J. Phys. chem. 1982, 86, 552-560
552
tance n* can be obtained again from a Born cycle asz8 A c * d = (N@o2/t)(1/n - l / O * ) (12) which yields A V d , if one assumes both a and a* to be independent of pressure:
We can calculate n* from the experimental values of the activation volume for ion-pair dissociation; a value of about 15 A is obtained for all media studied. This value for a* yields also the correct values for the activation volume of recombination, i.e., the process of bringing two ions together from infinity to a distance n*. This value of 15 A for n* also leads to values of the other activation parameters in the right order of magnitude, although no conclusions can be obtained here since a temperature dependence of n* may intervene. In conclusion, we may state that the results reported earlier together with the results obtained in this pressure study show that the ion pair of TBAP in media of low
polarity dissociates through an “activated” state which should be described as a loosely bound ion pair. The ions within this loosely bound pair bear the complete shell of electrostricted solvent as if they were free, although they are still within a distance at which there is a strong electrostatic interaction. Our results are therefore a convincing indication that the important factor in ion-pair dissociation-at least in nonsolvating media-is the dielectric structuring of the medium around the pair. It is also clear that the processes of charge creation and charge separation are about equally important in ionic dissociation processes in poorly solvating media.
Acknowledgment. We acknowledge with pleasure the stimulating discussions with Professor L. De Maeyer, Professor M. Van Beylen, and Dr. L. Hellemans. The expert advice of Dr. K. Heremans on the high-pressure techniques is especially acknowledged. Financial support by the Fonds voor Kollektief Fundamenteel Onderzoek (Grant No. 2.0051.77) and by the Belgian Government (GeconcerteerdeOnderzoeksacties, Convention 76/81 11-4) is gratefully acknowledged.
Nonideal-Associated Vapor Analysis of Hydrogen Fluoride Rlchard L. Redington Department of Chemistty, Texas Tech Universw, Lubbock, Texas 79409 (Received July 6, 198 1; I n Final Form: October 2, 198 1)
A nonideal-associated vapor model is proposed for hydrogen fluoride in the 19.5-56 “C temperature range. Thermodynamic parameters for the model are determined by fitting vapor density, heat capacity, excess entropy, excess enthalpy, and infrared absorption data. In this temperature range the cyclic hexamer becomes the most abundant oligomer at total pressures ranging from about one-third to one-half of the saturation values at 19.5 and 56 “C, respectively. In this model significant populations of higher cyclic oligomers accompany the buildup of hexamer. The series is truncated at the dodecamer and at saturation pressures P12is about 7% of Ps. Vapor nonideality contributes about 6% to the deviation of one mole of saturated HF vapor from the ideal gas law. Values for HF second virial coefficients are proposed from the analysis, but the curve for monomer-monomer interactions is the only that is appreciablysensitive to the presently available data. The experimentaldimerization and cyclic hexamerization enthalpies obtained for this model are -17.9 f 1and -165 f 2 kJ/mol, respectively. The estimated hydrogen bond energy of the dimer is -23 f 2 kJ/mol. The estimated monomer-monomer second virial coefficient is 3.25 dm3/molf 10% at 19.5 “C. The oligomer entropies are very large, suggesting the presence of low-frequency skeletal modes and molecular nonrigidity.
Introduction In crystalline hydrogen fluoride’ the molecules are linked together by strong hydrogen bonds to form an infinite chain structure with Fa-F distances of 247 pm and F--F-.F angles of 120.1O. Similar hydrogen bond associations seem prevalent in the liquid phase (cf. Raman spectroscopic studies2) and many e~periments”~ show that oligomers persist in the vapor phase as well at temperatures below about 80 OC. The existence of vapor phase hexamers is unquestioned but widely disparate estimates for their abundance, and for the abundance5 of other oligomer (1)M. Atoji and W. N. Lipscomb, Acta Cryst., 7, 173 (1954). (2)I. Sheft and A. J. Perkins, J. Znorg. N u l . Chem., 38,665 (1976). (3)R. L. Jarry and W. Davis, Jr., J. Phys. Chem., 57, 600 (1953). (4)W.Strohmeier and G. Brienleb. 2. Elektrochem.. 57. 662 (1953). (5jE.U.Franck and F. Meye; 2.kektrochem., 631 57i (1959). (6)D.F. Smith, J. Chem. Phys., 28, 1040 (1958). (7)E. U. Franck and W. Spalthoff, Naturwissenschaften, 22, 580 (1953). (8)R. A. Oriani and C. P. Smyth, J. Am. Chem. SOC.,70,125 (1948). 0022-3654/82/2086-0552$01.25/0
species ranging from dimer through nonamer, have been reported. Structural data are absent except in the case of dimer, determined to be a bent chain: and the hexamer, determined to be cyclic.’O Both chain and cyclic structures have been postulated for the other possible oligomers. Estimates of thermodynamic properties, including the hydrogen bond energy of the dimer, are not known accurately and properties of the oligomers including collisional energy exchange rates, lifetimes, and formation or dissociation channels are unknown except for limited work on the dimer.ll No model has been proposed for hydrogen fluoride vapor that can quantitatively account for the composite experimental information that is Presently available. The experimental intractability of HF and a focus on explaining (9)T.R.Dyke, B. J. Howard, and W. Klemperer,J. Chem. Phys., 56, 2442 (1972). (10)J. Janzen and L. Bartell, J. Chem. Phys., 50, 3611 (1969). (11)G.E.Ewing, J. Chem. Phys., 72,2096 (1980).
