EDWARD J. KING
1220 with HBF4 was about half that with HC1. Such data might indicate importance of anion in rates of hydrolyses. However, an approximate 400-fold increase in [BF4-] from the HBF4- to the Zn(BF4)2-catalyzed hydrolyses increased IC1 only 1.5 times, and the relative effectiveness in opening the second oxirane dropped from sevenfold at 40"to threefold at 75". The AH*, AS*, and AG* values at 40" for these hydrolyses are included in Table IV. The AH* values for the first ring opening in the presence of HC1 and Zn(BF4)2were equal and greater than AH* for HBF4 catalysis. The AH* values for the second ring openings followed the order HC1 > HBF4 > Zn(BF4)2. Only with Zn (BF4) was the AH* for the second ring opening less than for the first ring opening. Large negative A S * values with all catalysts indicated S Nmechanisms ~ in this pH range. The largest negative value of AS* was obtained with HBF4 for the first ring opening. For the second ring opening, more negative AS* values were obtained with both fluoroborates than with HC1
catalysis. The AG* values for the first oxirane openings were essentially the same for all catalysts and approximately equal to the AG* value for the second ring opening with Zn(BF4)2only; with HC1 and HBF4, AG* was about 1 kcal/mol greater for the second ring opening. All data indicate that with HBF4catalysis, the rate of hydrolysis of BDO is first order with respect t o [Hf]. When [H+] is as low as pH 3.25, the large negative AS* values indicate an S N mechanism ~ for HBF4, HC1, and Zn(BF4)2 catalysis. This is similar to the report of Koskikallio and WhalleyQfor ring openings in the presence of low concentrations of catalyst. It is only in the more acidic solutions that an SN1 mechanism predominates.
Acknowledgment. The authors wish to thank Mr. Emory Coll for assistance in use of the analog computer. (9) J. Koskikallio and E. Whalley, Trans. Faraday Soc., 55, 815 (1959).
Volume Changes for Ionization of Formic, Acetic, and n-Butyric Acids and the Glycinium Ion in Aqueous Solution at 25" by Edward J. King1 Department of Physical and Inorgantc Chemistry, University of New E'nolnnd, N . S . W . , Auslralia
(Recetved July 5 , 1 9 6 8 )
Partial molar volumes of glycine hydrochloride and of molecular formic, acetic, and n-butyric acids and their sodium salts in aqueous solution a t 25" have been obtained from pycnometric and dilatometric measurements. The volume changes associated with ionization of the acids are -6.80 for the glycine cation, -8.43 for formic acid, -11.50 for acetic acid, and -14.22 ml mol-' for n-butyric acid. The packing density, defined as the ratio of van der Waals volume to partial molar volume of a solute species, is a convenient property to use in comparing solutes with different structures and volumes. Above a certain minimum size all molecules or ions of a given type have the same packing density, and these values are close to the packing densities of randomly packed spheres. The effect of chain length in the acid on the volume change for ionization follows from the existence of such plateau densities. Both the electrostrictioii of solvent and the dead space or void volume between solute and solvent make important contributions to the volume changes for ionization reactions.
Introduction The partial molar volumes of solutes in solution merit investigation because of the insight they provide into solute-solvent interactions, For chemical reactions in solution the effect of the solvent is manifest in the accompanying volume change. The volume change for ionization of a molecular acid HX is expressed by AVao = Vo(Hf)
+ V o ( X - ) - V"(HX)
(1)
when all solute species are in their standard states. The Journal of Physical Chemietry
According to the simplest electrostatic theoryj2 such volume changes are negative because of the electrostriction of water produced by the ions
Ne2 1 AVOo=
- 'i;;(;;f
1
dlne.
;;>(F) T
(2)
(1) Address correspondence to the author at Barnard College, BOB West 120th Street, New York, N. Y. 10027. (2) E. J. King, "Acid-Base Equilibria," Pergamon Press, Oxford, 1965, pp 204-209.
ACIDS VOLUMECHANGES FOR IONIZATION OF FORMIC, ACETIC,AND ~-BCTYRIC where N is Avogadro's number, e is the proton charge, is the dielectric constant of water, and r H and rx are the radii of H+ and X - ions. The theory is not successful in predicting relative volume changes for a closely related set of acids.2 The Born theory also predicts for ionization reactions a simple proportional relation between volume changes and entropy changes.2 Relationships between AVao and As," have been investigated by H e ~ l e rK , ~~ u z m a n nand , ~ Conway.6 T o investigate such relationships or even to interpret changes in volume for individual acids, more accurate values of AV," are required than are usually available. Most published values6 are scarcely known to better than f0.5 ml mol-'; for acetic acid reported values range from -9.2 to -12.5 ml mol-'. The experimental problem is the accurate determination of apparent molar volumes in very dilute solutions. The best techniques are Wirth's differential sinker method' and the dilatometric method of Hepler, Stokes, and stoke^.^)^ Measurements by the latter technique are reported in this paper for four carboxylic acids.
e.
