Dielectric Relaxation of 1:l Electrolytes in THF and DEC
The Journal of Physical Chemistty, Vol. 82, No. 17, 1978
1943
Dielectric Relaxation of Some 1:l Electrolytes in Tetrahydrofuran and Diethyl Carbonate D. Saar, J. Brauner, H. Farber, and S. Petruccl” Departments of €iectrica/ Engineering and Chemistry, Polytechnic Institute of New York, Brooklyn and Farmingdaie Campuses. New York 11201 (Received March 23, 1978)
Complex permittivities in the frequency range of 0.3-67 GHz are reported for the systems 0.1 M LiN03,0.05 M sodium picrate, 0.1 M Bu4NN03in THF at 25 “C and for the systems 0.1 M LiC104,and 0.1 M LiSCN in diethyl carbonate (DEC) at 25 “C. The permittivities of the electrolyte solutions can be described within experimental error by the sum of two Debye relaxation processes. The high frequency process is comparable to the one observed for the pure solvents while the low frequency process is due to the presence of the electrolytes. Apparent dipole radii a, for the electrolytes in the THF solutions are calculated from the Debye relation using the macroscopic viscosity of the solvent. Charge to charge separation distances a, are calculated from the Bottcher relation which relates the change in dielectric strength (eo - eml), for a dipolar relaxation process, with the apparent dipole moment. These two parameters a, and up determined for the present systems and two previous systems (LiC104and NaC104 in THF) are linearly related to each other (correlation coefficient r2 = 0.89). The a,‘s also correlate linearly (9= 0.91) with the sums of the crystallographic radii. These relations reinforce our previous conclusions that the solute relaxation process in THF is mainly due to the rotation of ion-pair dipoles. For DEC solutions the appearance of a single relaxation process for the solute (instead of a distribution of relaxation times as the presence of two molecular species would suggest) is discussed in terms of the structure of the quadrupoles. Recent Raman and IR spectra in the literature suggest a centrosymmetric structure for (LiSCNI2 in ethers. This structure if Dresent for the carbonate solutions would Dredict the absence of a auadrupole dielectric relaxation process in accord with our results in dimethyl carbonate (DMC) and DEC.
Introduction In the past, complex permittivities of electrolytes in solvents of low permittivity have been measured in order to characterize the dynamic behavior of the electrolytes under the influence of microwave electric fields and to study the relaxation behavior of the solute molecular species. The general picture which had emerged from the previous studies carried out in this lab~ratoryl-~ was that the solute dielectric relaxation process was mainly caused by the rotational relaxation of the ion-pair dipoles. These species were also present in the largest concentration. However, these studies were confined to some alkali perchlorates in THF1i2 and to lithium salts in DMC.3 It was of interest to try to generalize the above conclusions for electrolytes of different nature and structure in the same solvent. Therefore, we have now extended our studies to include LiN03, sodium picrate, and Bu4NN03 in T H F in the same concentration range as used previously. The study of Li+ salts in DMC had shown a correlation between the change in the dielectric strength of the system eo - em1 due to the solute relaxation process and the calculated concentration of ion-pair dipoles, C,. It was of interest to extend this work to another solvent, namely, DEC. It has been shown4that although DEC is practically isodielectric with DMC, Li+ salts form ion-pair dimers or quadrupoles to a much larger extent in DEC. To this end we have studied LiC104 (0.1 M) and LiSCN (0.1 M) in DEC by dielectric relaxation methods to ascertain whether the structural information inferred by Raman and IR spectra is reflected by the molecular dynamics of the systems. Experimental Section The equipment and procedure have been described e1sewhere.l~~ T H F (Fisher) was distilled under nitrogen over liquid potassium with only the middle portion collected. DEC (Aldrich) was distilled under reduced pressure. The LiN03 (Fisher) was dried in an oven at 101 “C to constant weight. 0022-3654/78/2082-1943$01 .OO/O