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, NO. 4, 1982 553
Vapor Analysis of HF
a single type of data have both contributed to the discordance of the extant proposals. Ideal gas behavior has always been assumed for computations on HF vapor although nonideality for the saturated vapor has been discussed previously.12 I t is understandable that ideality is assumed when dealing with a limited data base as the number of parameters needed to describe a nonideal-associated vapor system is large. However, both association and nonideality effects are large for HF vapor and neither should be excluded; the available composite data base provides sufficient scope for an initial assessment of the combined effects. In the present paper a set of second virial coefficients and oligomer equilibrium constants is proposed with simultaneous consideration given to vapor d e n ~ i t y , ~heat * ~ Jcapacity: ~ infrared,6J4-16 entropy," and structuralgJOdata. This is part 2 of a series on HF vapor phase properties. In part 1,18 among other things, classical, rigid molecule, second virial coefficients for the monomers were calculated using six recently proposed intermolecular potential energy functions. The overall analysis provides a self-consistent picture for the partial pressures and other oligomer properties in the 19.5 to 56 "C temperature range. Data Base and Nonideal Vapor Association Model The present analysis of HF vapor composition utilizes a wider data base than was used by previous authors. It consists of seven vapor densit913 (e.g., "association factor", Z) and heat capacity5 isotherms in the 19.5 to 56 "C range, infrared spectra6J4J5and laser absorbance measurements,16 and excess entr~py'~J'and excess enthalpy13 values at saturation pressures. Much of this information is tabulated by Vanderzee and R~denberg,'~ who analyzed the thermodynamic excess properties of HF vapor and recommended smoothed values for the complete thermodynamic data base. They did not, however, present a model for the vapor phase composition. Their tables include otherwise unpubli~hed'~ vapor density data from Argonne National Laboratories that extend the accurate isotherms of Strohmeier and Briegleb4to the saturation pressures, and also provide the 19.5 and 23 "C isotherms used here.20 This data base is analyzed assuming that the HF monomer and oligomers obey the virial equation of state. The objective is to find the set of thermodynamic equilibripm constants K , and second virial coefficients B,, that produces the optimum calculated fit to the experimental data points. The empirical equation of state for one formula weight of HF vapor is written as
P V = RT/Z
(1)
where 2 is experimentally determined by measuring vapor density as a function of T and P. 2 is separated into contributions for nonideal gas behavior and molecular association as (12)J. W. Armitage, P.Gray, and P. G. Wright, J. Chem. SOC.,1796 (1963). (13)C. E.Vanderzee and W. W. Rodenberg, J. Chem. Thermodyn., 2, 461 (1970). (14)D.F. Smith, J. Mol. Spectrosc., 3, 473 (1959). (15)P. V. Huong and M. Couzi, J. Chim.Phys., 66,420 (1969). (16)J. J. Hinchen and R. H.Hobbs, J.Opt. SOC.Am., 69,1546(1979). (17)J.-H. Hu, D.White, and H.L. Johnston, J. Am. Chem. SOC.,75, 1232 (1953). (18)R. L. Redington, J. Chem. Phys., 75,4417 (1981). (19)P. R. Seufzer and J. J. Katz, 126th National Meeting of the American Chemical Society, New York, N.Y.,1954; P44M, 1954, as quoted in ref 13. (20)The value of 19.5"C was used for the boiling point of liquid HF. The true value is 292.90 K; I. Sheft, A. J. Perkins, and H. H. Hyman, J. Znorg. Nucl. Chem., 35, 3677 (1973).
2=
z,z,
(2) where both 2, and 2, are unity for an ideal, unassociated vapor. The virial equation of state is written P V = nRT(1 + n B / V ...) (3)
+
where n is the total of moles of monomer plus oligomers in one formula weight (20.01 g) of HF vapor and B is the composite second virial coefficient
B = CBqrX,Xr
(4)
,J
and X,and X,are the mole fractions of oligomers (HF), and (HF),. 2, is the association factor, i.e. 2, = l / n = Cqn,/n Q
(5)
where n, is the number of moles of oligomer (HF),. From these equations it is seen that 2;l
= (1
+ BPZJRT)
(6)
which can be solved for 2, if B is given for set T and P values. Reactions for the oligomerization equilibria are written qHF
F'
(HF),
(7)
Fugacities for oligomers (HF), are denoted f, and the thermodynamic equilibrium constants are written
K , = fq/fP = Y , P q / Y , ~ P l ~
(8)
To obtain partial pressures P, and thermodynamic quantities it is necessary to obtain estimates for the fugacity coefficients 7,. With the definition
B, = CB,,X,
(9)
B = CB,X,
(10)
r
eq 4 is rewritten as 9
It is assumed that partial pressures P, add to give the total pressure P. Since X , = nq/n, when eq 10 is substituted into eq 3 the result is a summation over q of terms P,V = n,RT(1 + nB,/V) (11) Application of eq 1, 2,5, and 11to the standard equation for the fugacity of a gas that obeys the second virial equation of state gives In f,/P, = In 7, = nB,/V = B,/VZ, (12) Association factor 2, and mole fractions X , are easily obtained after evaluating n, through eq 11 and 1 2 n, = (V/RT)P,/(1
+ In YJ
(13)
I t is evident from these equations that an iterative process must be used to determine the vapor phase properties for each set of equilibrium constants K , and second virial coefficients B,, that are tested. Thus, given values for the K, (calculated from the input enthalpy and entropy parameters) and B,, constants, the B, values of eq 9 are computed from initially estimated mole fractions (e.g., ideal gas values). Using these B,'s and initial estimates for 2, the y4 values are computed from eq 12. Partial pressures are then calculated via eq 8 at the particular total pressure P of interest. The computed partial pressures and fugacity coefficients are then used to recalculate the mole fractions and 2, using eq 13 and 5. 2, is calculated by solving eq 6 after finding B through eq 9 and 10. This procedure is repeated at each T-P data point until the y, Z,, and 2, values are stable. The product of
554
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The Journal of Physkal Chemlstry, Vol. 86,No. 4, 1982
2, and 2, yields the calculated Z to be compared with the experimental value. At T and P values corresponding to data points the oligomer populations calculated using the cyclical procedure were used to calculate values for specific heat, excess entropy, excess enthalpy, and IR absorption2' as well as 2. The K, and B,, input parameters were adjusted by using a weighted least-squares fitting routine to adjust the 287 calculated values to approximate agreement with the observed data points. In all calculations the experimental T, P, and V values were used. The heat capacity at constant pressure is given as
a
D
= -[Cn,H,] aT 4
=
where the standard state (ideal gas) monomer enthalpy is taken as the zero reference point (HIo= 0). The (an,/aT), terms are responsible for the large experimental heat capacities observed for the HF vapor.5 These derivatives were determined numerically using np values calculated via the cyclical procedure at temperatures 0.5 "C above and 0.5 "C below the experimentally used values. The ideal gas enthalpy and entropy values calculated from the input parameters were corrected for nonideality at each T and P of experimental interest. Helpful equations for nonideal gas mixtures are given by Beattie and Stockmayer.22 The enthalpy nonideality correction for a gaseous mixture is Hcor
= (B - B?RTn2/V
(15)
with B' = T(aB/dT),and n = Z;l. This equation was used directly at saturation vapor pressures to make excess enthalpy nonideality corrections. However, to make heat capacity nonideality corrections through the first term of eq 14 Hcorwas separated into components
H,""' = n,(B,
- B,')RTn/V
(16)
Nonideality corrections for the (dH,/aT), terms were neglected. The excess enthalpy is given as
HE = H - Hlo = CnqAHqo+ Hcor P
The excess entropy is given as SE = S - S I O = C nqSqo- R E n , In X , + R In P 4
4
(17)
+ Po* - Sl0 (18)
with scor
= -R(B - B?n2/V
(19)
The nonideality corrections for enthalpy and entropy are not neglible for HF vapor because of its large virial coefficients, as will be seen in the results tabulated below. Extensive calculations of the properties of HF vapor using the above procedure led to the conclusions that: (a) oligomers above the hexamer are needed to account for the vapor phase properties of HF, and (b) constraining equations are needed to ensure that relative values for the entropies, enthalpies, and virial coefficients all take (21) R. L. Redington, part 3, J. Phys. Chem., following paper. (22) J. A. Beattie and W. J. Stockmayer, Chapter 2 in "Treatise on Physical Chemistry", Vol. 11, H. S. Taylor and S. Glaastone, Ed.; D. Van Nostrand Co., Inc., New York, N.Y., 1951.
