J. Phys. Chem. 1981, 85, 3730-3733
3730
agree with2 (sr)mar= 11.15 t 1) where r is the length of the side of the fccub unit cell and s = (4asin e)/X. The expected sin can be calculated from the concentration of diffracting centers and the wavelength of electromagnetic radiation being diffracted. Daly and Hastings did not report the concentration of diffracting spheres directly but they did note that interpreting the observed el on the basis of a bccub structure gave lattice constants which were "typically smaller than those anticipated from the prepared sphere concentrations". They did not report how much smaller but they explained this observation on the basis of evaporation from the preparations, which should have been minimal. We will assume here that the O1 observed, if interpreted on the basis of a bccub structure, gives the concentration of spheres. From this concentration we will calculate the angle which would have been expected from eq 1. Based on these criteria the unit cell length of the bccub structure is a = (N2/2)(h/(sin Ol)DH) (2) where a is the bccub unit cell length (bccub for 110 planes), X is the wavelength, and (sin O1)DH is based on the experimental value of el given by Daly and Hastings. The volume of this unit bccub cell, u3, contains two spheres. At this concentration, the volume containing four spheres (the number of spheres in a fccub unit cell) is 2a3,and the length of a side of the fccub unit cell (r in eq 1)is ( 2 ~ ~ ) ' / ~ . The r in eq 1 is r = (2a3)1/3= [(21/3)(21/2)/2](h/(sind1)DH) (3) Substituting r from eq 3 and the definition of s into eq 1 gives
where (sin Ol)LB is that predicted from eq 1. Equation 4 tells us that the first angles observed by Daly and Hastings are, within a small experimental uncertainty, those expected for a loose fccub lattice. The first angles are therefore compatible with either a crystalline bccub structure or a loose fccub structure. Though not included in ref 2 the second maximum in I vs. rs for a loose fccub lattice occurs at (rs)- = -20.5. The ratio of (sin e2),/(sin el)LBis therefore 20.5/11.15 = 1.84. For the three entries in DH Table I the average ratio of ~ with a standard deviation of (sin sin 8 1 ) is~ 1.79 0 = 0.08. The ratio is within one standard deviation of the loose fccub value. The ratio is also within one standard deviation of the incorrect value 31/2but well outside the correct value of 21/2 for a bccub lattice. (3) Bahe, L. W.; Parker, D. J. Am. Chem. SOC.1975, 97, 5664-70. E d W s Note: I%. Da& and Hastings have expressed their agreement wlth the conclusions drawn In this comment. Department of Chemistry UniversW of Wisconsin- Milwaukee Milwaukee, Wisconsin 5320 1
Lowell W. Bahe
was found that the NH4HS04solution densities calculated from the data for the molarities of NH4HS04,[NH4HS04], and water, [H20],in Table I of Irish and Chen2 differed substantially from available data. The objective of this communication is twofold: (a) to determine the source of this discrepancy, and (b) to ascertain whether the extent of bisulfate dissociation in NH4HS04solutions is accurately represented by the data in Irish and ChenS2The validity of the NH4HS04molarity and extent of bisulfate dissociation data in that table will be determined by comparing predicted and measured NH4HS04solution densities. Although we focus on resolving the NH4HS04solution density discrepancy, we also illustrate a general technique by which the densities of aqueous ammonium sulfatesulfuric acid solutions can be calculated from dissociation data. The molal volume of an electrolyte, &, can be calculated by do
where do is the density of water, [Ei] is the electrolyte molarity, d is the solution density, and Ma is the electrolyte molecular weight? For a mixture of electrolytes the mean molal volume, a, can be predicted by Young's rule
where 4i for each electrolyte is taken at the same molar ionic strengths4 From eq 2, a generalized expression can be derived for predicting the molal volume of an ammoniated sulfate salt, (P(("~)x(H),(SO~),+,/~)
where x , y are the ammonium and hydrogen stoichiometric coefficients of the sulfate salt, P = [NH4+]/([NH4+]+ [",SO4-]), and y = [H+]/([HSO,] + [H+]). The total sulfate salt concentration conveniently cancels out of the numerator and denominator in eq 3. The ionic strength, I,, of the sulfate solution is given by 1, =
(
4Px + 4YY - l ) ~ ~ N H 4 ~ x ~ H ~ , ~ s o (4) 4~x+~/21 X + Y
Received: March 10, 1981; In Final Form: May 27, 1981
Typically, ionic molal volume data are not available except for very dilute solutions so the molal volumes of electrolytic salts must be used. Equation 3 can be rewritten as Prediction of the Density of Ammonium Bisulfate Solutions
Sir: The concentration dependence of the density of ammonium bisulfate solutions has recently been t3tudied.l It 0022-3654/81/2085-3730$01.25/0
(1) Stelson, A. W.; Seinfeld, J. H.Atmos. Enuiron. In press. (2) Irish, D. E.; Chen, H.J.Phys. Chem. 1970, 74, 3796. (3)Millero, F.J. Chem. Rev. 1971, 71, 147. (4) Young, T.F.; Smith, M. B. J. Phys. Chern. 1954, 58, 716.
