J . Phys. Chem. 1989, 93, 3308-33 11
3308 elements. They have two or more ligancies. IV. Discussion
Recently, Walsh and c o - ~ o r k e r s ~have ~ - ' ~extended the group additivity scheme in order to predict the heats of formation of alkylsilanes. At the same time, they have calculated the heats of formation using second-order many-body perturbation theory (MP2) and 6-31G(d) basis set at the self-consistent field (SCF)
geometries. These results are listed in Table 11. As shown in Table I, the estimated ArHo[C-(Si)(C)(H),] = -3.0 kcal. It is very interesting that the agreement between our group additivity and the ab initio based values is very good, within 0.3 kcal-mol-I. This gives further support to the estimated values of group parameters from eq 5.
Acknowledgment. This work has been supported by a grant from the National Science Foundation (CHE-8714647).
Thermochemistry of Inorganic Solids. 9. A Quantitative Relation between the Lattice Enthalpy and the Near-Neighbor Distance of Cubic Crystals Mohamed W. M. Hisham and Sidney W. Benson* Donald P. and Katherine B. Loker Hydrocarbon Research Institute, Department of Chemistry, University of Southern California, University Park, Los Angeles, California 90089- 1661 (Received: September 6 , 1988; In Final Form: October 28, 1988) The general two-parameter equation AHL'(M,Xb) = m / r M X+ c has been found to correlate the lattice enthalpy, AHLO, and the near-neighbor distance, rMX,of cubic crystals M,Xb The parameters m and c depend on the particular main group or subgroup and valence state. For 11 different classes of compounds examined, the deviations are always within the accuracy of the A H L O reported. For any given group average deviations are within 11 kcal/mol with maximum deviations never exceeding 1 3 kcal/mol.
Introduction
Ionic models have been used extensively for over 60 years to describe solid salts.' They require a knowledge of crystal structure and ionic radii for each valence state of each element. In addition, they require ionization potentials for each metal, electron affinities for each nonmetal, heats of atomization for each element, and repulsive parameters for every metal-anion pair. The data base required is far in excess of that required for bond additivity. Despite this data input, errors in estimated value for ArHocan range up to 10-12 kcal/mol for even 1-1 salt types. In addition, corrections have to be made for anion-anion repulsion when the ratio of radii r-/r+ exceeds about 2.3. This requires additional repulsive parameters and neglects any van der Waals attractions between ion pairs. One also finds that ionic radii are not constant but show small systematic changes, many of which are correctible but only by using ad hoc anion-anion repulsions and their associated parameters. Finally, the description of 2-2 salt types such as C a S or BaO lead to unrealistic models of unstable, doubly charged anions such as 0,- and S2-for whose existence no direct measurements are known. While 0 has an electron affinity of only 35 kcal/mol, 0,- has an estimated instability of about 240 kcal/mol, making it more of a bookkeeping quantity than a chemical entity. The types of correlations we present in this paper suggest a different approach to ionic models. Let us take 1-1 salts of the alkali metals as an example. In Figure 1, we show a plot of the lattice enthalpy AHLO of the cubic crystals MX against the reciprocal of the near-neighbor distance rMX-'.AH,' corresponds to the enthalpy change in the following process at 298 K ( T o ) : QL
+ MX(cr)
-.
W g )
+ X-(g)
(1)
where Q L = + A H L O . It is related to the potential energy change at absolute zero, ELo by
AHLO = ArH'(M+,g) + AfHo(X-,g) - AfHo(MX,cr) (2) = ELo + 5 R T - E,' - E,ho (3) For most of the alkali-metal salts the thermal energy EthE 6RT0, ( 1 ) O'Keeffe, M . Srrucrure and Bonding in Crystals; Academic Press: New York 1981; Vol. 1. p 299.