plausible values. The composite data fitting favors the presence of oligomers as large as dodecamers, but their direct observation is lacking and their populations are somewhat model dependent. The present model is justified below, insofar as possible, but it is clear that the availability of additional, improved data is of top priority before highly accurate populations for the various oligomers that are present in the high pressure ranges can be obtained. The constraint scheme described below for the enthalpy and entropy values allows 14 parameters to be simultaneously adjusted via the weighted least-squares data fitting procedure. Four of these parameters define the standard enthalpies of formation, nine parameters fix the standard entropies of formation, and the remaining parameter helps to define the B,, virial coefficient curve. The various data sets were weighted so that all the calculated points fall, at worst, to within a few percent of the experimental values. The calculated values frequently are within the quoted experimental errors, but the entire body of data could not be simultaneously fitted this well. Results: Thermodynamic Parameters A. Enthalpy Values. The standard enthalpies of formation for the oligomers are fixed using four adjustable parameters in relationships that enforce a uniform enthalpy "additive" factor, Le., an increase in the average hydrogen bond enthalpy with an increase in q. This additive effect is suggested theoretically by ab initio calcul a t i o n ~and ~ ~experimentally by the decreasing Fa-F bond distance in the sequence of dimer to hexamer to crystal. The assumed enthalpy relationships are L I H ~ , ,=~ ,q*Ah*s,;A H o q , c h = (9 - l).Ah.tq (20) for the cyclic and chain series, respectively. The "additive" effect is obtained by setting s4 = 1 and t2 = 1 - h3,with P
s4
=1
4
+ hl r=5 c2"';t,= 1 + h2 23-r r=3
(21)
for the higher q values. The average hydrogen bond enthalpy is made slightly greater for large cyclic oligomers than for the corresponding chains by shifting the entire set of t values to make t9 = s7. The !our adjustable enthalpy parameters are Ah, hl, h2, and hB. The indicated constraint scheme appears to yield better fits to the experimental data than alternative schemes possessing either faster or slower variations of the average hydrogen bond enthalpy with q. The set of standard enthalpies of formation determined using the 14 parameter least-squares refinement with the virial parameters chosen below is listed in Table I, along with the standard entropies of formation and the thermodynamic equilibrium constants at 26 "C. The additive effect on the average hydrogen bond energy is clearly seen in this table. In Table I1 the present AH2", AH4", values are compared with several previous estimates based on ideal gas analyses involving selected portions of the overall data. The introduction of gas nonideality in the data fitting procedure necessarily contributes to the reduced magnitudes of the new enthalpy values. B. Dimerization Enthalpies for Water and HF. At low pressures AH2" dominates 2, and the Bll second virial coefficient curve dominates 2,. In these calculations the most accurate fits to the low-pressure vapor density data as well as to the overall HF data base resulted when AH2" values with small magnitudes were coupled with Bll virial (23) J. E. Del Bene and J. A. Pople, J. Chem. Phys., 56, 2296 (1971).
The Journal of Physlcal Chemistry, Vol. 86, NO. 4, 1982 555
Vapor Analysis of HF
N rl
rl rl
0 rl
Q,
m 3
E"
M t-
;3'
?
rl
tY 0
W
u?