0 1981 American Chemical Society
The Journal of Physical Chemistty, Vol. 85,No. 24, 198 1 3731 ,
(
,
,
I
I
/
,
,
l
i
l
,
l
I
- ( N H 4 1 ~ S 0 4 -NH4CI - H C I
-
.--. It
a
+ -
I *I
-&
-
a2)$(H+,"4S04-)
(5)
where al and a2 are constants. Equation 5 was written including terms for NH4S04-ion pairs. The thermodynamic literature disagrees over the presence of NH4S04ion pairing. The Raman spectroscopy work of Irish and Chen,2 Chen and Irish,5 and Young et a1.6 indicates the amount of NH4S04-ion pairs is small via their internal consistency checks. Indirectly, Reardon' infers a significant amount of NH4S04-ion pairing exists. In agreement with the Raman data, we will assume the amount of NH4S04- ion pairs is small. Thus, p = 1 and a2 = 0. Equation 5 simplifies to
Figure 1. $ (H+,H+,SO?-) ionic strength dependence.
YOUNG ET AL
+
CHEN b IRISH KLOTZ b ECKERT
L
8- 4 0 1
34 0.01
I
1
1
0.1
i
l
l
I
I
I
,
10.0
1.0
1"
With sulfuric acid solution densities, sulfuric acid dissociation data, and an expression for the hydrogen sulfate molal volume, $(H+,H+,S042-),the hydrogen bisulfate molal volume, $(H+,HS04-), can be calculated by @(H2S04)+ (27 - l)$(H+,H+,SO?-) 2 - 2y (7) where @(H2S04)is the sulfuric acid molal volume. Equation 7 is derivable from eq 6 with al = 0, x = 0, and y = 2. The hydrogen sulfate molal volume can be determined by an approach analogous to that of Lindstrom and Wirth8 d(H+,HS04-) =
4(H+,H+,S042-)= $(X+,X+,SO?-) - 29(X+,C1-) + 2$(H+,C1-) (8) where the molal volumes, $( ), are determined at constant ionic strength, $(X+,X+,S042-),$(X+,Cl-), and $(H+,Cl-) are the molal volumes of X2S04,XC1, and HCI, respectively, and X+ is the Na+, NH4+,or K+ cation. (It is assumed that NH4C1, NaCl, KC1, HCl, (NH4)2S04,Na2S04, and K2S04totally dissociate within the concentration range of interest.) The (NHd2S04and H2S04molal volumes can be calculated with density data from Beattie et al.: the K 8 0 4molal volume from Dunn'O and Wirth," and (5) Chen, H.; Irish, D. E. J. Phys. Chem. 1971, 75, 2671.