The sodium picrate (Kodak) and Bu4NN03(Kodak) were dried under -0.1 Torr at 30 “C for 1 week. The LiC104 and LiSCN were purified as described previ~usly.~
Results Figures 1 and 2 show the real part of the complex permittivity E’ and the imaginary part of the complex permittivity E” plotted vs. frequency f. Both figures include a Cole-Cole plot of e’’ - E,” vs. E’,where E,” represents the (dielectric) loss due to ionic conduction. The data are for the systems sodium picrate (0.05 M), LiN03 (0.1 M), and Bu4NN03(0.1 M) in T H F a t 25 “C. The filled in points at f = 137 GHz are literature data for pure THFn5 In both figures the solid lines represent the functions E’ and E” vs. f as calculated from the sums of two Debye functions: 6’ = Em2 + (€0 - f m l ) / [ l + ( f / f R ~ ) ~+] ( E m 1 - Eco2)/[1 + (f/fR,)’] (1) -
=
(€0
cml)(f/fR1)/[l (6-1
+ ( f / f R ~ ) ~+ ]
- tmz)(f/fRz)/[l
+ (f/fR,)’] (11)
In the above eo is the static permittivity, eml is the permittivity at f >> fR1, em2 is the high frequency permittivity (f >> fRz), and fR1 and fR2 are the relaxation frequencies. The parameters €0, em1, E,’, fR1, and fR2are given in Table I, together with the fitting procedure. E: is calculated from the expression E,” = 1.8 X 10l2X / f where the conductivity of the solutions, X,was measured separately a t 25.00 “C. The conductivity apparatus consisted of a General Radio impedance comparator (equipped with an impedance reference arm), a Kraus conductance cell, and a precision thermostat as described earlier.6 The values of X are also given in Table I. In addition, the parameters eo, E,, and fRzused to fit the data for pure DEC via one term Debye relaxation functions [eq I11 and IV] are presented in Table I. The fitting of E’ = E m + (€0 - €,)/[I + (f/fR)’] (111) f”
=
(€0 - E m ) ( f / f R ) / [ 1
63 1978 American Chemical Society
+ (f/fd2]
(IV)
The Journal of Physical Chemistry, Vol. 82, No. 17, 1978
1944
Petrucci et ai.
TABLE I : Calculated Relaxation Parameters e o , E , ~ , e,?, f R , and f R According to Eq I and 11 for LiNO, (0.1 M), Sodium Picrate (0.05 M ) , and Bu,NNO, (0.l1M) in TfiF at 25 ‘Ca solvent THF THF THF
electrolyte LiNO, (0.1 M ) NaPic (0.05 M ) Bu,NNO, (0.1 M )
e,
€0
8 .O 8.4 9.05
7.2 7.3 7.4
DEC €0
LiClO, (0.1 M ) LiSCN (0.1 M )
DEC
e,*
1
3.75 3.35
f R 1,
2.4 2.4 2.4
2.84
2.31
e-
e-2
1
3.0 2.90
GHz
f R 2,
GHz
2.5 1.0 0.75
60 55 55
fR
fR2
X , n-’ cm-I
6.83 x lo-‘ 1.33 x 10-5 2.05 x 10-4
16
2.35 2.35
0.9 0.8
14 12
a The parameters eel, e m ,, e m 2 , f R l , and f ~have , been obtained by programming eq I and I1 with variable values of the above parameters as to have a minimum in the expression: (C le’calcd - elexpt I t i e “ c a l c d - ~ ’ ’ ~ ~ ~ This ~ 1 ) procedure . was also described previously. 9
4
0
7
-
-
8
C = 0 0 5 M *;25”C 7
f
6 5 4
3
f (“32
-
e-
I Cole-Cole plot
LNO, OIOM n T H F t = 25°C 2 3 2 4
c
(
*->
27
-
00
2
3
1
5
6
~
8
9
e-
Figure 1. Real part of the complex permittivity er and imaginary part of the complex permittivity err plotted vs. the frequency f (GHz) for sodium picrate (0.05 M) and LiNO:, (0.1 M) in THF at 25 O C . Cole-Cole plots of err vs. E‘ for the same system. 6;’ is the conductance contribution to E” (see text). The filled points at f = 137 GHz refer to literature data for pure THF (ref 5).
