimensional liquid-vapor as well as liquid-solid coexistence, accompanied by a two-dimensional critical point.3 In the next section, after a brief overview of the McQuistan-Hock model, we present our calculations and results. The third and last section is devoted to our conclusions. 11. Theory and Calculations
In a recent paper,' McQuistan and Hock developed an exact solution for the distribution function of q indistinguishable particles on a 2 X N lattice, where N is a positive integer number. The grand canonical partition function is written as 'Present address: Departments of Chemistry and Physics & Astronomy, State University of New York at Buffalo, Buffalo, N Y 14260.
0022-3654/87/2091-0414$01.50/0
where
and z = eh
in which
p
(6)
is the chemical potential of the adsorbed particles, @
= (ken-' with T the absolute temperature and kBthe Boltzmann
constant, V , is the interaction between two nearest-neighbor vacant sites, VI is the interaction between two nearest-neighbor adparticles, and Vois the interaction between a particle and the surface. The number of unique ways q indistinguishable particles can be arranged on a rectangular 2 X N lattice to form n , , occupied nearest-neighbor pairs and n,vacant nearest-neighbor pairs is given by AIN,q,n,,nll] in eq 2 by making use of a 15-term recursion relation. In a subsequent paper,4 Hock and McQuistan computed the adsorption isotherms for what we might call the "rigid" lattice in which there is no interaction between two nearest-neighbor vacant sites: V , 0. They showed that the coverage as a function of the spreading pressure does not exhibit a first-order phase transition. The McQuistan-Hock model is very interesting because of its simplicity, because it takes into account various interaction terms ( Vo, V,, and VII ) and because it is exactly solvable. We will show here that the model is applicable to realistic systems by comparing the critical temperatures obtained from it with the critical tem(1) McQuistan, R. B.; Hock,J. L. J . Math. Phys. 1976, 25, 261. (2) Lander, J. J.; Morrison, J. J. Surf. Sci. 1967, 6, 1. (3) Thomy, A,; Duval, X. J . Chim. Phys. Phys.-Chim. Biol. 1969, 66, 1966; 1970, 67, 286, 1101. (4) Hock,J. L.; McQuistan, R. B. J . Math. Phys. 1985, 26, 2392. (5) Vidali, G.; Cole, M. W.; Klein, J. R. Phys. Rev. B Condens. Mutter 1983, 28, 3064. (6) Rauber, S . ; Klein, J. R.; Cole, M. W . Phys. Rev. B Condens. Mutter 1983, 27, 1314. (7) Klein, J. R.; Cole, M. W. Furuduy Discuss. Chem. Soc., in press, and ref 27, 29-32 therein.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 2, 1987 415
Heat Capacities of Rare Gases Adsorbed on Graphite peratures measured in systems such as rare gases adsorbed on graphite. In the McQuistan-Hock model, the grand canonical partition m, is given by function (eq l ) , as N
-
Z(N,B,ML)%
(7)
where
and c3 is given by eq 24d. From eq 18 and from the recursion relation forfq(N,@) as given in ref 4, we can compute the initial condition for A,(p,p). W e obtain, provided V, > - 2 ~ 1 3 ,that A0