coefficients reaching large negative values at 19.5 OC. The thermodynamic parameters listed in Table I1 arise from a "best compromise" virial coefficient curve chosen from a set of calculations that covered a range of different Bll curves. The results for HF dimerization appear to be well justified by comparisons with recent experimental and theoretical results for water dimerization. Curtiss, Frurip, and Blanderu recently reported, on the basis of accurate vapor phase thermal conductivity measurements, that AH20 = -15 f 1.3 kJ/mol for water dimerization. Making use of quantum mechanically calculated dimer vibrational frequencies,%they estimated that U2 = -23 f 3 kJ/mol. The new AH20 value is more accurate than the decade-old value reported by Gebbie et al.,26and the error limits on the new U2value just cover the approximately -26 kJ/mol estimate obtained by Watts2' on analyzing the virial coefficient data of Kell et al.% (If explicit water dimerization, which Watts ignored, should be significant to the virial data analysis it would reduce the magnitude of the derived U2). The value U2 = -23 f 3 kJ/mol for water compares well with theoretical estimates such as -23.6 kJ/mol obtained by Dill, Allen, Topp, and P ~ p l using e ~ ~ a 6-31G* basis set and -22.6 kJ/mol reported by Chipmanm Dill et alSBalso calculated the dimerization energy for HF using the 6-31G* basis set and obtained -24.5 kJ/mol. This value is 0.9 kJ/mol larger than U2for water and the difference is assumed for discussion to be 1 f 1 kJ/mol. The value AH20 - U2 = 5 f 2 kJ/mol is estimated below for HF. Combining these two estimations with the value U2= -23 f 3 kJ/mol for water yields U2 = -24 f 4 kJ/mol and AH20 = -19 f 6 kJ/mol as encompassing the most likely values for HF dimer. These ranges bracket the values AH20 = -17.9 f 1 and U, = -23 f 2 kJ/mol, which are chosen exclusively from the present data fitting process as being the most reasonable experimental values for HF dimerization. C. Entropy Values. The oligomer entropies are determined using nine parameters that are adjusted by least-squares refinement. The basic entropy relationships, excluding AS2', are
+
ASoq,cy= ecy As-q-u, d
M
+
ASoq,* = ech As-q-u,
+ R In q
(224 (22b)
where R is the ideal gas constant and As, ecy,and echare adjustable parameters. Configurational entropies are absorbed into eq 22 except for the ring opening term R In q. Defining ug = 1, the higher uq factors are taken as up = uq-l
+ el/(e2 + q - 6)
(23)
with a similar series N
uq = uq+l- e3/(e4 + q
- 2)
(24)
for the lower q values. The chain and cyclic sequences are coupled as uq = uq(l + exp(-e5q))
(25)
(24)L. A. Curtiss, D. J. Frurip, and M. Blander, J. Chem. Phys., 71, 2703 (1979). (25)L. A. Curtiss and J. A. Pople, J. Mol. Spectrosc., 55, 1 (1975). Harris,and (26)H.A.Gebbie, W. J. Burroughs, J. Chamberlain,J-E. R.G. Jones, Nature (London),221,143 (1969). (27)R. 0. Watts, Chem. Phys., 26,367 (1977). (28)G. S.Kell,G. E.McLaurin, and E.Whalley, J. Chem. Phys., 48, 3805 (1968). (29)J. D. Dill, L. C. Allen, W. C. Topp, and J. A. Pople, J . Am. Chem. SOC.,97,7220 (1975). (30)D. M. Chipman, J.Phys. Chem., 83, 1657 (1979).
558
The Journal of Physical Chemistry, Vol. 86, No. 4, 1982
Redington
TABLE 11: Experimental Enthalpy Valuesa __
AH^', method
-AH,"
ring
-AH,", chain
-AH,", ring
-AH,", chain
nonideal gas
17.9 27 33 21 2s
97.1
77.5
164.8 169 167 167 168
137.6
PVT PVT IR
CP a
102 79 84
Units are kJ/mol for reactions qHF
-3
ref present work mon-dim-hex, ref 38 extended equil, ref 38 ref 6 , 1 4 ref 5
140
(HF),
TABLE 111: Statistical Calculations for (HF), Oligomers oligomer S O t r , J/K mol SorOt, J/K mol S o"ib, J/K mol S'conf, J/K mol S",,calcd, J/Kmol So obsd, J/Kmol A#', - U,, kJ/mol a
Chain oligomers.
2a
3"
4a
4
155 82 14b 0 256c 249
159 106 56 9 32lC 305
163 114 122 21 399 392
163 112 85 9 35OC 333 9
-
5a
5
6
7
8
9
10
11
12
166 122 185 34 473 472
166 117 136 21 423 409 16
168 125 193 34 486 481 17
170 136 247 49 555 547 24
171 147 307 65 626 619 24
172 158 363 82 696 693 33
173 169 425 101 767 767 32
174 180 363 120 837 841 41
175 191 407 140 913 915 39
Four group vibrational frequencies are used to calculate Sovib as given in text; however, see footnote
c. C These entropy values are for vibrational distributions with the group frequencies (363, 1 8 7 , and 50 cm-' ) taken (1,1, 0),(2, 3, 0 ) and (2, 4, 0 ) times for dimer, trimer, and cyclic tetramer, respectively. The best So, estimate, calculated using three observed frequencies, is 256 J/K mol.
which both constrains them and allows effective uncoupling of the two sequences for the tetramers and pentamen. These are the only oligomers that develop comparable populations for the cyclic and chain molecules in the temperature-pressure range under consideration and four of the entropy parameters are effectively devoted to characterizing these four oligomers. The set of nine parameters used to fix the overall entropy values are then As, ecy, ech,e1-e5, and AS2'. The e2 value was restricted to positive values and it was determined to be near zero. The purpose of e2was to decrease the large heavy oligomer populations that develop in the data fitting insofar as possible by reducing their Sqovalues, but it was ineffectual. Unconstrained data fitting tends to favor expanding relative populations for the highest oligomers at saturation pressures; the formula adapted for uqforces a monotonic decrease in their relative populations as q increases from the value q = 6. The saturation partial pressures converge adequately when the series is truncated at q = 12. It is possible that a different form would converge more rapidly than eq 23, leading to a dominant hexamer and series truncation at q = 8 or 9, but a satisfactory one was not found in this research. The derived standard entropy values are listed in Table I. Their large magnitudes suggest appreciable molecular nonrigidity. D. Statistical Calculations. A degree of corroborative evidence for the large entropy values and for truncation of the oligomer series at high q values is provided by the statistical calculations listed in Table 111. These make use of oligomer geometries calculated using potential energy function KM0l6 and four vibrational frequencies reported in Raman spectra of the low-temperature solid.31 The four frequencies include a previously unreported band near 50 cm-' that is assigned to the out-of-plane skeletal motion of the H F chains. Similar frequencies are found for HF oligomers in infrared spectra of vapor6 and mat r i x - i s ~ l a t e dsamples ~~ and of solid-state samples.33 (31) A. Anderson, B. H. Torrie, and W. S.Tse, Chem. Phys. Lett., 70, 300 (1980). (32) D. F. Hamill and R. L. Redington, to be published. (33) J. S. Kittleberger and D. F. Hornig, J. Chem. Phys., 46, 3099 (1967).