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TABLE I: Near-Neighbor Distances ( r ) ,Lattice Enthalpies (AH,'), and Atomization Energies (AH,:) of the Alkali-Metal Halides, Hydrides, and Metals M,XI, r/A AHIO/kcal mol-l AH,,'/kcal mo1-l
LiF LiCl LiBr LiI NaF
NaCl NaBr
NaI KF KC1 KBr KI RbF RbCl RbBr RbI CsF CSCl CsBr CSI TlBr LiH
NaH KH RbH
CsH LiLi
NaNa KK RbRb
cscs
2.01 2.57 2.75 3.00 2.31 2.81 2.98 3.23 2.68 3.14 3.29 3.54 2.82 3.28 3.43 3.66 3.01 3.47 3.62 3.83 3.44 2.04 2.44 2.83 3.02 3.19 3.04 3.72 4.54 4.95 5.31
250.1 205.8 195.4 182.0 221.8 188.2 179.6 168.0 197.5 171.5 164.6 154.8 189.4 165.4 159.0 150.4 180.8 159.6 155.1 145.8 174.9 218.8 192.5 170.0 162.9 155.8 186.2 157.2 131.2 123.6 115.4
204.2 164.9 148.7 128.2 181.7 153.1 138.7 120.0 175.8 154.8 142.1 125.2 171.5 152.5 140.3 124.6 169.4 153.2 142.9 126.5 111.8 91.3 87.2 83.9 83.3 76.2 51.4 42.6 38.6 36.4
while E,', the zero-point energy of MX, is usually less than 1 kcal/mol a t 298 K. Hence, AH,' and E L o at 298 K differ by nearly a constant 2 kcal/mol. ELo may be evaluated from any lattice model using a suitable function to represent the repulsive energy. If we employ a r-" type of repulsion, V ( r ) = Br-", then Ae2 ELo = TMX 1 -). ! (4)
-(
0 1989 American Chemical Society
Thermochemistry of Inorganic Solids
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3309
If we use an exponential repulsion, V ( r ) = B exp(-r/pMx) EL =
Ae2 rMX
-PMX/~MX)
where A is the Madelung constant and e is the electronic charge. We note the two types of repulsion give equivalent forms for ELo when n = r M X / P M X . Since AHLO and ELo differ by almost a constant, we have chosen to plot AHLO against l / r M xfor all the alkali-metal halides. The data used for the plots are given in Table I. Unless otherwise stated, thermochemical data used here are taken from NBS Tables2 and from ref 3, and values are given per mole of metal atom and at 298 K in kcal/mol. Near-neighbor distances of cubic crystals are taken from ref 4, and values are given in angstroms. The result is an excellent straight line through all 20 points with an average deviation of about 0.5 kcal and a maximum deviation of 2.5 kcal. If we compare the plot with eq 4 or 5, we note that ELo will be linear function of I / r M Xonly if P M X / r M X is a constant for all salts or if n M X is the same for all salts. (Actual Born models use parameters n that vary from about 6 (LiX) to 13 (CsX) while P M X / r M X vary from I .6 to 0.70.) The slope is 448 kcal A/moI for the mostly NaCI-type lattices. The intercept is 29 kcal/mol instead of zero. If we correct this by 2 kcal/mol for thermal and zero-point energies, we obtain ELo = 3 1 kcal/mol for an infinite lattice, a physically unacceptable feature. This could in principle be corrected by making the repulsive parameter n M X an increasing function of rMXso that the intercept is forced through the origin. However, it would require the introduction of at least two more parameters and lose the simplicity of the apparently universal function, eq 4. The slope for M X in Figure 1 yields a value for n M X = 4.42, which is reasonably close to common values used for the repulsive parameters. We also show in Figure 1 the values of A H L O of the alkali-metal hydrides plotted against rMXfor the crystal (data in Table I). All the points fall on an excellent straight line, with a slope of 369 kcal A/mol and an intercept of 44.0 kcal/mol. If we represent the relation by the expression given by eq 4, then the corresponding value for nMH is 2.62. Relative to the value of nMX for the halides it is a plausible value, suggesting that H- is a softer or more compressible particle than the halides X-. This is in line with the smaller electron affinity of H and its larger effective radius. Once again making n M H an increasing function of r M Xwould ensure passage through the origin. Because the zero-point energy is larger for the hydrides, the difference between ELo and AHLO is closer to 6 kcal rather than 2 kcal:
Encouraged by the simplicity of the empirical results for the salts and hydrides, we have plotted in Figure 1 the lattice. enthalpies of the alkali metals, now considered to be ionic salts of the formula M'M-. It can be seen that the empirical straight line reproduces the data given in Table I with a maximum deviation of 1 kcal/mol and an average deviation of less than 0.5 kcal/mol. The empirical relation takes the form AH~"(M+M-) =
500
- + 21.5 ~ M M
(7)
From the slope and eq 4 we find ~ M M = 7.33, suggesting a stiffer repulsion than for the salts. The alkali metals have the bodycentered lattice, and it may be that there is an additional anion-anion as well as cation-cation repulsion in these lattices because of the close approach of the next-nearest neighbors. Once (2) NBS Tables. J . Phys. Chem. Ref.Data 1982, 1 1 , Suppl. 2.