+ 4y1,h, + 2y21,h,2 A2 = ~ q ( 1 + 2~1,)+ 2 q h , ( 2 ~ ~ +~ ~yzq 1 , + x +l/y) + 1,b2(2Xy31, + 4y3? + x + 4) + 2 y t l ~ 3 ( + i y3s) AI = 21,
(21b) (214
Let us define
and ql is the smallest root of
ZN 3
Cbjd = 0
s(7)
= Yl/dO
CzqAq = u(z)/v(z)
(9)
j=O
(22)
q-0
where
In eq 8
3
u(z) =
CCjZJ j=o
and S T d
= (as/aa),J,,
(11)
The coefficients a,(x,y,z) and bj(x,y,z) in eq 9 and 10 are. given in eq 9 and 10 of ref 4. The spreading pressure p can be obtained as 1 20s’ p=-ln28 sz where 0 is the coverage as N
-
m,
Here the dis are given by eq 20 and the cis are given by
CI
(24a)
+ Yd4Y - XY - 1)1
(24b)
+ ~ y q ( 2- x ’ ) ] (24c) + 4 yv(l - x)](xy - 1) + (4 - y - 2x) x ~2
c j = x$t3([9y1,
and
= tltwo
co = Y1,
= qt2[xdo
(X
41,) = (as/az)xJ,ll
(13)
+ y2q) - 1) (24d)
If we rewrite eq 22 as
In order to study the behavior of the heat capacity as a function of the temperature, keeping the spreading pressure p and the number of adparticles q as constant, it is most convenient to work within the isothermal-isobaric ensemble, whose partition function will be denoted by A = A(q,&p). The Gibbs free energy, entropy, and heat capacity are given by G(q,B,p) = - - In A
- xy3q
co + C l Z + c,z2
u(z) -=K+z 0 )
1
+ Dlz + D2z2+ D3z3
(14)
and Dj = dj/do we obtain where C(q,B,p) has a maximum at the critical temperature ( p = Bc), Le., the solution of the equation 2A(A’)2
+ 3BAA’A”
- 2B(A1)3- pA2Atf - 2A2A’’ = 0
(17)
In eq 15-17, the prime denotes the derivative with respect to 8.
3
Aq = Cx(zj)z;q,
= Aq(Ps) = N=l CFq(N9B)?N
6(z) = 0
x(4 = (18)
N=l
where 1, e-2flp. Multiplying by 24, summing over q from 0 to 2N, and using eq 3 and 4 given in ref 4, we obtain the following recursion relation for A,(&p) doAq = -dlAvl
- d2Aq-2 - d3Av3 + ~ 3 6 , 3
(29)
(30)
and
m
= fl %fqW,@)tlN
< q I2 N
where the z i s are the roots of the equation
In order to find A(q,p,p), we start from its definition A(q,B,p)
0
/=1
(19)
where 6q,3is the Kronecker delta
W / W )
(31)
Here the prime denotes the derivative with respect to z. It is worth noting that, for the special case in which x = y = 1, C2 = D3 = 0 and the solution of eq 29 gives z* =
-7
f 1,’/2
-
(32) 1,t and the positive root is always smaller than the absolute value of the negative root, so that for large q
do = 1 - y31, dl
= -v(Xydo
+ y31, + 1)t
d2 = -1,(x3dO + xyq)t2 d3
= xq2(xy - l)[x2do + y1,(2xy
(33)
(20b) (20c)
- l)]t3
(20d)
and the heat capacity per adparticle is C = kBa2/sinh2CY
(34)
416 The Journal of Physical Chemistry, Vol. 91, No. 2, 1987 TABLE I: Adparticle-Surface ( V , ) and Adparticle-Adparticle ( VI]) Interactions in kelvin Ne AT Xe
-378 -1 103 -1912
-34.6 -1 20 -236
15.7 55 118
14.5 51.0 102.0
.
"From ref 5. bFrom ref 6. 'From ref 7 . dFrom this study at coverage value of 0.5. ( T , is the critical temperature.)
0.0'
Ne
Ar
50
Xe 100
TIK I Figure 1. Specific heat for Ne, Ar, and Xe adsorbed on graphite using the interaction parameters given in Table I and the coverage values of 0.1, 0.3, and 0.5.