Crystal frequencies31 at 50, 287, and 363 cm-' are therefore considered as apt group frequencies for the low frequency modes of both the cyclic and chain oligomers. The skeletal bending modes for cyclic oligomers with even q values are partitioned equally between the 187 and 50cm-' frequencies, while the extra mode for oligomers with odd q is placed at 187 cm-l. The nominal F-F stretching modes are similarly partitioned between 363 and 187 cm-l. The FH-F bending modes, which contribute little to the entropy, are all given the value 600 cm-' as a compromise for the four crystal fundamentals in the 550-950-cm-' range. Configurational entropy for the cyclic oligomers is R In [ ( q - 1)!/2] when the H atom positions, which must be considered separately, are not included. As seen in Table 111, the entropies calculated this way for q I5 are in close agreement with the values determined by data fitting. The simple group frequency concept seems to fail for the cyclic tetramer, which is planar, and which can be expected to lack the very low frequency skeletal modes. Its entry in Table I11 results from shifting one 50-cm-' skeletal mode to 187 cm-' as an indication of the trend of its skeletal vibrations towards higher frequencies. The HF oligomers possess a double minimum potential energy surface because of the two energetically equivalent H atom positions. Hollenstein, Bauder, and Gunthard34 recently studied the problem -of calculating partition functions for nonrigid molecules that possess similar multiple conformations. They present, with numerical examples, several ways to approximate the partition function associated with the coupled rotation-nonrigid vibration motion. Use of their crudest approximation is necessary here: i.e., a symmetry number of unity is taken with harmonic vibrations and the equilibrium molecular geometry of one conformation. Statistical entropies for the chain oligomers were calculated using the same four group frequencies plus the values 363 and 187 cm-' for the pendant H atom. However, the entropies generated by the constraint formulas for the chain oligomers are very large and to reproduce them for tetramer and higher oligomers as accurately as (34) H. Hollenstein, A. Bauder, and H. H. Giinthard, Chem. Phys., 47, 269 (1980).
The Journal of Physical Chemistry, Vol. 86, No. 4, 1982 557
Vapor Analysis of HF
for the cyclic sequence reasonably requires a larger proportion of the low-frequency modes. The nominal Fa-F torsions are all assigned the value 50 cm-l for the chain oligomers and the nominal FFF angle bend modes are divided equally between 50 and 187 cm-l for even q, with the extra mode for odd q placed at 50 cm-'. In addition, the extra F-F stretch for odd q molecules is placed at 187 instead of 363 cm-l. As noted for the cyclic tetramer, the group frequency pattern used for the large oligomers yields high entropies when applied to small oligomers that lack the lowest vibrational frequencies. The entropy values listed in Table I11 use 363 and 187 cm-' values for the dimer calculation, and two 363 and three 187 cm-' values for the trimer. Argon matrix-isolation spectra have been obtained32for HF and bands a t 3828,3702,561, and 381 cm-' appear to be dimer fundamentals. In addition, a dimer band has been reported in the v a p ~ r at ' ~381 ~ ~cm-'. ~ These values, together with the group frequency of 187 cm-' and the observed geometry: yield S2"= 256 J / K mol and AH2" -U, = 3.2 kJ/mol. For comparison, the above HF stretches used with the low frequency set 600,600,363, and 363 cm-' yields 250 J / K mol and 4.4 kJ/mol, while the low frequencies 588, 519, 266, and 171 cm-' calculated using a 6-31G basis set by Curtiss and PopleMyield 260 J / K mol and 3.2 kJ/mol. These values are all in the range AH2" - U2= 5 f 2 kJ/mol that is estimated as resonable from the virial coefficient curves obtained by data fitting. The best statistical entropy, 256 J / K mol, is about 7 J / K mol larger than the observed value of 249 J / K mol. Previous experimental AS2" values yield much larger S2"values, e.g., to 280 J / K mol, but the present 7 J / K mol disagreement is still poor. The discrepancy must reside in the need for more critical data for estimating the dimer thermodynamic properties and in the approximations used in the simple statistical calculation. It appears unlikely that the lowest dimer vibrational frequency is much higher than that used in the statistical calculation (187 cm-l). The difference between the enthalpy and the dissociation energy is written in general form as AH9" - U, = (4 - 7q/2) + E,, + ZP,,where E,, is the vibrational energy of (HF),, ZP, is the zero point energy difference between (HF), and q monomers, and translational and rotational equipartition of energy is assumed. The monomer vibrational frequency is 3962 cm-'. The AH," - U, values calculated for the cyclic oligomers using the above four group frequencies with 3426 and 3280 cm-l for the (hydrogen bonded) HF stretches21B2are also listed for reference in Table 111. E. Virial Coefficient Values. The second virial coefficients for HF are discussed in part 1,18and a plot of the present experimentally estimated Bll (monomer-monomer) curve is given there along with several calculated curves. The experimental curve is written B,, = &-a(t-19.5) + Be-@(t-19.6) (26) which is intended to approximate Bll from the HF boiling point of 19.5 "C to about 200 "C. At temperatures above 100 "C the equation is dominated by the B,P exponential while the A,a exponential begins contributing significantly at about the same temperatures that oligomer populations start to build up. Sets of B and /3 values were pre-selected and held fixed during the 14 parameter least-squares refinements. The B,P exponential curve allows an estimate of the hydrogen bond energy, U2,to be made as it matches classical, rigid molecule, virial coefficient points calculated (35) D. F. Smith, J. Chem. Phys., 48,1429 (1968).
(36)L. A. Curtiss and J. A. Pople, J . Mol. Spectrosc., 61, 1 (1976).
TABLE IV : Some Nonideality Parameters for HF Oligomers Bll,adm3/mol B,,, dm3/mol B,,, dm3/mol B , , dm3/mol B , g dm /mol B' = T(dB/dT),, dm3/mol 71
71 7 6
56°C
32°C
19.5"C
-0.789 -0.263 -0.061 -0.010 -0.439 -2.457 0.923 0.973 0.994
-1.817 -0.606 -0.140 -0.022 -0.824 -2.703 0.925 0.974 0.994
-3.250 -1.083
-0.250 -0.040 -1.313 -3.373 0.916 0.970 0.993
Virial coefficients calculated from eq 26 and 27. Parameters: A = -2.476 dm3/mol,(Y = O.O6239/"C, B = -0.774 dm3/mol, p =- -O.O1010/"C, 7 = -0.3662. B, B ' , and 7 are calculated at saturation pressures using eq 1 0 and 1%. a
.F I-
[HFI
t,
/
/ I
19.5"
ATM
1
2
3
Flgure 1. Calculated HF monomer partial pressures plotted as a function of total pressure. The vertical strikes at each saturation pressure give the contribution of hexamer to the oligomer partial pressures.