Experimental Section Materials. All solutions were prepared from distilled water that had been passed through a monobed ionexchange column so that its specific conductance was below 4 X 10-7 9-1 cm-I. May and Baker reagent grade hydrochloric acid was diluted with water to the required concentration and standardized by Stokes' conductance method.1° Carbonate-free solutions of sodium hydroxide were prepared by the method of Marsh and Stokes," stored in stainless steel tanks, and standardized against the hydrochloric acid. I n manipulations of the sodium hydroxide solutions, carbon dioxide was excluded by a stream of argon that had passed successively through a tube of Carbosorb and a bottle of aqueous sodium hydroxide solution. Samples of Analar formic and acetic acids were separately subjected to three fractional freezings; about 20% was discarded each time. Stock solutions were prepared by dilution of the purified acid with water and were standardized against the sodium hydroxide solution. Most of the solutions of sodium formate were prepared from the solid salt. Although May and Baker anhydrous sodium formate contained less than 0.004% Fe, C1, "3, and heavy meta1s,l2 there was a small amount of insoluble material in it. It was recrystallized twice from water and washed with alcohol and acetone. After first being dried in air, some of the salt was heated to 110' and the rest to 200'; there was no essential difference between the two products. The purity of the salt was checked by quantitative conversion into sodium sulfate;13 best results were obtained by controlling the temperature of final ignition a t 700' in an electric furnace. One sodium formate
1221
solution was prepared by neutralization of the stock formic acid solution with sodium hydroxide, May and Baker sodium acetate trihydrate was used without further purification; it contained less than 0.004% Fe, C1, NHB,or heavy metals.12 The purity was checked by titration with standard hydrochloric acid and by conversion into sodium sulfate; the two results differed by 0.06%, within the combined errors of the analyses. With the exception of one solution, which was prepared by neutralization of the stock acetic acid solution, all sodium acetate solutions were prepared from the solid salt. A sample of British Drug Houses reagent grade nbutyric acid was distilled through a 35-cm Vigreux column, and the middle fraction was collected: bp 159.4-159.5' (uncor) at 675.6 mm, nD at 20.2 f 0.1' was 1.39791 as compared with a literature valueI4 of 1.39788 f 0.00005, Aqueous solutions of the acid were prepared by dilution of this material and were standardized against sodium hydroxide solution. Sodium butyrate solutions were prepared by neutralization of the stock solution of the acid. The glycine, a product of the H. M. Chemical Co., Santa Monica, Calif., contained less than 0.004% Fe, C1, "I, and heavy metals, and its purity, established by formol titration, was 99.95%,. Solutions of its hydrochloride were prepared from solid glycine and standard hydrochloric acid. Analyses were carried out by weight titration. The course of each reaction was followed with a Radiometer pHM4 meter using glass and saturated calomel electrodes. The end points were determined with a precision of +0.01 to &0,0370by the intersection of two straight lines according to the Gran method.I6 The absence of carbonate in the stock solutions of sodium hydroxide was evident from the graphs.lB Measurements. Densities of solutions with concentrations greater than 0.2 214 were measured with Ostwald pycnometers having volumes of about 50 ml. (3) L. G. Hepler; J. Phys. Chem., 69, 965 (1965). (4) W. Kausmann, A. Bodansaky, and .J. Rasper, J . Amer. Chem. SOC.,84, 1777 (1962). ( 6 ) R. E. Verrall and B. E. Conway, J. Phys. Chem., 70,3961 (1966). (6) Reference 2, pp 199-201. (7) H. E. Wirth, J . Amer. Chem. Soc., 59, 2549 (1937). (8) L. G. Hepler, J. M. Stokes, and R . H. Stokes, Trans, Faraday Soc., 61, 20 (1965). (9) L. A. Dunn, ibid., 62, 2348 (1966). (10) R. H.Stokes, J . Phys. Chem., 65, 1242 (1961). (11) K. N.Marsh and R. H. Stokes, Aust. J. Chem., 17, 740 (1964). (12) M. P.Stoddard and M. 9. Dunn, J.BioC. Chem., 142,329 (1942). (13) 0. Redlich and L. E. Nielsen, J . Amer. Chem. Soc., 64, 761 (1942). (14) R. R. Dreisbach and R. A. Martin, Ind. Eng. Chem., 41, 2875 (1949). (15) G. Gran, Analyst (London), 77, 661 (1952). Certain reflnements in the method were necessary for such titrations as that of sodium acetate with hydrochloric acid. These will be published elsewhere. (16) F. J. 0. Rossotti and H. Rossotti, J. Chem. Educ., 42, 376 (1965).
Volume 73,Number 6 May 1069
EDWARD J. KING
1222 ~~~
Table I: Densities and Apparent Molar Volumes of t h e Concentrated Solutions at 25" c,
mol 1.-1
p. g ml-1
&, ml mol-1
c,
cy
mol. I.-'
Formic acid (Mr = 46.026 amu) 0.9892 0.7789 0 7623 0.6143 0.5289 0.5018 0.3818 0.2996 0.26913 0.24614 0.20413
1.008320 1.005976 1.005775 1.004122 1.003160 1 002846 1.001486 1.000543 1I000200 0.999935 0.999450
I
I
34 760 34.699 34.714 34.656 34.623 34 629 34.574 34.550 34 516 34.508 34.492 I
I
I
1.007278 1.004559 1.002764 1.001702 1.000877 1.000404 0,999632
%-Butyric acid 1.5539 1.0776 0.8046 0.6039 0.5012 0.3909
1.001926 1IO01018 1.000136 0.999413 0 999032 0.998604 I
52,016 51.976 51.943 51.932 51.906 51.890 51.885
0 9866 0.7718 0,6314 0 4923 0.4764 0 26217
0.00373 0.00435 0.00498 0.00551 0.00607 0,00648 0.00736
1.0350 0.9300 0.5499 0.5354 0.4036
0.00311 0.00367 0.00432 0.00499 0 00547 0.00619
0.6035 0.4839 0.4076 0.2576
I
&, ml mol-'
I
I
I
1 037644 1.029033 1.023311 1.017644 1.016948 1.008107
26 965 26.680 26,531 26,307 26.366 26.005
I
I
Sodium acetate (Mz = 82.041 amu)
(Mz= 88,107 amu) 85.234 84.695 84.553 84.482 84.445 84.443
ml-1
Sodium formate (Mz= 68.008 amu) 0 0154 0.0172 0.0174 0.0193 0.0207 0.0212 0.0242 0.0269 0.0282 0.0296 0.0325
Acetic acid (Me= 60.053 amu) 1.2457 0.9095 0.6887 0,5589 0.4581 0.4005 0.3073
p, g
1.039338 1.035174 1.019858 1.019282 1.013887
41 .322 41.191 40..726 40.673 40.494
Sodium n-butyrate (Ma = 110.089 amu) 1.021006 1.016332 1.013337 1.007416
70.641 70.501 70.403 70.152
I
Glycinium chloride (Mz= 111.529 amu) 0.7415 0.6980 0.5010 0.28050
1.028684 1.026879 1.019049 1.009312
69.101 69.034 68.689 68.102
0.0696 0.0720 0.0855 0.1161