(3) Hisham, M. W. M.; Benson, S. W. The Estimation of the Enthalpies of Formation of Solid Salts. Molecular Structure and Dynamics; Liebman, J., Ed.; VCH Publishing: New York, in press. (4) Wyckoff, R. W. G. Crysral Srrucrure; Interscience: New York, 1961; Vol. 1.
'
+/io-
Figure 1. Plot of lattice enthalpies AHH,'(kcal/mol) against l/rMx, the anion-cation distance in crystal. Values are for alkali-metal halides, hydrides, and NH4C1, NH4Br, NH4I (see text). M M refers to pure metals considered as 1-1 salts M'M-. TABLE II: Near-Neighbor Distances ( r ) and Lattice Enthalpies (AHL') of Halides of Alkaline-Earth Metals, Silver, and Ammonium"
SrCI, BaCI2 CaF2 SrF2 BaF2 AgI AgBr
3.02 3.18 2.36 2.51 2.68 2.81lC 2.882
514.7 490.6 629.8 596.7 563.3 213.3 215.8
AgCl AgF NH4F NH4CI NH4Br NH41
2.778 2.457 2.705* 3.348 3.505 3.784
218.9 232.1 201.5 170.0 162.9 151.6
Includes AfH"(NH4',g) = 150.6 (5). Calculated from room-temperature density (1.009 g/cm3) and hexagonal structure (ZnS). Because of stronger H bonds between NH4+ and F the contributions of zero-point energies may be significantly different from the other halides. *Has a ZnO lattice.
again, the finite intercept could be removed by making n M M an increasing function of r M M . In Figure 1 we have also plotted the data for NH4CI, NH4Br, and NH41, all of which have body-centered cubic (BCC) lattices like CsCI. AfHo(NH,+,g) = 150.6 k ~ a l / m o l . ~As can be seen, the points all fall right on the line for the BCC alkali metals. The largest deviation is 1 kcal. NH4F is poorly characterized as a hexagonal crystal. Assuming a Wurtzite (ZnO) type lattice with coordination of 4 and using the reported density at 25 OC, we have calculated a near-neighbor distance of 2.705 A. This puts it about 4 kcal/mol below the line as shown by the bracketed point in Figure I . We have also examined halides of the alkaline-earth metals and silver (data are given in Table 11). It is interesting to note that as shown in Figure 2, these compounds also exhibit a linear relation given by general equation A: AHLO(MX~)=
m
-+ c rMX
( 5 ) Jenkins, H. D. B.; Morris, D. F. C. Mol. Phys. 1976, 32, 231. Their value is derived from a linear relation between the coefficients in the JonesDole viscosity of ions in dilute solution and the estimated heats of hydration of +1 ions. The latter depends on AJf"(M+,g). Kebarle (Kebarle, P. Ann. Reu. Phys. Chem. 1977, 28,445) quotes a directly measured value of 153.9 kcal/mol, which would raise all the points by some 3.3 kcal/mol.
Hisham and Benson
3310 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
TABLE III: Near-Neighbor Distances ( r ) and Lattice Enthalpies (AHLO') of Alkali-Metal Oxides, Sulfides, and Selenides
LizO Na20 Kz0 RbzO Li2S NazS
2.000 2.403 2.787 2.919 2.472 2.826
KZS
470.8 390.2 332.2 315.2 433.2 378.2
3.200 3.313 LizSe 2.600 NazSe 2.948 KzSe 3.324
336.8 320.4 428.0 372.8 340.2
Rb2S
TABLE I V Near-Neighbor Distances ( r ) and Lattice Enthalpies (AH,,'') of Alkaline-Earth-Metal Oxides, Sulfides, and Selenides M,Xb
500v I
0.35
'/& r
AHL0'(M&)/ kcal mol-'
MgO 2.107 CaO SrO BaO
laC1,
470' 0.30
r/A
MgS CaS
I 0.4 0
2.405 2.580 2.762 2.602 2.845
M,Xb SrS BaS Case SrSe Base MnO
705.1 612.1 569.5 529.1 644.0 575.6
r/A
uLo'(MzXb)
kcal mol-'
3.010 2.194 2.960 3.12 3.300 2.222
/
540.9 506.7 548.3 520.1 485.7 694.1
0.45
TABLE V Parameters Involved in Eq A
Figure 2. Plot of lattice enthalpies against l/rMX,the anion-cation distance in crystal. Values are for some alkaline-earth halides. class of compd alkali-metal halides (MX)
parameters m/kcal A mol-' c/kcal (n, eq 4)' mol-' 448.1 (4.42) 29.0
alkali-metal hydrides (MH)
369.3 (2.62)
44.1
alkali-metal metal (MM)
500.4 (7.33)
21.5
1250.0 (3.96)
98.8
266.7 (1.85)
123.5
+500.4 (7.33)
21.5
alkaline-earth-metal halides (MX2) silver halides ammonium halides
dev in AH,', kcal mol-' max = 1.8 av = 0.6 max = 1.7 av = 0.4 max = 1.5 av = 0.7 max = 1.9 ave = 1.3 max = 0.6 av = 0.3 max = 0.3 av = 0.2
'Values of n, the repulsive parameter from eq 4, are in parentheses. I
pb (50 I 0.3
I
I
I
/Pb /
I
I
0.4
0.5
'300
' I
1
Figure 3. Plot of partial lattice enthalpies against reciprocal of anioncation distance in crystal for alkali-metal oxides, sulfides, and selenides.