at 275 and 450 K using the KMO potential function scaled to various U2 values.le The KMO function reproduces several experimental features of the dimer, and it is based on a large basis ab initio calculation of the HF-.HF interaction energy. Constant A in eq 26 is held fixed during the least-squares refinements, but cr is optimized. A AHH,"value is obtained from the general least-squares data fitting refinement for each Bll curve, and the most acceptable least-squares data fits yield AH,"- U2= 5 f 2 kJ/mol, where U2is estimated from the B,P exponential as indicated. The best fits occur in the range with Bll (19.5 "C) = -3.25 dm3/mol f 10% and Bll calculated from the unscaled KMO function found to be satisfactory for representing the B,/3 exponential. The Bll and AH," values are strongly coupled, with large Bll values at 19.5 "C yielding small AH2" values. The optimum choice of AH2" = -17.9 f 1 kJ/mol correlates with the above range for Bll. The B virial coefficients and their evaluation is also discussedrin part 1.18 They are parameterized to yield B12 = 0.333B11, with the complete set of B,, values given by B,, = BlleY(q+')
(27)
The value for y is therefore 0.3662. The quality of the data fit is not very sensitive to y or to other manipulations of the B,, virial coefficient set; however, freely optimized data fits tend to minimize the B,, values and to make them positive. In lieu of critical data able to fix their values more accurately, they are given the modest values calculated from eq 27 as preferable to zero or to positive values because of the large negative values for Bll of HF and for many other molecules reported in the literat~re.~' (37)J. H. Dymond and E. B. Smith, "The Virial Coefficients of Clardendon Press, Oxford, 1969.
Gases",
558
The Journal of Physical Chemistry, Vol. 86, No. 4, 1982
Redington
TABLE V : Calculated and Observed“ Properties of Saturated HF Vapor Z,, calcd Z,, calcd Z , calcd Z , obsd -SE, calcd, J / K mol -SE. obsd -HE, calcd, kJ/mol -HE, obsd M , calcdb AHcor(eq 15), calcd, kJ/mol Scor(eq 19), calcd, J/K mol
56°C
44°C
38°C
32°C
26°C
23°C
2.594 1.059 2.747 2.906 61.23 58.49 19.27 20.41 4.65 0.29 1.25
2.847 1.056 3.005 3.045 65.42 63.09 20.59 21.24 4.70 0.17 0.85
2.968 1.056 3.133 3.125 67.19 65.44 21.13 21.65 4.70 0.12 0.70
3.103 1.057 3.278 3.206 68.93 67.86 21.67 22.06 4.64 0.10 0.60
3.222 1.059 3.410 3.322 70.39 70.37 22.11 22.46 4.56 0.08 0.53
3.285 1.060 3.481 3.376 71.08 71.63 22.32 22.87 4.52 0.08 0.53
a Observed Z, SE,and HE values from Vanderzee and R ~ d e n b e r g . ’ ~ Slope of infrared log are 4.6 i 0 . 1 ; “entha1py”from these pointsis-111 i. 4 kJ/mol vs. -111 kJ/mol (calcd). I
~~
k
I
plotted as a function of total pressure at 32 OC.
I
3.360 1.062 3.567 3.429 71.90 73.12 22.57 22.89 4.45 0.06 0.49
2.7% 2.2% 2.6% 1.7%
vs, log P. Observed values16
I
I
I
I
I
lHFi
0
04 1 0 ATM Figure 2. Calculated HF ollgomer partial pressures P2-P5 and P I ,
I
o!
19.5“C a v % d e v
04
32°C
10 ATM
Figure 3. Calculated HF oligomer partial pressures P5-P,2 plotted as
A few virial coefficient values for the (HF), oligomers are listed in Table IV, where the parameters of eq 26 are also given. In addition, the composite virial coefficients B of eq 4 and several fugacity coefficients are listed for the saturation pressures. F. Oligomer Partial Pressures, Association and Nonideality Factors. Oligomer partial pressures calculated using the parameters of Tables I and IV are plotted in Figures 1-3. As seen from the equilibrium contant values presented in Table I, cyclic oligomers are more important. For tetramers the chain population equals the cyclic population at 56 OC,but there is a 2.5-fold abundance of cyclic tetramers at 19.5 “C. Figure 1 shows the calculated monomer and hexamer partial pressures in a form that emphasizes the substantial contributions of oligomers to the total pressure over the entire 19.5-56 “C temperature range considered here. Figure 2, drawn for 32 OC but scalable from Figure 1 at other temperatures, shows that dimer through tetramer populations are always small in the vapor, and that the dimer and pentamer partial pressures equalize at approximately one-third of the saturation pressures. Figure 3 reduces the scale of Figure 2 20-fold and shows the rising populations of large cyclic oligomers as the pressure is increased to saturation values. As shown in Tables IV and V, the fugacity coefficients yp (eq 12) and the nonideality factor 2, span nearly the same values at all seven temperatures as pressure is changed from zero to the saturation values. The nonideality factors 2, are near 1.06 at the saturation pressures. Interactions involving monomers contribute the most to 2, and Table IV shows that the monomer fugacity coefficients are about 0.92 at the saturation pressures. The oligomer partial pressure distribution arising from the present analysis differs substantially from previous studies in its emphasis on the buildup of populations of large cyclic oligomers at saturation pressures. Realistic
a function of total pressure at 32 O C . The partial pressures decrease monotonically from P , through P , * .
constraints placed on the relative AHqoand relative Sqo values leads to the natural appearing patterns of Pq partial pressures plotted in the figures. The present comprehensive analysis is contrasted with several earlier analyses in the following section, which focuses briefly on results for the separate data sets.