where CY p pp/2. We note that this is the same result that would have been obtained for a system of independent, distinguishable particles, each of which has only two accessible states: the particle is on the site or it is not. For the more general case in which VI, # 0, we have solved eq 30 numerically choosing values for the interaction constants that are likely to model rare gas/graphite systems. In Table I, the chosen values are displayed together with the literature reference from where they have been taken. Several studies of the Ne, Ar, Xe/graphite systems have reported a broad anomaly in the specific heat at temperature values given in Table I, namely at 15.7 K for Ne, 55 K for Ar, and 118 K for Xe. In Figure 1 we present the specific heat as a function of temperature for values of the coverage of 0.1, 0.3, and 0.5. We see that maxima in the specific heat are observed in the range of the experimental critical temperatures. 111. Discussion and Conclusions
From Figure 1 we can see that maxima in the heat capacity signatures are found around 10-15 K for Ne, 40-52 K for Ar, and 80-102 K for Xe. Let us compare these results with some experimental findings and with some other model calculations. Steele and Karl8 reported peaks in the specific heat measurements of Ne adsorbed on graphitized carbon black powder, where at half coverage they found a peak at 16.1 K. Antoniou' in turn found a peak at 12 K. Huff and Dashlo made the first set of measurements of Ne adsorbed on Grafoil, for submonolayer coverages in the temperature range of 2-20 K, and found peaks between 12 and 15 K. The anomalies they obtained have been confirmed by Rapp et al.," who found, besides a first peak at 13.6 K, a second one at around 16 K which moves up in temperature as the coverage increases and then disappears at higher coverages. Recent experimental studies of neutron s ~ a t t e r i n g , ' ~heat J~ capacities,14and synchrotron X-ray ~cattering'~ on submonolayers of Ar on graphite were all interpreted as being consistent with (8) Steele, W.A.; Karl, R. J . Colloid Interface Sci. 1968, 28, 397. (9) Antoniou, A. A. J . Chem. Phys. 1976, 64, 4901. (10) Huff, G.B.; Dash, J. G. J . Low Temp. Phys. 1976, 24, 155. (1 1) Rapp, R. E.; de Souza, E. P.; Lerner, E. Phys. Rev. B: Condens. Matter 1981, 24, 2196. (12) Taub, H.; Carneiro, K.; Kjems, J. K.; Passell, L.; McTague, J. P. Phys. Rev. B Solid State 1977, 16, 4551. (13) Tiby, C.; Lauter, H. J. Surf. Sci. 1982, 117, 277. (14) Chung, T. T. Surf. Sci. 1979, 87, 348. (15) McTague, J. P.; Als-Nielsen, J.; Bohr, J.; Nielsen, M. Phys. Rev. B: Condens. Matter 1982, 25, 7765.
Battaglia et al. TABLE II: Values of the Ratio R and from Experiments" system I D (CN = 2) H L (CN = 3) MQH (CN = 3)
R 4.80 2.63 2.35
= I V,,/T,I
from Various Models
svstem EXP SL (CN = 4)
R 2.1 3 1.76
~~
"EXP = experiment, 1D = linear lattice, H L = honeycoinb lattice, MQH = McQuistan-Hock lattice, and S L = square lattice. For EXP and MQH the average values are reported. The coordination number is indicated in parentheses.
a continuous melting process. Migone et a1.I6 reported heat capacity measurements of submonolayer Ar on graphite foam, with anomalies at 47.2,49.5,and 55 K, and interpreted the first peak as a "weak" first-order transition. For Xe the situation is similar: melting of Xe physisorbed on graphite has been interpreted as both continuous17J8and first-order melting.I9 The interesting aspect of the McQuistan-Hock model is that it does not allow for any first-order transition. Yet, when applied to "realistic" systems, it predicts maxima in the heat capacity signatures at temperatures very close to the experimental ones. A simple model that forbids a first-order transition but displays maxima in the heat capacity curve is the one-dimensional lattice with nearest-neighbor interactions. Unfortunately, this model predicts those maxima at temperature values well below the experimental data. A two-dimensional model such as the square lattice predicts second-order critical temperatures above the experimental data, while the honeycomb lattice gives better agreement with experiments. In Table I1 we report the ratio R p ~ V l l / Tasc predicted ~ by various models, as well as the experimental average value R of this ratio as obtained from Table I [I(R- i?)/RI < lo%]. The predictions from the McQuistan-Hock model are also reported as averages because, as can be seen from Figure 1, R is not exactly constant. From this table it is interesting to see that the McQuistan-Hock model predicts values of T, that are in good agreement with the experimental measurements. The model does not allow for first-order transitions, nor does the linear-lattice model with nearest-neighbor interactions. On the other hand, the linear-lattice model predicts maxima in disagreement with experiments. In good agreement with experiments are also the anomalies predicted by the square-lattice and by the honeycomb models, the difference between the two being what we call the coordination number (CN in Table II), Le., the number of nearest neighbors attached to each site: the experimental T,'s lie in between the predictions from the models with C N = 3 and C N = 4. Our conclusions are the