25
15
c
05
ali
d.2
d.5
1:o
i.0
Ib
i.0
-
2b
f (GHz)
io
IAO 2b0
5k
Cole cole Plot
3
8qNNO3 OIMlnTHF t ‘ 254c
ct 2 W
2’
I
\ I 4
5
7
6
8
9
E’
Figure 2. Real part of the complex permittivity er and imaginary part of the complex permittivity errplotted vs. the frequency f(GHz) for Bu4NN03(0.1 M) in THF at 25 OC. Cole-Cole plot of err - e;r vs. er for the same system. The filled points of f = 137 GHz refer to literature data for pure THF (ref 5).
0
Figure 3. Real part of the complex permittivity E’ and Imaginary part of the complex permittivity E” plotted vs. the frequency f (GHz) for LiClO., (0.1 M) and LiSCN (0.1 M) in DEC at 25 ‘C. Cole-Cole plots of E” vs. for the same systems.
the DEC solvent by a single relaxation function is adequate up to about 35 GHz with positive deviations appearing at high frequency due to an absorption process in the submillimeter near-infrared region of wavelengths. Details of this aspect will be described in a later paper. For the time being, our interest being centered on solute relaxation, we will ignore the systematic deviations of E” at 50-70 GHz. Figure 3 shows the quantities E’and E” vs. the frequency f and E” vs. e’ for LiC104 (0.1 M) and LiSCN (0.1 M) in DEC a t 25 “C. The solid lines correspond to values calculated from eq I and I1 where e/ has been neglected. The associated parameters are reported in Table I.
Discussion Literature5l7 reports assigns a single Debye type relaxation process to pure THF with parameters eo = 7.36, E , = 2.30, and fR = 62.9 GHz. From Figures 1 and 2, and Table I it is evident that the high frequency relaxation process in the THF solutions can be associated with the solvent molecules. The second relaxation process occurring at lower frequencies in the solutions is specific, both in terms of the change in dielectric strength eo - tml and relaxation frequency fR1, for the electrolyte considered. In the following we will relate apparent dipolar radii obtained from the relaxation parameters eo - eml and from the relaxation time, T [where T = (27rfR)-l], to each other and to radii calculated from crystallographic data. These relationships should clarify the nature of the molecular process associated with the dielectric relaxation attributed to the solute. First, we present equations which will permit the calculation of a,,, the apparent distance between the centers
Dielectric Relaxation of 1:l Electrolytes in THF and DEC
The Journal of Physical Chemistry, Vol. 82, No. 17, 1978 1945
TABLE 11: Values of Ion Pairs and Triple Ions Formation Constants K , and K , in THF Taken from the Literature, Concentration of Ion Pairs C, [ (1 - u - ~ u J T ) Charge ~], t o Charge Separation u p Calculated According to Debye, Sum of the Crystallographic Radii ( r + t r - ) Taken from the Literature and from Molecular Models electrolyte in THF LiCIO, NaC10, NaClO, LiNO, NaPic BU,NNO,
KT
C,
M
K,,
M-I
4.84 X l o 7 9 . 9 3 ~107 9.93 X l o 7 590 x 107
0.05 0.1 0.05
0.10 0.05 0.10
x 107
0.7
9
M-’
ref
C,, M
153 150 150 182
a
0.0492 0.0985 0.0495 0.0998 (0.5) 0.0933
155
b b c c
lo%,,
lo%,,
cm
cm
(r+t r-)108
ref
3.11 2.63 3.02 2 .oo 4.36 4.10
4.18 3.78 3.93 3.53 4.77 5.22
2.25 2.62 2.62 1.82 4.5 6.16
d,e
e e
f
g
h
*
0.60 A is the radius of the Lit ion. e 0.97 A is the radius of the Na+ a Reference 9a. Reference 2. Reference 9c. ion; 1.65 A is the C1-0 distance in the perchlorate ion. 1.22 A is the N - 0 distance in the nitrate ion. g 3.5 A is the calculated radius of the picrate ion from molecular models (distance from the oxygen to the center of the benzene ring). $ u 4 ~ += 4.94 A , from R. A. Robinson and R. H. Stokes, “Electrolyte Solutions”, Butterworths, London, 2nd ed, 1959,
p 125.