The pycnometers were calibrated with water, which was assumed to have a density1' of 0.997075 g ml-l. Each pycnometer was rinsed six times with solution and then filled by siphoning solution from the stock bottle. The pycnometers were adjusted in a water bath controlled to 25.000 =t 0.002°. At least two determinations of density were made at each concentration. Vacuum corrections, which took into account temperature, pressure, and humidity, were applied to all weighings. The average precision of the density determinations was h4.0 ppm. Of the 43 determinations, 33 had a precision of f 5 ppm or better; none had a precision over f 1 0 ppm. Most of the concentrated solutions were diluted in a dilatometer to obtain apparent molar volumes for concentrations below 0.05 M . The techniques of earlier workers were followed.**9
Calculation and Results The apparent molar volume 9u(initial) of a solute in one of the concentrated solutions can be calculated from the density of the solution p , the density of water The Journal of Phyaical Chemistry
the molar concentration c, and the molecular weight of solute Mz
p*,
The data and calculated values of the apparent molar volumes are given in Table I along with values of the degree of ionization a! of the weak acids, which are required for subsequent calculations. Suppose that dilution in the dilatometer of a sample of the concentrated solution, which contains nz moles of solute, produces a volume change Av. The apparent molar volume of the solute in the final solution is given by8 &(final) = cbp(initial) Av/nz (4)
+
The final concentrations, volume changes per mole, and calculated values of (final) are given in Table 11. (17) B. B. Owen, R. White, and J. 5. Smith, J . Amer. Chem. Soc., 78, 3561 (1956):G.9. Kell, J. Chem. Eng. Data, 12, 66 (1967).All volumes in this paper are expressed in old milliliters (1.000028 cms) to conform to the usage of earlier workers.
1223
VOLUMECHANGES FOR IONIZATION OF FORMIC, ACETIC, AND %-BUTYRIC ACIDS Table 11: Dilatometric Data and Apparent Molar Volumes for the Dilute Solutions at 25" Av/nl, ml mol-1
c, mol 1 . - 1
&, ml mol-1
e, mol L - 1
01
Av/nz, ml mol-1
Sodium formate
Formic aoid
-0.784 -0.808 -0.898
0.027587 0.021258 0.013995
0.0822 0.0930 0.1130
33.976 33.906 33.731
-1.574 -1.338 -1.256 -1.115 -0.966 -0.800
0.027513 0.021525 0.017607 0.013695 0.013285 0.00731 1
-0.321 -0.350 -0.302 -0,423
0.0223 0.0223 0.0371 0.0395
51.695 51.666 51.601 51.467
i
-0.914 -0.279 -0.376
-1.700 -1.237 -1.197 -1.067
0.025889 0.015297 0.014892 0.011226
39.491 39.490 39.467 39.427
Sodium n-butyrate
n-Butyric acid
0.043205 0.016791 0.010869
25.391 25.342 25,275 25.192 25.400 25.205
Sodium aoetate
Acetic acid
0.03474 0.03474 0.012760 0.011140
&, ml mol-1
0.01867 0.02953 0.03657
84.320 84.203 84.067
-1.217 -1.057 -0.849
0.016787 0.013460 0.007162
69.424 69.444 69.303
Glycinium chloride
0.020626 0.019417 0.014201 0.0078027
-3.472 -3.427 -3.431 -3.540
0.3686 0.3784 0.4244 0.5221
65.629 65.607 65.258 64.562
We require the limiting values cpvm of the apparent molar volumes a t infinitesimal concentration of solute. For 1-1 strong electrolytes a suitable extrapolation function is18
cp' = cpv
- kc112 = +vm
+ bic
(5)
The limiting slope IC has the theoretical value for aqueous solutions at 25' of 1.868 cms L1l2 m01-3/2 whereas the coefficient bi is an empirical constant. Good linear extrapolations were obtained for the sodium salts of the aliphatic acids. 'Corrections for hydrolysis of these salts were negligible. Values of the constants and their standard deviations are nil mol-1
Salt Sodium formate Sodium acetate Sodium butyrate
25.056 f 0.022 39.227 f 0.011 69.188 & 0.016
bi, mlz mol-8
13 f 49 188 f 20 27 f 45
Range
0-1 M
0-1 M 0-0.6M
The value of the intercept for sodium acetate is in good agreement with 39.244 ml mol-I reported by Redlich and Nielsenla and 39.274 reported by Wirth.l9 For the weak, molecular acids the observed apparent molar volumes can be regarded to a first approximation as .the sum of independent contributions from the ions and moleculesz0 '
+
4v = 4% (1 - a)& (6) The sum of apparent molar volumes of the ions can be approximated by18
4i = cpi"
+ k(ac)"2 + beac
(7)
where c is the stoichiometric concentration of acid. The apparent molar volume of the molecular acid can be represented by
4u = 4,"
+ b,(l
- a)c
(8) By combining eq 6, 7, and 8 we can obtain a suitable extrapolation function for determination of the limiting apparent molar volume of the molecular acid cpum
4''
cpV
- aAV'
- ka(a~)'/~ =
9,"
+ bide + b,(l - a ) %
(9)
Values of a can be calculatedz1 from accurate conductancez2and viscosityz3data. Accurate conductance data for solutions of acetic acid with concentrations greater than 0.05 M are lacking, but CY can be estimated with sufficient accuracy from = a%/( 1
- a)
(10) where Qa is taken to be 1.80 X mol The term k a ( a ~ ) 'in / ~ eq 9 is very small, and the term Qa
(18) 0.Redlich and D. M. Meyer, Chem. Rev., 64, 221 (1964). (19) H. Wirth, J . Amer. Chem. Soc., 7 0 , 462 (1948). (20) H. 9. Harned and B. B. Owen, "The Physical Chemistry of Electrolytic Solutions," 3rd ed, Reinhold Publishing Corp.. New York, N. Y., 1958, pp 404-406. (21) Reference 2, p 30. (22) (a) Formic and n-butyric acids: B. Saxton and L. 9. Darken, J . Amer. Chem. Soc., 6 2 , 846 (1940);(b) acetic acid: D. A. MacInnes and T . Shedlovsky, ( b i d . , 54, 1429 (1932). (23) (a) Formic and acetic acids: E. W. Washburn, Ed., "International Critical Tables," Vol. V, McGraw-Hill Book Co., New York. N. Y . , 1929, p 20; (b) n-butyric acid: 0. R . Bury and J. Grindley, J . Chem. Soc., 1003 (1936). VoZums 78,Number 6
May 1969
1224
EDWARD J. KING
bia2c should be still smaller.24 Thus, the intercept 4," can be evaluated by treating 4'' as a linear function of (1 - a)?c. (In all but the most precise work on carboxylic acids, it suffices to plot 4'' against c alone.) For formic acid is 34.690 f 0.0076 ml mol-' and b, is 217 f 16 m12 mol-2. The corresponding values for acetic acid are 51.928 f 0.0048 ml mol-l and 107.6 f 8.0 mla This value of 6," for acetic acid is in excellent agreement with 51.942 ml mol-' previously reported by Redlich and Nielsen.la The apparent molar volumes of n-butyric acid do not conform to eq 9. As shown in Figure 1 , with increasing concentration c, 4'' decreases at first to a minimum value, then rises steeply. Had no data been obtained a t concentrations below 0.5 M , this peculiar behavior of n-butyric acid would have been missed altogether, and a large error would have been made in 4,". Dimerization of the molecular acid, which is well established for aqueous solutions, is the most probable cause of this behavior, Reported values26 of the dissociation quotient &d
(11)
[HX]z/[HzXz]
range from 2 to 11 mol dimerization
1.-1.