Results: Data Comparisons A. Vapor Density Data. The present analysis began by considering only two or three oligomers because of the apparent success for analyzing vapor density data using an ideal vapor monomer-dimer-hexamer at low to moderate pressures and a simple monomer-hexamer m ~ d e at P saturation ~ ~ pressures. Maclean, Rossotti, and R o s s ~ t t showed i~~ that adding a single oligomer to the dimer and hexamer populations at modest pressures does not lead to a better fit to the vapor density data; contributions from two or more additional oligomers must be balanced for any noticeable changes in the fit. Prior to this work, Briegleb and StrohmeierMhad introduced eight oligomers to interpret the same vapor density data. The present data fitting calculations show inescapably that many oligomers are required when fits to higher pressure vapor density data and to other experimental information are considered. The model overcomes the lack of correlation between monomer-hexamer calculations fit to low pressure3s vs. saturation p r e s s ~ r data. e~~~~ The accuracy of the Argonne vapor density datalgwhich was joined smoothly to the Strohmeier and Brigleb data4 (38)J. N.Maclean,F. J. C. Fbsotti, and H. S. Rossotti,J. Inorg. Nucl. Chem., 24,1549 (1962).
(39)R.W.Long, J. H. Hildebrand, and W. E. Morrell,J. Am. Chem. Soc.,65, 182 (1943). (40) G. Briegleb and W. Strohmeier,2. Elektrochem.,57,668(1953).
The Journal of Physical Chemistry, Vol. 86, No. 4, 1982 559
Vapor Analysis of HF
TABLE VI: Association Factors and Heat Capacities at 3 2 "C (J/K mol)
p,
Z, obsd
0.040 0.158 0.298 0.434 0.473 0.479 0.530 0.591 0.668 0.711 0.735 0.804 0.852 0.949 0.975 1.000 1.132 1.213 1.255 1.540
1.004 1.016 1.036 1.080 1.098
atm
1.110 1.150 1.229 1.338 1.414 1.460 1.621 1.743 1.960 2.026 2.088 2.415 2.605 2.698 3.206
z,
z,
z,, c,
calcda
calcd
1.004 1.015 1.036 1.086 1.111 1.115 1.158 1.228 1.342 1.418 1.465 1.608 1.716 1.946 2.010 2.071 2.388 2.580 2.675 3.278
1.001 1.003 1.003 1.013 1.051 1.073 1.076 1.114 1.177 1.281 1.351 1.395 1.528 1.628 1.843 1.903 1.959 2.258 2.439 2.529 3.103
c
calcd obsdb caEd 1.012 1.022 1.033 1.036 1.036 1.040 1.043 1.048 1.050 1.051 1.053 1.054 1.056 1.057 1.057 1.058 1.058 1.058 1.057
46 103 285
33 71 227
490 632 761
438 591 758
900 874
884 830
836
792
a Twelve Z points between 0.066 and 0.395 atm omitted in tabulation Zcdcd - Z o b d is within k0.002 (the quoted Monomer value =. experimental error) for all 1 2 points. 2.5R =. 20.8 J / K mol. Experimental error is about 15% on the steep rise and fall portions of the C, curve.
by Vanderzee and Rodenberg13is not known. The middle to high pressure Z data are not perfectly fitted by the present model when simultaneous good fits to other types of data are made and a possible contributing factor may reside in the history of experimental difficulties that has accompanied HF research. As a small indication, the adjusted Z values recommended by Vanderzee and Rodenberg for saturation pressures (and used here) are selfconsistent with the other (smoothed) data, but they are larger than the original values published by Jarry and Davis3by about 4% at 19.5 "C and by about 1% at 56 "C. Similarly, Vanderzee and Rodenberg needed to slightly shift the heat capacity data of Franck and Meyers5while seeking self-consistency. The average difference between the calculated and observed Z values is 0.016 for all 204 data points, 0.0011 for , < 1.05 and 0.0037 for the 113 the 82 data points with ,Z Briegleb and Strohmeier points. These are quoted as accurate to *0.002. The Z values were weighted as in the least-squares refinement to favor a fit to the Briegleb and Strohmeier data, particularly at low pressures. The calculated and observed Z values at saturation pressures are included in Table V along with the calculated 2, and 2, values. Calculated 2, Z, Z, and C, values are listed as a function of total pressure at 32 "C in Table VI. Better fits to any portion of the observed data can be made at the expense of worse fits to the other portions, and this table is representative of the quality of the analysis of Z and C data. The close fit to the low pressure Z data helpecf to optimize the AH,"value. B. Heat Capacity Data. At low temperatures the enthalpies of the oligomerization reactions cause the HF heat capacity curves to rise and pass through high maxima as a function of pressure. The heat capacity maximum is almost thirty times the monomer C, value at 19.5 "C. The experimental error on the steep rise and fall portions of these curves is quoted as up to 15%.5 The 61 C, data points used here were read from curves drawn through the tabulations of ref 13. The pressures correspond to those at observed Z values. The C, points were e q d y weighted in the least-squares refinement, but sufficiently lightly that
their combined weighted residuals amounted to about one-tenth those of the combined weighted residuals of the Z data. The present analysis preserves the previously reported importance of the cyclic he~amerization~ to the C, curve, but it emphasizes otherwise completely different oligomerization equilibria. The calculated C, curves rise somewhat slowly at low pressures but may reproduce the experimental maxima closely. They drop off a bit too rapidly after the maxima. The average value of the 61 data points is 553 J / K mol. The average (unweighted) residual for these points is 7.2 J / K mol, or about 1% of the average data point. The largest residual is 83 J / K mol, which corresponds to a 11% error on the steeply falling slope of the C, curve at 19.5 "C, however, there are larger percentage errors than this. Table VI contains the calculated and observed C, values at 32 "C. The heat capacities can be fitted better, but only at the expense of a good fit to the Z values-which appear to be more accurately determined experimentally. C. Excess Entropy and Excess Enthalpy. Calculated values for these quantities are compared with the tabulations of Vanderzee and Rodenberg in Table V, along with the enthalpy and entropy nonideality corrections at saturation pressures as given by eq 15 and 19. Vanderzee and Rodenberg estimate the uncertainty in the H E values at 1% (i.e., 0.2 kJ/mol) and the SEvalues at 0.4 J / K mol ( 0 5 0 . 