where AHL"(MX2) is the lattice enthalpy of the crystal MX2 and r M Xis the near-neighbor distance. m and c are constants. The results are summarized in Table V, together, with those for alkali-metal compounds illustrated above. For the CaFz type structures of the alkaline-earth halides we can calculate from values of m a value of the repulsive parameter n (eq 4). It is 3.96, very close to the value found for the alkali-metal halides. Unfortunately, t o examine metal oxides, sulfides, selenides, etc., we require AfHovalues for bivalent anions such as 02-and S2-,which do not exist.l However, to explore the relation, we plot AHLO' against l / r M X(where AHL'' = aAfHo(M"+,g) - AfHO(M,Xb,cr)). For these types of compounds, data are given in Tables 111 and IV and plotted in Figures 3 and 4. The result is a straight line and can be represented by general equation B: m' AHLO'(M,X,) = - + C' rMX
where m'and c'are constants for each class of compounds. The results obtained for this equation are summarized in Table VI, and for any class of compounds the average deviation is less than
480
BO
0.30
I
1
I
I
0.35
0.40
0.45
0.50
+/A='
Figure 4. Plot of partial lattice enthalpies against reciprocal of anioncation distance in crystal for oxides, sulfides, and selenides of some alkaline-earth metals.
1 kcal/mol with the maximum deviation of 3 kcal/mol. Values of the repulsive parameter n (eq 4) are also shown in parentheses.
J . Phys. Chem. 1989, 93, 331 1-3313
TABLE VI: Parameters Involved in Eq B uarameters m‘/kcal A mol-l c’lkcal class of compd alkali-metal oxides ( M 2 0 )
(a eq 4)
moP
986.7 (2.44) -21.3
alkali-metal sulfides (M2S)
1044.4 (2.66)
9.2
alkaline-earth-metal oxides (MO)
1540.0 (2.97)
-30.8
alkaline-earth-metal sulfides (MS)
1920.0 (5.8)
-95.2
alkali-metal selenides (M2Se)
1042.9 (2.66)
22.7
alkaline-earth-metal selenides (MSe) 1787.5 (4.36) -54.3
dev in AHLO’, kcal mo1-I max = 1.5 av = 1.1 max = 3.0 av = 1.5 max = 4.7 av = 3.2 max = 4.0 av = 1.9 max = 4.3 av = 3.7 max = 1.9 av = 1.6
Discussion The empirical, two-parameter equations illustrated above describe an accurate correlation for determining AHLO or AHLO’ and hence AfHofrom the lattice parameter for cubic crystals. Alternatively, if we know AfHo,then it is possible to use these equations to determine the nearest-neighbor distance with better than 0.5% accuracy. In certain cases such as the oxides of alkaline-earth metals the maximum deviation obtained is more than 2 kcal, and these may be attributed to experimental errors involved in the measurements of AHLO. Although a linear relationship is illustrated mainly for groups IA and IIA metal compounds having cubic structures, we have found that halides of any metal cations and ammonium with cubic structures can be related linearly. The results are given in Table
3311
11. However, it is not possible to relate compounds of other metals with those of the main-group metal compounds, although both may have the same cubic structures. For example, NaCl type MnO does not fit in the line of group IIA metal oxides (see Figure 4). Similarly, CsCl type TlBr does not fall on the line for the group IA metal halides (see Figure 1). If one examines the enthalpies of atomization (values are given in Table I) of the alkali-metal halides or hydrides as functions of rMx,they show the qualitative but not the quantitive behavior observed for the lattice enthalpies. While these do not rule out a covalent description of the M X salts, it does tell us that no simple, general covalent model will describe the entire group.