Let p be the degree of
sum of contributions from the ions, monomer molecules, and dimer molecules.
6 = adi
+ (1 -
Q
- P)&
(14)
The apparent molar volume of the dimer t$d should be more than twice that of the monomer 4%because the number of hydrogen bonds between acid molecules and solvent is decreased by chain dimerization. Assuming that the formation of one hydrogen bond causes a decrease in apparent molar volume2e of about 5 ml mol-', we expect I$d to be 5 ml mol-' larger than 24,, or about 175 ml mol-'. With this value of +d and with &d = 3 mol I.-', we can shift the minimum in the plot of 4'' against c to much higher concentration (broken line in Figure l ) , and with other choices of these constants we could eliminate the minimum altogether. I n the extrapolation of apparent molar volumes to infinite dilution, greatest weight was given to the results a t low concentrations which are comparatively insensitive to uncertainties in &d and &. The intercept, for n-butyric acid, is 69.188 f 0.016 ml m ~ l - ~ . ~ ' for glycinium chloride differs The evaluation of from that for the other three acids because the acid is a cation and its degree of ionization is comparatively large. A suitable extrapolation function is
4'" = &, - k d 2 - aAVa0 = Its value can be calculated from
+ *P&
(Pum
+b , ~
(15)
The term aAVao is large, so that a must be estimated carefully. The concentration quotient
T h e observed apparent molar volume of the acid is the
Qi
= [G*][H+]/[GH+]
= a2c/(1
- a)
(16)
is not known for solutions of pure glycinium chloride, but it may be approximated by the quotient &I" for GHCl in a solution of sodium chloride with the same concentration c as the pure glycinium chloride solution.28 In accord with eq 15, rp'" was found t o be and slope b, are 67.828 f linear in c ; the intercept 0.0083 ml mol-' and 171.4 f 7.0 m12 m o P . For ionization of a molecular acid HX H+ X - , the change in partial molar volume associated with the reaction when all solutes are in their standard states
+
(24) For hydrogen and formate ions, to a flrst approximation
833
-
'\ \
I
0
I
-
+ b(NaI.ICO8) - b(NaC1)
From known values of the b coemcients br is estimated to be -1080
I
Figure 1. Determination of the limiting partial molar volume n-butyric acid. The broken curve includes a correction for dimerization of the acid using +d = 175 ml mol-' and &d = 3 mol I.-'. The JOUTnd of Physical Chernialw
bi = b(HC1)
.--/ / /
of
mlz mol-2. The term b@c for formic acid solutions with concentrations up to 1 M does not exceed 0.0002 ml mol-1. (25) E. E. Schrier, M. Pottle, and H. A. Scheraga, J. Amer. Chem. Soc., 86, 3444 (1964); see also ref 2, p 31. (26) 8. D. Hamann in "High Pressure Physics and Chemistry." Vol. 2, R. 8. Bradley, Ed., Academic Press, New York, N. Y., 1963, Ohapter 7. (27) Although formic and acetic acids are known to form chain dimers in concentrated aqueous solution,zs their degrees of dimerizabton are smaller than those of butyric acid and satisfactory extrapolations can be made without correcting for dimerization. (28) E. J. King, J. Amer. Chem. Soc., 67, 2178 (1946); KI*in that paper is called QI" here-see also ref 2, p 270.
1225
VOLUMECHANGES FOR IONIZATION OF FORMIC, ACETIC,AND WBUTYRIC ACIDS is given by AVO = +,"(NaX)
- &"(HX)
+ C+,"(H+) - @,"(Na+)l
(17)
Values of the first two terms on the right have already been given and the difference in volumes of the hydrogen and sodium ions is 1.204 ml mol-I at 2Z0.29 For ionization of the glycine cation GH+ G* H+, the corresponding volume change is
+
+
AVO = cbum(HC1) (6u"(G*)
- @,"(GHCl) (18)
The value of the last term on the right is given in the preceding paragraph; for hydrochloric acid9 at 25') 4," is 17.815 ml mol-1 and for glycineaoat 25') 4," is 43.217 ml mol-'. Volume changes for ionization of the four acids are given in Table I11 together with previously reported values. The standard deviations of the four results of this investigation, as estimated from the standard deviations of the apparent molar volumes, are f 0 . 0 3 ml m o F , but the actual uncertainty in the values reported here is undoubtedly larger, perhaps h0.05 ml mol-'. It will be noted that for acetic acid, the volume change of -11.50 ml mol-' reported here is in excellent agreement with the values reported by Redlich and Nielsen's and Wirth.lg
Formic acid Acetic acid
n-Butyric acid Glycine cation as.