of unit positive and negative charge for the ion-pair dipole where a, = p / e , p is the apparent dipole moment of the ion pair and e is the electron charge. The apparent dipole moment p is calculable from the Bottcher expression8 47rLC,
Y
ho
-
hm1
=
I
(1 - c ~ f ) ~ ( 2+~ 01)3hT
(V)
A+ + BAB
+ A+
-
-+
AB
AgB+
Kp KT
Now K, = (1 - u)/a2C where a is the degree of dissociation of the pairs and C is the stoichiometric concentration of the electrolyte. SimilarlylO [AB21 [A81 UTC KT = -=-[AB1PI [AB][AI [(I - a - ~ ~ T ) C I ( U C ) where UT is the fraction of ion pairs which form triple ions according to the equilibria for triple ion formation shown above. Obviously then, C, = [AB] = (1 - a - 3aT)C. The (2,s’ were calculated for all the THF solutions except the sodium picrate solution where no data for K and KT are available. In the latter case C, was assume2 equal to C, the total solute concentration. These values are presented in Table 11. The (1 - af12factor in eq V was neglected which implies a zero polarizability or a rigid sphere model for the ions in the ion-pair dipole. Thus in the rigid sphere model for the constituent ions a, is equal to a, = p / e , the charge to charge separation distance. The values of the p’s and a i s are reported in Table 11. As an alternate method for the calculation of an apparent radius for the relaxing dipole we use Debye’s equationll 7’
47ra73 kT
‘
=-
for l i i eiectrolyles in THF a1 25’C
t
6-
*,”
E
5-
0 c
4-
32-
Figure 4. Correlation between a,, the dipole radius calculated from the relaxation time, and a,, the charge to charge distance calculated from the apparent dipole moment. Correlation between a, and the sum of the crystallographic radii ( r + 4- r-).
laxation time. This can be related to the experimental relaxation time 7 by the Powless-Glarum expressiod2 7’ = [(2t0 ~ ~ ~ ) / 3The ~ ~ values ] 7 . of aTcalculated from eq VI are given in Table 11. Figure 4 shows the values of a, vs. a,. The solid straight line is that calculated by a linear regression analysis with a correlation coefficient r2 = 0.89. We would not expect a slope of one in this plot in view of the crudeness of the models assumed and the fact that a, is derived from a property related to the distribution of charge in the ion pair while a, is derived from a property related to the rotation of the ion pair in the solution. The ion pair may be solvated to some extent and this would modify the size of the rotating species. In spite of all this a correlation as shown in Figure 4 does emerge. Further evidence that the solute relaxation process is due to the rotational relaxation of ion-pair dipoles is given by the linear relationship between the a,‘s and the sums of the crystallographic radii (r+ + r-) also shown in Figure 4. The solid straight line was calculated from a linear regression analysis with a correlation coefficient of r2 = 0.91. Attempts to rationalize the observed correlation between a, and the sum of the crystallographic radii in terms of a libration of an ionic lattice in the alternating electric field of the microwaves, while not impossible, is not likely to be justified. Explanations of this sort would be more reasonable at higher concentrations than those studied, where the pair to pair distances would become the same as the interionic distances in a single pair. At the maximum concentration studied here, C = 0.1 M (neglecting other species), the volume associated with a single pair. V = 1/(C X 10-3L) = 1.66 X cm3, Taking V = 4/37rr3
+
KT
AB+B-+AB,
radii ond crystoilographic rodii
opparenl dipole moments for lil
t
0
where C, is the concentration of the ion-pair dipoles, L is Avogadro’s number, f is the reaction field factor, and a the polarizability. To use eq V we need to calculate C, which can be done from values of the equilibrium constants for ion-pair formation and triple-ion formation, K, and KT, respectively. The values Kp and KT are in the literatureg and are presented in Table 11. The equilibria involved are
(VI)
where a, is the radius of a spherical dipole immersed in a continuum of viscosity 7 and 7‘ is the microscopic re-
Correlotion between Debye ieloxoiion
radii and radii COIcuiated f r o m electrolyles i n THF 01 25°C.
t
10-3(3~nu2) . ”.