(6) Young, T. F.; Maranville, L. F.; Smith, H. M. In "The Structure of Electrolytic Solutions";Hamer, W. J., Ed.; Wiley: New York, 1959; Chapter 4. (7) Reardon, E. J. J. Phys. Chem. 1975, 79, 422. (8) Lindstrom, R. E.; Wirth, H. E. J. Phys. Chem. 1969, 73, 218.
Flgure 2. 9 (H+,HSO.,-) ionic strength dependence. The solid line is the least-squares fit to the data of Kiotz and Eckert,*' I, I0.14, and Irish and Chen.*
the Na2S04molal volume from Dunn,'O Geffcken and Price,12 Beattie et a1.,9 and Gibson.13 The NH4Cl molal volume can be calculated with density data from Beattie et al.? Pearce and Pumplin,14and Stokes,15the NaCl molal volume from Millero,16 Wirth,17 Dunn,l* Kruis,lg and Beattie et the KC1 molal volume from Wirth,ll Geffcken and Price,12Dunn,18Kruis,lg and Beattie et al.: and the HCl molal volume from Beattie et al.,9 Wirth,l' Millero et and Dunn.lo The ionic strength dependences of the molal volumes can be obtained by fitting polynomials to the data. The ionic strength dependences of $(H+,H+,S0,2-)calculated from the NH4+,K+, and Na+ cation data are shown in Figure 1in addition to the tabulated values taken from Lindstrom and Wirth8 and Klotz and Eckert.21 The K+ and Na+ data agree quite well whereas the NH4+data disagree with the others at low ionic strengths. The three different c#I(H+,H+,SO~~-) ex~
~~
(9) Beattie, J. S.;Brooks, B. T.; Gillespie, L. J.; Scatchard, G.; Schumb, W. C.; Tefft, R. F. In "International Critical Tables of Numerical Data, Physics, Chemistry and Technology",Washbun, E. W., Ed.; McGraw-Hill: New York, 1933; Vol. 111, p 51. (10) Dunn, L. A. Trans. Faraday SOC.1966,62,2348. (11) Wirth, H. E. J. Am. Chem. SOC. 1937, 59, 2549. (12) Geffcken, W.; Price, D. 2. Phys. Chem. B 1934, 26, 81. (13) Gibson, R. E. J. Phys. Chem. 1927,31,496. (14) Pearce, J. N.; Pumplin, G. G. J. Am. Chem. SOC. 1937,59,1221. (15) Stokes, R. H. Aust. J. Chem. 1975,28,2109. (16) Millero, F. J. J. Phys. Chem. 1970, 74, 356. (17) Wirth, H. E. J. Am. Chem. SOC.1940, 62, 1128. (18) Dunn, L. A. Trans. Faraday SOC.1968,64,1898. (19) Kruis, A. Z. Phys. Chem. B 1936, 34, 1. (20) Millero, F. J.; Hoff, E. V.; Kahn, L. J. Solution Chem. 1972,1, 309. (21) Klotz, I. M.; Eckert, C. F. J. Am. Chem. SOC.1942, 64, 1878.