v w
=
vi - V H
(20)
Bondis' has compiled a set of van der Waals radii that can be used, along with bond lengths, to calculate the van der Waals volumes of molecules in crystals. I have assumed that the same method of calculation can be used for solutes in solution. Covalent and hydrogen bond lengths were taken from standard compilation^.^^ Not infrequently, the choice between alternative sets of bond lengths for a particular molecule was somewhat arbitrary, but the variation in volume produced by using different sets of lengths was generally within 0.2 ml m o F . The use of bond lengths that pertain to the crystalline state for solutes in solution can be questioned, and the hydrogen bond corrections, in particular, are probably more uncertain than the intrinsic volumes. Packing Densities of Molecules. The packing density of molecules in a crystal is defined3' as the ratio of their van der Waals volume to the molar volume of the solid. If the void volume of the crystal is represented by V v , the packing density can be expressed by VW/(Vw V u ) . It represents the fraction of the crystal volume actually occupied by the molecules. Likewise, the ratio of van der Waals volume to molar volume of a liquid is its packing density. For solutions the volume of the solute molecules must be compared not with the total volume of the solution but with the effective volume occupied by solute. We shall be concerned with solutes only at infinite dilution or in the hypothetical solution of the standard state. In either case the partial molar volume of water is the molar volume of pure water. The standard partial molar volume of the solute is the difference between the volume of the solution and the volume of pure water used to make it, and it represents the effective volume of the solute in solution. The packing density of a solute in solution is accordingly defined by
+
Table 111: Volume Changes for Ionization of t h e Acids at 25" -AVO, ml mol-1
Acid
represented by -VH and the last term, -V,, is the change in volume of the solvent induced by the solute. The first and third terms together will hereafter be called the van der Waals volume
8.43 (this investigation) 8.0," 8.8,b 9.20 11.50 (this investigation) 9.2,d10.3; 11.46,t11.47,g11.6," 12.1,h12.2,d12.5a 14.22(this investigation) 13.7a 6.80 (this investigation) 7.1e
D. Hamann and S. C. Lim, Aust. J. Chem., 7, 329 (1954).
* S. D.Hamann and W. Strauss, Trans. Faraday Soc., 51,1684 (1955). L. DistBche and P. Disteche, J . EZectrochem. Soc., 112, 350 (1966). d B . B. Owen and S. R. Brinkley, Jr., Chem. Rev.,29, 461 (1941). e H. H. Weber, Biochem. Z., 218, 1 (1930)(values a t 30" and an ionic strength of 0.1M ) . f H.Wirth, J. Amer. Chem. Soc., 70,462(1948). go. Redlich and L. E. Nielsen, ibid., 54, 761 (1942). h A . J. Ellis and D. W . Anderson, J . Chem. SOC.,1765 (1961).
d =
vw/v0
(21)
It represents the fraction of the partial molar volume occupied by the molecules or ions of the solute. Packing densities in aqueous solution at 25' were calculated not only for the carboxylic acids but also for
Discussion The partial molar volume of species X may be expressed as a combination of four contributions
Po=
vi
+ v,- P H - v,
(19)
Here V i is the intrinsic volume of particles of X, and V , is the dead space or void volume between them and the solvent. The decrease in volume that occurs when particles of X form hydrogen bonds with the solvent is
(29) L. A. Dunn, University of New England, N . S. W., personal communication, 1967: see also ref 9. (30) ,H. D.Ellerton, G. Reinfelds, D. E. Mulcahy, and P. J. Dunlop,
J. PhUS. Chem., 68, 398 (1964). (31) A. Bondi, J. Phys. Chem., 68, 441 (1964). (32) "Tables of Interatomic Distances and Configurations of Molecules and Ions," The Chemical Society, London, Special Publications No. 11, 1958, and No. 18, 1965; G. S. Pimentel and A. L.
McClellan, "The Hydrogen Bond," Freeman, San Francisco, Calif., 1960, Chapter 9. Volume 73,Number 6 Mag 1060
EDWARD J. KING
1226
“t
I
I
40
t
60
I
I
BO
I
I
I00
V,. ml mol”.
Figure 2. Packing densities of molecular solutes. Substances in each class are listed in order of increasing van der Waals volume:
0,HIO, CHaOH, CzHSOH, YL-C~H~OH, i-CsH,OH, n-CaHoOH, CCIHBOH, s-C~H~OH, n-CaH11OH; @, CH4, CZH6, CsHs, CeHe; HsNCONHz, HOCHzCONHz, HzNCSNHs, CHaCHOHCONHz; I, NHR,CHaNH1, (CH&NH, C ~ H C N(CHa)aN, , piperidine, (C2Hs)sN; X, He, Hz, 0 2 , Nz, CO, Ar, Con; -I-, H&O(CHzCHzO)z CHs, H ~ C O ( C H ~ C H ~ O ) ~ V, C H HCOzH, Z; CHpCOgH,n-CaH?COzH;and for comparison, A,HCOz-, CHsC02-, and n-CsH~C02-. Lines labeled DRP and LRP correspond to packing densities for dense and loose random packing. The curve represents packing densities given by eq 22.