X
Corraiotion between Debye relaxotton
1946
Petrucci et al.
The Journal of Physical Chemistry, Vol. 82, No. 17, 1978
TABLE 111: Calculation of A e = e o - E , = SpCp t SqCqfor LiSCN (0.1 M), and LiClO, (0.1 M) in DEC and for LiBr and e o from the Microwave Data (0.1 M ) and LiSCN (0.09 M) in DMCa, Comparison between e o at 2MHz from solvent DEC DEC DEC DEC a E,
electrolyte 0.1 M LiSCN 0.1 M LiC10, 0.1 M LiBr 0.09 M LiSCN
is the solvent permittivity.
CP 0.017 0.021 0.021 0.037 e o = A e t e,
6P
CP
6q
A€
14.3 14.7 8.7 14.6
0.0414 0.0395 0.0395 0.0260
8 .O 15.0 1.8 10.8
0.577 0.902 0.254 0.830
e,(DEC) = 2.84; e,(DMC) = 3.09.
this volume corresponds to r N 16 A where 2r represents the minimum distance between the centers of a given pair and some adjacent pair. The value of the calculated 2r is between 7 and 16 times the values of afi quoted in Table 11. In a previous paper3 we reported a roughly linear correlation between the quantity co - cml and C,[3c0/(2c0 + l ) ] for LiBr, LiSCN, and LiC104 in DMC in accord with eq V. In that correlation the contribution to to- t m l due to rotational relaxation of any ion-pair dimers (quadrupoles) was neglected because of the small dipole moments of these dimers for LiBr and because of their low concentration for LiC104.4 For the present systems of LiSCN and LiC104 in DEC, the association to quadrupoles should be more extensive and the dipole moments of these quadrupoles should be relatively large,4as indicated by the values of Jq. The 6 ’s represent the contribution per unit concentrations of t i e quadrupoles to the increase of the permittivity of the solution with respect to the solvent, At. Then it would appear that the approximation of neglecting the contribution of the quadrupoles in the quantity c0 - E , is no longer tenable for the present systems. Still it is not possible to detect the presence of more than one single Debye relaxation process, within experimental error, for 0.1 M LiSCN and LiC104 in DEC. It is not conceivable that the two species ion-pair dipoles and quadrupole-dimers have the same relaxation time. On the other hand, the presence of the latter species has been detected by IR spectra (-2040-cm-l band).13 Since the parameter cm2 is comparable for the electrolyte solution and the solvent DEC (and the size of the quadrupoles is larger than the one of the dipoles), one might suspect that the rotational relaxation of these species occur at lower frequency than our lowest experimental frequency. That this suspicion is unfounded is easily provable by recalculating the static € 0 ) ~(at f = 2 MHz) from Chabanel’s work4and comparing it with our calculated ~0)s. Menard and Chabane14 defined the dielectric effect Ac Ac = F,C, + 6,C, (VIU where Ac = co - c8 with cs the permittivity of the solvent and Cp and C, are the concentration of the pairs and quadrupoles, respectively. 6, and 6, are the dielectric increments due to the presence of unit concentrations of the two species. The calculation of A€ is possible from the tabulated4 6,, 6,) and KqP4 where K, refers to the equilibrium: 2[ABI + [ A m 1 - uq uq/2 K, =
a,/2
C(1 - uq)2
(VIII)
and C, = (1- u )C, C, = (uq/2)C. Table I11 reports the results of this c&ulation for LiSCN and LiC104 in DEC as well as for LiBr and LiSCN in DMC (the values for K,, ,a and 6, for LiC104 in DMC were not r e p ~ r t e d ) .It~ is
€0
b
3.42 3.74 3.34 3.92
eoextrapc
3.35 3.15 3.45 3.95
Data from Table I and ref 3.