3732 The Journal of Physlcal Chemistry, Vol. 85, No. 24, 1981
pressions will be used to calculate the sensitivity of 4(H+,HS04-)to d(H+,H+,SOt-): With the sulfuric acid dissociation data of Chen and Irish5 and Young et a1.,6 $(H+,HSO4-)can be calculated from eq 7. The 4(H+,HS04-)ionic strength dependences from Lindstrom and Wirth8 and Klotz and Eckert21are shown in Figure 2 in addition to the 6(H+,HSO4-)ionic strength dependences calculated from the data of Chen and Irish5 and Young et al.6 The individual data sources show distinctively different 4(H+,HS04-)ionic strength trends. The error bars in Figure 2 represent the standard deviation of 4(H+,HSO,) evaluated from the K+, Na+, and NH4+data and the points are the mean values. Below an ionic strength of 0.14, the results of Klotz and Eckert21 were recalculated by using the expressions for 4(H+,H+,SOt-)in this paper. The data in Figure 2 were evaluated in order to ascertain the best representation for 4(H+,HS04-). The data of Chen and Irish? Young et a1.,16 and Klotz and Eckert,21 I , I 0.14, are thought to be the best quality. Above an ionic strength of 0.14, the Klotz and EckertZ1data are poorer since they assumed the classical ionization constant to be invariant. Lindstrom and Wirth8 assumed 4(H+,HS04-)= 4(H+,H2P04-)plus an adjustment factor and thus, +(H+,HSO4-)shown in Figure 2 is actually 4(H+,H2P0,) modified. The best representation for 4(H+,HSO,) is obtained from the data of Chen and Irish6 and Klotz and Eckert,21 I, I 0.14, since these data smoothly extrapolate to each other, whereas, the data of Young et aL6 and Klotz and Eckert,21Z, I0.14, are discontinuous. The data of Chen and Irish5 and the recalculated results of Klotz and Eckert21were fit via a leastsquares technique to a polynomial 4(H+,HSOL) = 33.25 + 9.031,1/2- 2.7731, + 0.41341,3/2 (9)
where the standard deviation is f0.12. The solid curve in Figure 2 is eq 8 and will be used to represent the ionic strength dependence of 4(H+,HS04-). To calculate the NH4HS04molal volume, we must know the ammonium bisulfate molal volume, 4(NH4+,HS04-). The ammonium bisulfate molal volume can be calculated from +(NH4+,HS04-) = +(H+,HS04-) @(NHd+,Cl-)- 4(H+,C1-) (10)
Comments 7
-
1
1.7
I
I
I
I
I
I
6.0
7.0
8.0
DATA IRISH 6 CHEN
1.6
1.5
A
TANG
A
BEATTIE ET AL.
THEORETICAL PREDICTIONS 0 IRISH b CHEN
T E
1.4
0
,--1.3 t v) z
x
1.2
1.1
I .o a9
0.0
1.0
2.0
3.0
4.0
NH, HSO,
5.0
9.0
MOLARITY
Flgure 3. Density of NH4HSO4solutions. The lines are the authors' representation of the data of Tang" and Beattie et a1.8 and Irish and Chen.*
From the ionic strength dependences of 4(NH4+,H+,SOz-),4(H+,HSO,), and 4(NHd+,HSO4-)and bisulfate dissociation data for NH4HS04solutions, the molar volume of NHJIS04, @(NH4HS04), and the density of NH4HS04solutions can be calculated. From eq 12 @(NHIHSO4)= Y~(NH~+,H+,S+ O ~(1~-- )Y ) ~ ( N H ~ + , H S O(13) ~-) With the dissociation data of bisulfate in ammonium bisulfate solutions of Young et al.6 and Irish and Chen? the molal volume of ammonium bisulfate can be calculated by using eq 13.22 After the ammonium bisulfate molal volume has been calculated, the density can be determined from [("4),(H)y(~04),+y,21 d= 1000 (M(NH3,(H)~(SOJ.+,jn-
do@((NH4),(H)y(s04)~+y/2)) + do (14)
+
The ionic strength dependence of +(NH4+,HSO;) can be determined with the 4(H+,HS04-)ionic strength dependence calculated from the sulfuric acid dissociation data of Chen and Irish6 and Klotz and Eckerta21 Analogous to the approach of Lindstrom and Wirth: 4(NH4+,NH4+,S04") and 4(H+,H+,SOt-)can be replaced by 4(NH4+,H+,S0d2-)= '/2(4(NH4+,NH4+,S042-) + 4(H+,H+,S042-))(11) By substitution of eq 11 into eq 6 and solving for al, one obtains
The..de;sity hasrbeen calculated for 0.206-6.571 M or 0.39-9.76 ionic strength ammonium bisulfate solutions. The solubility of ammonium chloride is 5.70 M, and thus, the molal volume data of ammonium chloride had to be smoothly extrapolated to 9.76 M.14 The calculated solution densities are compared to the densities calculated from the [NH4HS04]and [HzO]data in Table I of Irish and Chen2 and the measurements of Tang23and Beattie et al.g in Figure 3. The agreement between the predicted NH4HSO4solution densities from molal volume and dissociation data and the data of Tang23and Beattie et al? is good, whereas the solution densities calculated from the [NH4HS04] and [H20]data in Table I of Irish and Chen2 disagree with the theoretical predictions and the experimental measurements. Thus, the water molalities in the paper of Irish and Chen2 must be in error, since the solution (22) For Irish and Chen? the dissociation data of bisulfate in ammonium bisulfate solutions were taken directly from Table I. The polynomial presented in Chen and Irish6 was not used since it inaccurately represents the data of Irish and Chen2 at high NHdHSO4 molarities. (23) Tang, I. N., Brookhaven National Laboratory, personal communication.