following: (i) The McQuistan-Hock model has the advantage of being exactly solvable and of giving good agreement between computed and experimental temperatures at which the maxima in the heat capacity signatures occur. (ii) The model contains very few parameters, namely Vo,V,, and Vll; yet those parameters are sufficient to determine the "gross" features of the statistics of adsorption. This implies that the details of the various interactions occurring in real systems do not seem to play an important role. (iii) The model contains a parameter, Voo,which accounts for distortions of the substrate, an effect that has been invoked by various authors10J1to explain some details of the experimental heat capacity signatures. It would seem worthwhile to explore the dependence of the adsorption thermodynamics on V,. (iv) The impossibility of showing a first-order transition, on one hand, and the good agreement between the computed and measured critical temperatures suggest that the order of the transition seen at temperatures around the T, values (16) Migone, A. D.; Li, Z. R.; Chan, H.H. W. Phys. Reu. Lett. 1984,53, 810. (17) Heiney, P. A.; Stephens, P. W.; Birgeneau, R. J.; Horn, P. M.; Moncton, D. E. Phys. Rev. B Condens. Matter 1983, 28, 6416. (18) Rosenbaum, T. F.; Nagler, S. E.; Horn, P. M.; Clarke, R. Phys. Rec. Lett. 1983, 50, 1791. (19) Hammonds, E. M.; Heiney, P. A,; Stephens, P. W.; Birgenau, R. J.; Horn, P. M. J . Phys. C 1980, 13, 1301.
J . Phys. Chem. 1987, 91, 417-426 of Table I might not necessarily be of first order. (v) An important role seems to be played by the number of nearest neighbors attached to each site: the McQuistan-Hock model (with C N = 3) represents a logical extension from a one-dimensional to the complete two-dimensional system. It would therefore be interesting to examine the influence of the average number of nearest neighbors on the heat capacity as a function of temperature.
417
Acknowledgment. This research was supported by the Office of Naval Research, the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract F49620-86C-0009, and the National Science Foundation under Grant CHE-85 19053. Registry No. Graphite, 7782-42-5.
Heats of Hydration of Organic Ions: Predictive Relations, and Analysis of Solvation Factors Based on Ion Clustering Michael Meot-Ner (Mautner) Center for Chemical Physics, Chemical Kinetics Division, National Bureau of Standards, Gaithersburg. Maryland 20899 (Received: May 28, 1986; In Final Form: August 18, 1986)
Gas-phase proton affinities PA(B) and clustering data are used to predict and analyze the heats of solvation of protonated organic bases BH+. Prediction is based on simple empirical relations between PA(B), heats of clustering to four water molecules (AH00,4),and heats of hydration by the bulk solvent (AHog-,q(BH+)). These relations predict the heats of hydration of protonated oxygen and nitrogen bases within A2 kcal/mol on the basis of a single gas-phase parameter, PA(B). These relations can also predict the heats of solvation of the neutral bases B. To analyze the energy terms that contribute to the heats of hydration of ions, the following procedure is applied: (1) The experimental heat of clustering of four H 2 0 molecules to BH+ contains the contribution of ionic hydrogen bonding between BH+ and the inner solvent molecules. Therefore, the further bulk hydration of the clusters BH+.4H20, Le., AW,,JBH+.4H20), as obtained from clustering data and Born-Haber cycles, does not involve further ionic hydrogen bonding, and it can be decomposed to the sum of cavity-forming (AW,,) and dielectric energies (AHodlel), residual (neutral) hydrogen bonding from the inner cluster to outer water molecules (AH'RHB), and hydrophobic hydration of alkyl groups of the ions (AHohydph).(2) Calculated values of AH',,, AHodlel,and NoRHe are then used to isolate the contribution of AHohydph.(3) Finally, the experimental heats of hydration of the naked ions BH+ are decomposed to cavity, dielectric, hydrophobic, and ionic hydrogen-bonding (IHB) terms. The first two are calculated, and the hydrophobic term is used as obtained in the preceding step. AHoIHB is then calculated from the difference between the experimental heat of hydration and its other terms. The ionic hydrogen-bondingcontributions vary from -34.8 kcal/mol for CH30H2+to -6.1 kcal/mol for (C2H5)3NH+and increase regularly by about -10 kcal/mol for each protic hydrogen. The solvation energies (kcal/mol) of the alkyl groups increase with alkyl size and vary from -9.1 for CH30H2+and -9.7 for CH3NH3+to -30.2 for ( C - C ~ H ~ ) ~ O and H +-34.5 for (C2H5),NH+. m ' h y d p h of aromatic ions is small due to the small number of substituent hydrogens. The hydrophobic interactions of ions are not affected significantly by the presence of charge; Le., AHohydph of ions BH+ is similar to that of the corresponding neutral bases B. This is explained by loss of charge from the onium ions to the solvent through hydrogen bonding, which leaves the charge densities on the alkyl groups of ions similar to those in the neutral bases. By use of gas-phase data, the model accounts for the complexities of ion solvation in terms of simple trends as a function of ion structure.