Acknowledgment. This work has been supported by a grant from the National Science Foundation (CHE-87- 14647). Registry No. LiF, 7789-24-4; LEI, 7447-41-8; LiBr, 7550-35-8; LiI, 10377-51-2;NaF, 7681-49-4; NaC1,7647-14-5; NaBr, 7647-15-6; NaI, 7681-82-5; KF, 7789-23-3; KCI, 7447-40-7; KBr, 7758-02-3; KI, 768111-0; RbF, 13446-74-7; RbCl, 7791-11-9; RbBr, 7789-39-1; RbI, 779029-6; CsF, 13400-13-0; CsCI, 7647-17-8; CsBr, 7787-69-1; CsI, 778917-5; LiH, 7580-67-8; NaH, 7646-69-7; KH, 7693-26-7; RbH, 1344675-8; CsH, 13772-47-9; LiLi, 14452-59-6; NaNa, 25681-79-2; KK, 25681-80-5; RbRb, 25681-81-6; CsCs, 12184-83-7; TIBr, 7789-40-4; SrCI,, 10476-85-4; BaCI2, 10361-37-2; CaF,, 7789-75-5; SrF,, 778348-4; BaF2, 7787-32-8; AgI, 7783-96-2; AgBr, 7785-23-1; AgCI, 778390-6; AgF, 7775-41-9; NH4F, 12125-01-8;NH,CI, 12125-02-9;NH4Br, 12124-97-9; NH,I, 12027-06-4; Li20, 12057-24-8; Na20, 1313-59-3; K,O, 12136-45-7; Rb20, 18088-11-4; Li2S, 12136-58-2; Na2S, 131382-2; K2S, 1312-73-8; Rb2S, 31083-74-6; Liz%, 12136-60-6; Na2Se, 1313-85-5; K2Se, 1312-74-9; MgO, 1309-48-4; CaO, 1305-78-8; SrO, 1314-11-0; BaO, 1304-28-5; MIS, 12032-36-9; Cas, 20548-54-3; SrS, 1314-96-1; Bas, 21109-95-5; Case, 1305-84-6; SrSe, 1315-07-7; BaSe, 1304-39-8; MnO, 1344-43-0.
Phase Transition of Aqueous Solutions of Poly(N-isopropylacrylamide) and Poly(N-isopropylmethacrylamide) Shouei Fujishige,* Research Institute f o r Polymers and Textiles, 1-I Higashi, Tsukuba 305, Japan
K. Kubota, College of Technology, Gunma University, 1-5 Tenjin, Kiryu 376, Gunma, Japan
and I. Ando Tokyo Institute of Technology, 2-12 Oh-okayama, Meguro, Tokyo 152, Japan (Received: July 19, 1988; In Final Form: September 14, 1988)
When aqueous solutions of well-fractionated poly(N-isopropylacrylamide) samples are heated, the polymer molecular dimensions change abruptly at a critical temperature (-32 “ C ) , followed by aggregation of individual polymer chains dispersed in a state of globular particles to give an optically detectable phase transition. The transition occurs independently of either the molecular weight of the polymer (5 X lo4 to 840 X lo4) or its concentration (0.01 to 1 wt %). This behavior is reminiscent of the thermal denaturation of proteins in aqueous medium.
To elucidate the role of water molecules in thermal denaturation of biological polymers, various properties of aqueous polymer
solutions have been studied by many investigator^.'-^ Among those, aqueous solutions of relatively simple synthetic polymers
(1) Kauzmann, W. Nature 1987, 325, 763-764. (2) Klotz, E. M. Fed. Proc. 1965, 24 (Suppl. No. IS), S24-33. (3) Horne, R. A,; Almeida, J. P.; Day, A. F.; Yu, N. J . Colloid Interface Sci. 1971, 35, 77-84.
(4) Heskins, M.; Gillet, J. E. J . Macromol. Sci., Chem. 1968, A2, 144 1-1 455. ( 5 ) Molyneux, P. Water-Soluble Synthetic Polymers: Properties and Behaoior; CRC Press: Boca Raton, FL, 1983, 1984; Vols. I, 11.
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0 1989 American Chemical Society