3733
J. Phys. Chem. 1981, 8 5 , 3733-3734
densities calculated from the ammonium bisulfate molarities, [NH4HS04],agree well with the theoretical predictions from the data of Young et a1.6 and the data of Tang23and Beattie et al.9 Recent communication from Irish24included a polynomial fit to the NH4HS04density data used in Irish and Chen2 d = 0.99671 0.604706 X 10-l[NH4HS04] - 0.108981 X 10-2[NH4HS04]2+ 0.106394 X 10-4[NH4HS04]3(15)
+
Equation 15 agrees with the data of Tang23and Beattie et a1.: in support with our observation that the water molarities in Table I of Irish and Chen2must have been calculated incorrectly. The NH4HS04solution densities predicted from eq 15 agree with the theoretical predictions based on dissociation and molar volume data within 0.40% for Irish and Chen2 and 0.65% for Young et ala6 In addition to isolating the error source in the paper of Irish and Chen,2we have illustrated a technique by which the densities of ammoniated sulfate solutions can be calculated from dissociation data or vice versa. Specifically, this work discussed ammonium bisulfate solutions but eq 12 can be applied to any aqueous ammonium sulfatesulfuric acid solution. (24) Irish, D. E., University of Waterloo, personal communication.
A. W. Stelson J. H. Selnfeld”
Department of Chemical Engineering California Institute of Technology Pasadena, California 9 1 125
Recelved: April 14, 1981; In Flnal Form: July 15, 1981
Intermediates in the Gas-Phase Disproportionation of HO, Radicalst
Sir: I t is now well established from recent kinetics studi e ~ l that - ~ the gas-phase reaction HO2 + HO2 H202 + 02 (1) is more complex than a simple hydrogen atom transfer as first The observed pressure and temperature dependence of the rate constant has prompted Cox and Burrows2to suggest that the reaction proceeds through an intermediate, the H204 molecule, previously identified in the frozen products from electrically dissociated water vapor.g Of the two possible pathways for the decomposition of that intermediate, I or 11, the former is now ruled +
H
H
I
I1
out following the l80 isotope studies of Pu’iki et al.1° (1) E. J. Hamilton and R. R. Lii, Int. J. Chem. Kinet., 9,875 (1978). (2) R. A. Cox and J. P. Burrows, J. Phys. Chem., 83, 2560 (1979). (3) B. A. Thrush and J. P. T. Wilkinson, Chem. Phys. Lett., 66, 441 (1979). (4) R. R. Lii, R. A. Gorse, Jr., M. C. Sauer, Jr., and S. Gordon, J.Phys. Chem., 83, 1803 (1979). (5) S. N. Foner and R. L. Hudson, J. Chem. Phys., 36, 2681 (1962). (6) T. T. Paukert and H. S. Johnston, J.Chem. Phys., 56,2824 (1972). (7) C. J. Hochanadel, J. A. Ghormley, and P. J. Ogren, J.Chem. Phys., 56. 4426 ~ ~ (1972). - --,(8) A. C. Lloyd, Int. J. Chem. Kinet., 6, 169 (1974). (9) P. A. Gigugre and K. Herman, Can. J. Chem., 48, 3473 (1970). (10) H. Niki, P. D. Maker, C. M. Savage, and L. P. Breitentach, Chem. Phys. Lett., 73, 43 (1980).