Introduction Solvation by a polar, hydrogen-bonding solvent such as water affects extensively the thermochemistry of ions. The intrinsic molecular effects on ion thermochemistry, e.g., acid-base properties, are unraveled by gas-phase ion energetics. Gas-phase data are also useful in that, combining gaseous and aqueous heats of formation, it becomes possible to calculate the heats of solvation of ions (AHo,,,(BH+)).i*2 Even given this data, however, the various solvation factors, such as cavity forming, dielectric charging, hydrogen bonding, and hydrophobic solvation energies, are hard to identify, in part because of the large but unknown contribution of strong ionsolvent hydrogen bonding. The objective of this paper is to use recent clustering data to overcome this problem. This will allow then a quantitative analysis of the solvation terms for organic ions. To help evaluate ionic hydrogen bonding in the inner solvent shell, we obtained in preceding papers data on the hydrogenbonded clustering of a large series of onium ions and solvent molecule^.^*^ The present work will carry this analysis further (1) Taft, R. W.; Wolf, J. F.; Beauchamp, J. L.; Scorrano, G.; Arnett, E. M. J . Am. Chem. SOC.1978, 100, 1240. (2) Taft, R. Prog. Phys. Org. Chem. 1983, 14, 241. (3) Meot-Ner (Mautner), M. J . Am. Chem. SOC.1984, 106, 1257. (4) Meot-Ner (Mautner), M. J . Am. Chem. SOC.1984, 106, 1265.
by combining the clustering data with the overall ion solvation thermochemistry, and with theoretical models for the cavity and dielectric terms, to identify the hydrophobic and ionic hydrogen-bonding (IHB) contributions. This model will use gas-phase clustering data; ion solvation thermochemistry, which relies on gas-phase proton affinity data, as applied by Taft and Arnett et al.;i*2thermochemical cycles as used by Kebarle and Saluja et al.;536and also electrostatic solvation models as applied to onium ions by Ford and Scribner.' With this model, we wish to demonstrate a comprehensive application of gas-phase data to understand ion solvation. In the preceding paper^^,^ we observed unexpectedly simple relations between the following: (1) the proton affinity PA(B) and the attachment energy of the first water molecule to the ion, Le., AHoD(BH+.0H2);(2) the attachment energy of the first water molecule and the enthalpy for partial solvation by four water and the total solvation molecules, i.e., AH00,4; and (3) energy of the ions, AH',,(BH+). In the first part of this paper we shall combine these relations to present simple predictive relations for the heats of solvation of protonated oxygen and (5) Magnera, T. F.; Caldwell, G.; Sunner, J.; Ikuta, S.; Kebarle, P. J. Am. Chem. SOC.1984, 106, 6140. (6) Saluja, P. P. MTP Int. Rev. Sci.: Phys. Chem., Ser. Two 1976, 6, 1 . (7) Ford, G. P.; Scribner, .I. D. J . Org. Chem. 1983, 48, 2226.
This article not subject to U S . Copyright. Published 1987 by the American Chemical Society