a,
hydro~arbons,~~ alcohols,34nonpolar gases,36ammonia and amines,s6 amides,37and long-chain ethers.38 The packing densities of these molecular solutes a t 25’ are shown in Figure 2. For the small molecules in Figure 2, the packing densities are low and scatter widely. This is what must be expected, even aside from the scatter produced by large uncertainties in some of the partial molar volumes, when molecules of different sizes are accommodated in large cavities in the solvent. Similar behavior is shown by small molecules in the solid gas hydrates. Because there may be transitory, clathratelike cages of water molecules about solute molecules in solution, it is useful to examine the packing densities of guest molecules in the solid hydrates. We cannot expect the cage structures in solution to be the same as those in the solids, nor can we expect a molecule to have the same packing density in the two states. Nonetheless, the behavior of guest molecules in the solid gas hydrates serves as an illuminating, if extreme, reference state for that of solutes in solution. Accurate structural data are available for the solid hydrates.3g The water molecules which form the host lattice of these hydrates are situated at the vertices of polyhedra. In methane hydrate, for example, the hydrocarbon molecules are situated inside dodecahedra and tetrakaidecahedra. The dodecahedron has a volume of 102 ml mol-l, but after allowance is made for the volume occupied by the 20 water molecules on its vertices and edges, the cavity inside it has a volume of only 54 ml mol-’. The packing density of methane The Journal of Physical QhemistTy
in such cavities is 0.32, as compared with 0.47 for aqueous The tetrakaidecahedral cavity has a void volume of 81 ml mol-‘, and the packing density of methane in such cavities is only 0.21. In solid propane hydrate the molecules of the hydrocarbon occupy hexakaidecahedral cavities with void volumes of 108 ml mol-l, and the packing density of propane in these cavities is 0.35 as compared with 0.56 in solution. The tighter packing of solutes in liquid solution is not surprising in view of the greater density of the liquid water. In the solid hydrates with their open host frameworks the lattice density of watel.39 is only 0.79 g ml-l. The packing densities of a given class of solutes appear to approach a constant limiting value for molecules above some minimum size. We tentatively assume that such limits exist and refer to them hereafter as plateau densities. The limits cannot yet be established with certainty because of the lack of reliable partial molar volumes for big molecules or, in the case of carbohydrates, of uncertainty about the hydrogen-bonding correction to the van der Waals volumes. For hydrocarbons, primary alcohols, and carboxylic acids the plateau densities appear to be in the range 0.57-0.59, whereas for amines the limit is about 0.61. These values are lower than the packing densities of like molecules in crystals: 0.740 for closest packing of spheres, 0.681 for body-centered cubic lattices, and values near 0.70 for closest-packing in crystals of organic compound^.^^ Lower packing densities are expected for liquids, e.g., 0.63 for mercury (33) (a) W. 2. iMasterton, J . Chem. Phys., 2 2 , 1830 (1954); (b) R. Kobayashi and D. L. Katz, Ind. Eng. Chem., 45, 440 (1953). (34) (a) M. E. Friedman and 1%. A. Scheraga, J. Phys. Chem., 69, 3795 (1965); (b) D. M. Alexander, J . Chem. Eng. Data, 4, 252 (1959). (35) (a) I. Kritchevsky and A . Iliinskaya, Acta Physicochim. URSS, 20,327 (1945); (b) T.Enns, P. F. Scholander, and E. D. Bradstreet, J. P h y s . Chem., 69, 389 (1965); (c) I. Lauder, Aust. J . Chem.. 12, 40 (1959). (36) (a) R. E. Verrall and B. E. Conway, J. Phys. Chem., 7 0 , 3961 (1966); (b) B. E. Conway, R. E. Verrall, and J. E. Desnoyers, Trans. Faraday Soc., 6 2 , 2738 (1966): (c) S. D. Hamann and 9. C. Lim, Aurt. J. Chem., 7, 329 (1954). (37) (a) F. T. Gucker and W. L. Ford, J. P h y s . Chem., 45, 309 (1941); (b) F. T. Gucker and T. W. Allen, J . Amer. Chem. Soc., 64, 191 (1942). (38) (a) W. J. Wallace and A. L. Mathews, J . Chem. Eng. Data, 9, 267 (1964); (b) W. J. Wallace, C. S. Shephard, and C. Underwood, i b i d . , 13, 11 (1968). (39) G . A. Jeffrey and R. K. McMullan, Progr. Inorg. Chem., 8 , 43 ( 1 9 6 7 ) . (40) Somewhat tighter packing is predicted if we consider the void in the dodecahedron to be a spherical volume with a radius equal to the average of the maximum and minimum distances from center to vertices.89 This volume is 39 ml mol-1, which corresponds to a packing density for methane of 0.43,rather close to the value for solutions. Likewise, if the cavity of the tetrakaidecahedron is treated as an oblate spheroid,sB its volume is 69 ml mol-1 and the packing density of methane in such cavities is 0,25. The sphere and spheroid are probably good approximations to the free volume of the guest molecule, which may be rotating or oscillating, but they do not include the voids near the periphery of the polyhedra. (41) A. I. Kitaigorodskii, “Organic Chemical Crystallography,” Consultants Bureau, New York, N. Y.,1961.