evident that the two sets of eo’s, the static and the extrapolated ones are within experimental error. The absence of a dielectric relaxation spectrum for the quadrupoles can be rationalized by the possibility that dipole moment is negligible or zero a t variance with calculated value^.^ Recently Chabanel et al.14 have obtained the IR and Raman spectra of LiSCN in various ethers. They invariably detect a lowering of the force constant in the C-N bond of the SCN- ion when the ion pairs dimerize to form quadrupoles. In fact, the “free” SCN antisymmetric stretching frequency a t 2060 cm-l is lowered to 2040-2030 cm-l. Similarly, the IR active C-S stretching frequency is increased from 735 cm-l in the “free” ion (which occurs in solvents such as dimethylformamide) to 792 cm-l on quadrupole formation in dimethoxymethane. In contrast, the formation of the simple ion-pair LiSCN results in increases in both the C-N and C-S stretching frequencies with respect to the free SCN- ion. The decrease of the force constant in the C-N bond and the simultaneous increase in that of the C-S bond has been interpreted14 as being due to the formation of a centrosymmetric quadrupole S=C=N
/
Li. a.
N=C=S
I
Li
By symmetry the dipole moment of this species would be zero. Unfortunately, the carbonate solvents absorb in the same region as the C-S stretch in the thiocyanate ion which precludes the study of the thiocyanate-carbonate system in this region. However, for the C-N stretching frequency IR datal3 indicate the presence of quadrupoles in carbonate solvents, the band occurring at 2040 cm-l. [Raman spectra have confirmed this band although at 2052 ~ m - ’ . ~ Addition ] of a high permittivity solvent such as dimethylformamide to the LiSCN-ethyl acetate system also causes this IR band a t 2040 cm-l to disappear.13 If the ion-pair dimers or quadrupoles that are formed in the carbonate solvents are centrosymmetric and thus have a zero dipole moment as is indicated in the case of the ether solvents, then the dielectric data for the LiSCN and LiC104 in DEC would find a complete rationalization for their showing a single relaxation process. This then is to be attributed to the ion-pair dipoles as in the previous case for the DMC s ~ l v e n t . ~ Addendum According to the report of one of the referees, Chabanel et al. have recently arrived at the same conclusion. The apparent inconsistency between IR (pq = 0)14and dielectric (6, # 0) measurements4 was puzzling. A quantitative IR study of LiSCN solutions in DEC and in other solvents showed that the activity coefficients of ion pairs cannot be neglected in the 0.01-0.1 M concentration range. Thus the ideal solution model, even if it seems to work well, cannot give correct 6, values. The main reason is that these values arise from an extrapolation to infinite concentration. It is also concluded that 6, values are reliable and com-
The Journal of Physical Chemistty, Vol. 82,No. 17, 1978
Self-Association of Gases
parison between IR and dielectric measurements shows that 6, is close to zero for LiSCN. Consequently quadrupoles must be nonpolar in agreement with the result of the present paper.
1947
(7) J. P. Badiaii, H. Cachet, A. Cyrot, and J. C. Lestrade, J. Chem. Soc., Faraday Trans. 2 , 69, 1389 (1973). 18) C. F. J. Bottcher, “Theory of Electrical Polarization”, Elsevier, Amsterdam, 1973. (9) (a) P. Jagodzinski and S. Petrucci, J . Phys. Chem., 78, 917 (1974); (b) H. Farber and S. Petrucci, ibid., 80, 327 (1976); (c) H. C. Wang and P. Hemmes, J . Am. Chem. SOC.,95, 5119 (1973). (10) R. M. Fuoss and M. Accascina, “Electrolytic Conductance”, Interscience, New York, N.Y., 1959, Chapter XVIII, p 254. (11) P. Debye, “Polar Molecules”, Chemical Catalog Co., New Yo&, N.Y., 1929. (12) M. Davies in “Dielectric Properties and Molecular Behaviour”, N. Hill et al.. Ed.. Nostrand-Reinhold. London. 1969. D 298. (13) M. Chabanel, C. Menard, and G. Guiheneuf, C . A. Acad. Sci., Paris, 272, 253 (1971). (14) D. Paoli, M. Lucon, and M. Chabanel, to be submitted for publication.