7 % )at saturation pressures. The calculated values for H E and SEdiffer from the observed values by about 2.6% and 2.2%,respectively. D. Infrared Absorption Coefficients. The infrared analysis is discussed more extensively in part 3.21 Infrared band profiles were studied as a function of T and P by Smith>14 and laser absorbance measurements of the HF stretching region of the oligomers were preformed by Hinchen and Hobbs.16 The latter data were included in the data fitting process by assuming that the infrared absorbance is proportional to Cq=412Cq, where C, is the molar concentration of the (HF), oligomer. Measured absorbance, plotted as log (a)vs. log (P),yields a straight line with slope 4.6 f 0.1. van't Hoff analysis of the intercepts of these lines yields an "enthalpy" value of -111 f 4 kJ/mol. In the present data analysis slopes of log (C412C,)vs. log P, called M in Table 111, were best fitted to the value 4.6 at all seven temperatures and the derived "enthalpy" value was best-fitted to the value -111 kJ/mol. The infrared data were fitted to within the experimental error and this part of the analysis plays an important role in defining the relative partial pressure contributions of the various oligomers. The analysis shows that the slopes 4.6 are due to large oligomer populations rather than to an abundance of tetramer in a very low overall oligomer population. E. Other Data. As was discussed extensively above, the overall analysis favors multiple equilibria with a buildup of substantial populations of flexible cyclic pentamers, hexamers, and higher oligomers as the total pressure approaches saturation values. Small populations of cyclic tetramer and of chain oligomers are always present in the equilibria. The model favors population of chain tetramers comparable to those of cyclic tetramers. The infrared studies of Huong and Couzi15on deuterium-labeled samples were interpreted to favor the presence of chain tetramers in the vapor. Two additional data sets exist that favor strong intermolecular association in HF vapor, but these are not included in the present least-squares data fitting because they require many additional parameters beyond those
580
73e Journal of Physical Chemisfry, Vol. 86, No. 4, 1982
already accepted. The measurements are for thermal conductivity' and electric polarization? Rather similar to the present results, the polarization experiments were analyzed in terms of a general vapor model favoring polar oligomers in equilibrium with more abundant cyclic species.8 In the interpretation, as for the present model, lowering the temperature causes chain closure and relative increases in populations of cyclic oligomers. Quantitative reanalysis of these data is not attempted due to uncertainties such as the chain oligomer dipole moment values, but the results of the original analysis are conceptually in good agreement with the present conclusions. The thermal conductivity data were analyzed using a monomer-hexamer model.41 Again, the results are consistent with a dominating hexamer population, but too many unknown diffusion parameters are required to attempt a more generalized reanalysis of the data here.
Conclusions 1. A general nonideal-associated vapor model for HF vapor is developed that includes (HF),oligomers extending through the dodecamer. Except for trimer and tetramer, the dominant populations of each oligomer are cyclic. Calculated populations with q I4 never achieve large values in the 19.5-56 OC temperature range; however, cyclic oligomers with 5 Iq I12 all buildup substantial partial pressures as the total pressure rises to saturation values. Cyclic hexamer population exceeds the dimer population at total pressures about one-third of the saturated values and it is the most abundant oligomer thereafter. The present model is the first to quantitatively emphasize the very large cyclic oligomers. 2. Standard enthalpies, entropies, and free energies for the oligomerization reactions are determined by extensive weighted least-squares data fitting. These quantities are smooth functions of q as a result of constraints placed on their relative possible values. The dimerization enthalpy determined from this nonideal gas model is -17.9 kJ/mol. This value is lower than previous (ideal gas) estimates, but it is based on the composite HF data and is harmonious with recent theoretical and experimental results reported for the water dimer. The average hydrogen bond energy is found to increase gradually with q (the "additive" effect) and to level off at approximately -28 kJ/mol for the large oligomers. The standard oligomer entropy values are very large, which suggests very low frequencies for their skeletal vibrational modes. The large entropy values agree with other data suggesting fluxional molecules and they can be calculated approximately by using group vibrational frequencies, including one at 50 cm-', obtained from Raman and infrared spectra. 3. The first experimental estimates for the second virial coefficients of HF vapor species are presented. The ex-
Redington
perimental BI1monomer-monomer second virial coefficient curve runs between -0.79 at 56 "C and -3.25 dm3/mol at 19.5 "C (the boiling point of HF). These values are much larger than those observed for HC1 at the same temperature^.^, The analysis suggests that nonideality (as distinct from association) effects in HF vapor in this temperature range are dominated by the monomer-monomer virial coefficients and that there are only small contributions from the oligomers. At saturation pressures 2, is about 1.06, whereas 2 = Z.$, runs from 2.91 (56 "C) to 3.43 (19.5 "C). Water and other substances that possess large second virial coefficients may be experimentally difficult to study because of low vapor pressures at the temperatures needed to realize large vapor nonideality effects. This is no problem for HF; on the other hand, interpretation of the HF data is complicated by its extensive intermolecular association. The set of B,, oligomer virial coefficients is rather insensitive to the available data. Tentative magnitudes for these are proposed that emphasize an increasing repulsive (volume) contribution to B,, as oligomer size is increased, that do not distinguish virial coefficients involving equal numbers of HF units, and which roughly obey the standard "rule of thumb" B,, = (Bq$,,)1/2. 4. The oligomers (HF),-(HF),, provides an extensive family of vapor phase molecules with important theoretical and practical interest. The present indirect but fairly convincing demonstration of their abundance at easily accessible temperatures and pressures should serve to stimulate additional experimental and theoretical research on these molecules. For example, both static and dynamic experiments in conventional apparatus or in molecular beams can be performed to more fully elucidate their properties. The HF oligomers can serve as subjects for studies of the formation and properties of clusters, of large amplitude motion and its vibrational-rotational couplings, and of proton tunneling. The HF system parallels water as a subject for simulation studies of the hydrogen bonded liquid, and past studies have had to bypass the oligomers and move directly from considerations of dimer properties to those of the liquid.
Note Added in Proof. The calculated C, points discussed above are up to 1% smaller than values erroneously used to help choose the 17 parameters. This error is too small to significantly affect any results. Acknowledgment. Portions of this research were supported by AFOSR under Grant AFOSR-78-3616 and HEW under Grant No. 5 RO1 GM 24515-02. Computer time was furnished, by the Computer Center of Texas Tech University. (42)
(41)R. S. Brokaw and J. N. Butler, J . Chem. Phys., 26, 1636 (1957).
B. Schramm and U. Leuchs, Eer. Eunsenges. Phys. Chem., 83,847
(1979).