VOLUME
CHANGES
FOR IONIZATION O F
1227
FORMIC, ACETIC,AND n-BUTYRIC ACIDS
to a constant packing density, l/ma) independent of r, a t 25'. Berna142 has proposed that simple liquids a condition that does hold for large molecules or ions correspond in structure to a randomly close-packed but not for small ones. array of hard spheres. The density for loose random Densities of Ions, To calculate packing ~~ packing43is 0.601 and that for random close p a ~ k i n g ~ ~ , Packing densities we need the individual ionic partial molar is 0.634. These values are close to the plateau densities volumes. Zana and Yeager,4*using vibration potential of solutes in aqueous solutions. The comparison of measurements, have recently established that the randomly packed hard spheres with mixtures of solute partial molar volume of the hydrogen ion is -5.4 f and water molecules, which differ in size, shape, and 0.2 ml mol-' at 22', and this result agrees well with force fields, is not as far fetched as it may appear a t earlier estimates at 2jo.4e The convention is adopted first glance. Alder has shown4sthat a 1 : 1 mixture of that AVO for the ionization of water shall be indetwo kinds of spheres, which have volumes in a ratio of pendent of the assumed degree of hydration of its ions, as much as 5:1, has the same density for random i.e., of n and m in the equation packing as spheres of equal size. Differences in size of solute and water molecules may therefore have only (n m 1)HzO Ft H+(H20), OH-(HZO), a minor effect on packing densities as long as the solute Hence molecules are too large to fit into cavities in the water structure. Differences in molecular shape may also V'[H+(HzO),] = V'(H+) nV'(Hz0) (23) have a minor effect on random packing; we know that the packing densities of organic molecules in crystals For the hydronium ion (n = 1) the partial molar (about 0.70) do not differ greatly from those of atoms 18.07 or 12.7 f 0.2 ml mol-', a volume is -5.4 in metals (0.68 and 0.74). As for the difference in value comparable with that of the ammonium ion, force fields between solute and water molecules, we 12.4 ml mol-'. we have taken into account the most important interThe partial molar volumes of other ions are best molecular interaction, hydrogen bonding, in calculating obtained by the substitution method using the estabthe packing densities of solutes in solution. lished value for the hydrogen ion. For example, since The observation that different classes of molecular V'(H+) - V'(Na+) = 1.204 ml m01-',2~ the partial solutes have different plateau densities is not surprismolar volume of the sodium ion is -6.6 f 0.2 ml ing. Bernal and F i n n e found ~ ~ ~ that in a random closemol-'. For C1- and Br- ions we obtain 23.2 and 30.2 packed collection of spheres the packing density was an ml mol-l, respectively. Zana and Yeager's48 experiaverage of the densities around individual molecules, mental values for these three ions are -7.4 f 1.9, the extreme values being 0.57 and 0.70. Two different 23.7 =t: 1.2, and 30.2 f 0.2 ml mol-1. classes of molecular solutes might stabilize different The packing densities of ions are shown in Figure 3. distributions of density. It may also be noted that The smallest monatomic ions (Li+ and Xa+) have in the clathrate hydrates39 different molecules, even negative partial molar volumes, and their packing though they may be polar and capable of forming densities, which are also negative, are not included in hydrogen bonds, can occupy cavities in various ways: Figure 3. There is evidently extensive breakdown of some form no bonds with the water, others hang batlike the water structure extending some distance out from by hydrogen bonds from the host lattice, and still these ions as a result of the electrostriction of water others are built into the wall with their nitrogen or in their intense electric fields. The packing densities oxygen atoms taking the place of water molecules in the of potassium and fluoride ions, considered as spheres host lattice while their alkyl groups project into the with their crystallographic radii, are greater than cavity. unity. The anomalous behavior of fluoride i 0 n ~ ~ is 1 ~ ~ The smooth curve in Figure 2 represents packing almost certainly due to a contraction in van der Waals densities calculated as if the void or dead space about a volume as a result of hydrogen bonding with water. solute particle were equivalent to a spherical shell of An estimate of this contraction can be made by assumthickness a independent of the radius of the particle.46 ing an average 0-H---Fdistance of 2.74A as in da = C T / ( T a)]8 (22)
+ +
+
+
+
+
The empirical constant a is chosen to be 0.55 A. For small molecules eq 22 gives an approximate representation of the trend in packing density with molecular volume, but for large molecules the plateau density clearly cannot be 1.000 as predicted by the equation. A number of authors,47in applying the Born theory to ionic partial molar volumes, have corrected the crystallographic radius of the ions not with an additive term a but with a multiplicative factor m. This corresponds
(42) J. D.Bernal, Nature, 185, 68 (1960). (43) 0. D. Scott, (bid., 188, 908 (1960). (44) J. D. Bernal and J. 1,. Finney, Discussions Faraday SOC.,43, 62 (1967). (45) B. J. Alder, J . Chem. Phys., 23, 263 (1955). (46) E. Glueckauf, Trans. Faraday SOC.,60, 577 (1964); 61, 914 (1965). (47) For example, P. Mukerjee. J. Phys. Chem., 65, 740 (1961); L. G.Hepler. $bid., 61, 1426 (1957). (48) R. Zana and E. Yeager, J. Phys. Chem., 71, 521 (1967). (49) See the summary by P. Mukerjee, ibid., 70, 2708 (1966). ( 5 0 ) F. J. Millero, ibtd., 71, 4567 (1967). Volume 73,Number 6 M a y 1989
1228
Oa4t O 021 0
Figure 3.
EDWARD J. KING
30
60
.V,
I
90 rnl mol-'.
Packing densities of ionic solutes:
I,RNHa'; 0, RzNH2+; 0, RaNH';
1 I
I
le0
160
0 ,halide
+,
180
ions;
X, RaN'; HsO', Rb+, and Csf; A,RCOz-; and for comparison, V, RCOzH and 0, ROH. Lines labeled D R P and L R P correspond to packing densities for dense and loose random packing of spheres.
available for some very Iarge tetraalkylammonium so that a well-defined limit of 0.660 for large i0ns,6~8~' cations can be established. The existence of this limit lends weight to the assumption made tentatively in the preceding section that similar limits exist for molecules. For ions the plateau values of the packing density are approached from above; for molecules they are approached from below. Water is packed more tightly around the substituted ammonium ions than around large amine molecules, which have packing densities near 0.61. The positive charge of the ion could affect the plateau density by inducing the formation of a particuIar water structure or distribution of structures. The observations on clathrate hydratess9 provide no clues about particular close-packed water structures. In the clathrates the nature of the host lattice is not necessarily fundamentally altered when a guest molecule is replaced by an ion. This situation may not hold for solutions because the water molecules are under less constraint and the packing densities of the solutes are higher than in the clathrates. The packing of very large ions depends primarily on water-alkyl group interactions, which are most probably London dispersion forces,68 and the polarizing effect of the ionic charge on water is of minor importance. For a large ion like (C4H9)4N+,in which the positive charge is buried underneath the alkyl groups, we expect, on the basis of the Born theory,2 that the electrostrictive effect of the ion on water will be small. The positive charge is a t one end of the long-chain ion CHs( CH2) ,NH3+ where it is more exposed to solvent, so that we expect it to produce more electrostriction. Yet these two ions, after a correction is applied for hydrogen bonding, both have a packing density of 0.66. Although electrostriction is a precisely defined concept in macroscopic electrostatics, the electrostriction produced by an ion in solution can as yet be calculated only on the basis of crude models. Many workers have therefore used as an empirical measure of electrostriction the difference between the intrinsic and partial molar volumes of an ion.