References and Notes (1) H. Farber and S. Petrucci, J . Phys. Chem., 79, 1221 (1975). (2) H. Farber and S. Petrucci, J . Phys. Chem., 80, 327 (1976). (3) D. Saar, J. Brauner, H. Farber, and S. Petrucci, J . Phys. Chem., 82, 545 (1978). (4) D. Menard and M. Chabanel, J . Phys. Chem., 79, 1081 (1975). (5) S. K. Garg and C. P. Smyth, J. Chem. Phys., 42, 1397 (1965). (6) S. Petrucci, P. Hemrnes, and M. Battistini, J. Am. Chem. Soc., 89, 5552 (1967).
Self-Association of Gases. 1. Theory. Application to the Nitrogen Dioxide-Dinitrogen Tetraoxide Systemt David
R. Powell and E. T. Adams, Jr.”
Department of Chemistry, Texas A&M University, College Station, Texas 77843 (Received March 27, 1978) Publication costs assisted by The Robert A. Welch Foundation
Using the Gibbs-Duhem equation and some simple assumptions about activity coefficients, an equation of state for nonideal gas self-associations can be developed. This equation, which relates total pressure to the total concentration, temperature, and apparent number average molecular weight, is formally identical with an equation developed for osmotic pressure of a self-associating solute. Thus, one can use previously developed procedures in the analysis of gas self-associations. These procedures have been successfully applied to the NO2 data of Verhoek and Daniels.
Introduction Self-association of molecules in the gas phase is a well-known and long-studied phenomenon.l-17 Verhoek and Daniels,l as well as other^,^^^ carried out pressurevolume (P-V) measurements on NO2 gas at several temperatures. The analysis of the data obtained by Verhoek and Daniels1 led them to conclude that their data were best represented as a monomer-dimer association, i.e. 2NO2(g) + N@,(g) (1) Another well-known self-association involves H F gas. Hildebrand and his associate^^^^ believed that H F gas underwent a monomer-hexamer association at temperatures below 80 “C. Others believed this to be the case, but later studies indicated that the association might be more c~mplicated.~-~ The association of vapors of organic acids1°-13 has also been studied. The dimerization of formic acidlOJ1and acetic acid12J3are notable examples of this case. At high temperatures vapors of alkali metals,14 KBr, and other salts16-17 are also reported to self-associate. At present, the theories for the analyses of gas-phase self-associations have been largely based on methods developed by Verhoek and Daniels,l Kreuzer,ls and Rossotti and Rossotti.lg These methods suffer in that they must assume ideal behavior. Usually no attempt has been made to correct for nonideal effects. In addition, none of these methods explicitly takes advantage of the interrelation between the number average (Adnc)and the weight average (MWJmolecular weights, or their apparent values (M, and MwJ when nonideal behavior exists.*22 By using these interrelations it is possible to test for the presence or absence of a variety of self-associations, set up plots or Presented in part at the 32nd Southwest Regional Meeting of the American Chemical Society, Ft. Worth, Tex., Dec 1-3, 1976.
0022-3654/78/2082-1947$01 .OO/O
other tests to choose the best model, and evaluate the equilibrium constant or constants (Ki) and the nonideal term (BMl).21i22This is done using the experimental data and other data derived from it. In addition one can also evaluate the apparent weight fraction of monomer (f,) and use it in the analysis. These advantages were not possible with the other previously developed methods. The purpose of this paper is to show how one’can analyze both ideal and nonideal gas phase self-associations using methods of analysis very similar to those used for studying self-associations in solutions by membrane or vapor pressure osmometry.20B21We will show how one can obtain M,,, Mwa,and fa from measurement of the total gas pressure in a series of experiments at constant temperature. This method will be applied to the self-association of NO2 gas. In a subsequent paper we will discuss its application to the self-association of H F gas and to CH,COOH gas.
Theory At constant temperature the condition for chemical equilibrium for a self-association is npl = p,, (n = 2, 3, ...) (2) Here p, is the molar chemical potential of species i (i = 1, 2, ...). In order to develop a relation between the equilibrium total pressure, P, and the apparent number average molecular weight, M,,, one starts with the Gibbs-Duhem equation. Thus a t constant T
V d P = E n , dp,
(3)
1
can be rearranged to give (4)
Here ci is the concentration of species i in g/L. Now 0 1978 American Chemical Society