--.
\--
However, as noted by these authors, their results cannot distinguish between transition state I1 and a hydrogenbonded cyclic dimer (HO,),. Pending further experimental data a survey of the indirect evidence based on thermodynamics, kinetics, and structure shows that the cyclic dimer is by far a more likely intermediate than the covalent H204molecule, as regards both formation and decomposition. On thermodynamic grounds the formation of a cyclic (HO,), dimer resembles that of double molecules in the vapor of formic and acetic acid,ll with an enthalpy of dimerization of some 15 kcal mol-l. This value agrees nicely with the estimate from the kinetics of the HOz self-reaction.I2 I t also fits almost exactly twice the hydrogen bond energy calculated by Hamilton for the H20HOBc0mp1ex.l~ In contrast, the covalent H2O4 molecule is expected to be unstable with respect to dissociation into two H02, as shown by Benson14 from bond energy considerations. Similarly, calculations of the thermodynamic functions of the H2O4 molecule1s lead to an estimate of 6.5 kcal mol-l for its free energy of formation from H02 radicals, as compared with -4 kcal mol-l for the dimer, again by analogy with the double molecules of carboxylic acids.“ As for the decomposition process, the (HO,), complex is clearly favored by a rather low bond-dissociation energy, D(H-02) = 47 kcal mol-’, due to the favorable reorganization energy of the fragment^.^ In contrast, for the tetroxide D(H-O,H) must have nearly double that value again by analogy with hydrogen peroxide16for which D(H-02H) = 90 kcal mol-’. From the viewpoint of kinetics, the formation of a cyclic (H02)2dimer in the gas phase may be seen as a “sticky collision”following suitable 0-H-0 approach and dipole orientation. On the contrary, the covalent H2O4 molecule can only be formed in the condensed phase, thanks to intermolecular forces; that is by the “cage effect” at the surface of a glassy matrix, following the trapping at very low temperature (below 80 K) of a pair of adjacent H 0 2 radicals stabilized by hydrogen bonding.” This explains why it decomposes so readily at the slightest change of environment, for instance, upon devitrification of the glassy deposit at 160 K. In the gas phase, decomposition of the cyclic (HO,), complex is a three-center process, hence more likely than the four-center process for transition state 11. Yet another transition state, 111, yielding H20 and 03,
,-8”-y ‘0-0
11I
considered by Niki et al.,l0was effectively negated by their experimental results. Although thermodynamically possible, it appears still less likely with the (H02)2complex than with the H2O4 molecule which already contains the chain of three covalently bonded oxygen atoms. Lastly, from the structural standpoint the geometry of the hydroperoxyl radicalla is particularly suited to the formation of a strong dimer (IV)because of nearly colinear (11) G. C. Pimentel and A. L. McCellan, “The Hydrogen B o n d , W. A. Freeman, San Francisco, 1960, p 210. (12) B. A. Thrush, Acc. Chem. Res., 14, 116 (1981). (13) E. J. Hamilton, J. Chem. Phys., 63, 3682 (1975). (14) S. W. Benson, J. Chem. Phys., 33, 306 (1960). (15) P. A. GiguBre, Trans. N.Y. Acad. Sei., Ser. II, 34, 334 (1972). (16) P. A. Gigugre, “Peroxyde d’hydroggne e t polyoxydes d’hydroggne”, Val. 4 of “ComplBments au nouveau trait6 de chimie minerale de P. Pascal”. Masson, Paris, 1975, pp 67, 78, and 164. (17) J. L. Arnau and P. A Gigugre, Can. J. Chem., 53, 2490 (1975). (18) J. F. Ogilvie, J. Mol. Struct., 31, 407 (1976).
0022-3654/81/2085-3733$01.25/0 0 1981 American Chemical Society