crystalline KF.4H20,51 and the corrected packing density is shown in Figure 3. This calculation is highly sensitive to the choice of hydrogen bond length, as well as to the partial molar volume of the fluoride so that the corrected value is subject to considerable uncertainty. It is included in Figure 3 only to show that a correction for hydrogen bonding could be sufficiently large to bring the packing density of fluoride ion into a reasonable relation with those of other halide ions. The packing densities of small ions, like those of small molecules, show considerable individuality of behavior. All ions except the anions of the carboxylic acids have packing densities larger than that expected for loose random packing, and the densities decrease sharply as the ionic volume increases. This is what we should expect from the polarizing effect of the ionic charge on water. Thus, the small ammonium ion has a much higher packing density, 0.93, than that of the methylammonium ion, 0.75. (The difference in the number of hydrogen bonds formed by the two ions has been taken into account in calculating the van der Waals volumes.) It is remarkable that the packing densities of the straight-chain ionslS2 RNH8+, are usually almost identical with those of their isomeric - Yo= Y, (24) di-, tri-, and tetraalkyl-substituted ammonium ions.6~8~63 The straight-chain alkyl group may be so coiled in For a given class of ions d l those above some minimum aqueous ~ o l u t i o nas ~ ~to give the packing effect of size have the same packing density and the difference several alkyl groups directly substituted on nitrogen. Anions have lower packing densities than cations with the same van der J17aals volume. Carboxylate (51) G.Beurskens and G. A. Jeffrey, J . Chem. Phys., 41, 917 (1964). ions are particularly remarkable because their packing (52) J. E. Desnoyers and M . Arel, Can. J . Chem., 45, 359 (1967). (53) B. J . Levien, Aust. J. Chem., 18, 1161 (1965). densities are only slightly larger than those of mole(54) A. Bondi, J . Phys. Chem., 58, 929 (1954). cules. Since the charge density of the carboxylate (55) For a contrary view see E. Spinner, J. Chem. Soc., 4217 (1964); group may be low because of electron delocalizationlK6 B, 879 (1967). its polarizing effect on water could be unusually small. (56) B. E. Conway, R . E . Verrall, and J. E. Desnoyers, Trans. Faraday Soc., 62, 2738 (1966). The existence of plateau packing densities for large (57) F. Franks and H. T. Smith, i b i d . , 63, 2586 (1967). ions is evident in Figure 3. Although data are sparse (58) E. Grunwald and E. K. Ralph, 111, J. Amer. Chem. SOC.,89, for anions, good partial molar volumes are fortunately 4405 (1967).
v,
The Journal of Physical Chemistry
v,
VOLUME
CHANGES
FOR IONIZATION O F
V , - V0 becomes proportional to the van der Waals volume,
V,
- Vo =
(1
1229
FORMIC, ACETIC,AND n-BUTYRIC ACIDS
- l/d) V ,
(25)
ammonium ions and amines to be d+ = 0.66 and do = 0.61. Then, according to eq 26a AV,"[(CHs)aNH+]
- AVa'[(CHs)z"z+]
= 0.9 ml mol-' Because it includes both electrostatic and nonelectrostatic contributions to V , as well as the void v o l ~ m e , ~This ~ ~ prediction ~~ agrees well with the observed difference, this difference is not a direct measure of electrostric1.2 f 0.6 ml mol-1 (Verrall and C ~ n w a y or ) ~ 0.7 ml tion. It is common practice to allow for the void mol-' (Hamann and Lim) .360 Using the same plateau volume by expressing the intrinsic volume by Ar3, densities but different values of V , (0) in eq 26a we can where the constant A and the radius r are subject to successfully predict the relative v,olume changes for empirical adjustment.46*47 ionization of the pyridinium and anilinium Volume Changes for Ionization of Large Acids. us. dimethylammonium ion. The calculation fails for From the constancy of packing densities for large triethylammonium6 and piperidiniumS6c ions which molecules and ions we can draw an important conapparently have unusually low volume changes for clusion about volume changes associated with ionizaionization. tion reactions. Let do, d+, and d- be the packing As examples of molecular acids, let us compare ndensities of particular molecules, cations, and anions. butyric and acetic acids. From Figures 2 and 3 we Let SV, represent the difference between the van der estimate the plateau densities to be do = 0.57 and Waals volume of an ion and that of its conjugate d- = 0.615. The relative volume change, calculated molecule, Vw(0). The volume changes associated with eq 26b, is with the ionization of cation and molecular acids can AV," (CsHvC02H) - AV," (CHaC02H) then be expressed by ls9
=
AVao(cationacid) = Vo(H+) - d+ S V W
and SVW
AVao(molecularacid) = Vo(H+) - __
a-
-
(:
- ;)V,(O)
(26b)
The difference in reciprocal densities, which appears in these equations as the coefficient of the V w ( 0 )term, is always positive. For sufficiently large acids the packing densities have fixed values, independent of V , (0). Moreover, SV, is small and virtually constant for a given series of acids. The first two terms on the right-hand side of eq 26 are therefore constant and AV,' is a linear function of V w ( 0 ) . The volume change for ionization of a cation acid should increase as the size of the molecular form increases whereas the volume change for a molecular acid should decrease. These predictions are in qualitative accord with most of the known volume changes. There are no acids with either sufficiently large molecules or sufficiently reliable values of AVao to permit a direct, quaktitative test of the predictions. The best we can do is to predict relative volume changes for pairs of acids. Let us compare, for example, the and ( CHB)2NH2+. From Figures acids (CHa) 2 and 3 we estimate the plateau densities of large
-2.6 ml mol-'
in excellent agreement with the observed difference of -2.72 =t 0.02 ml mol-I. The changes in AV," with molecular size of acids can thus be accounted for by assuming constant packing densities for large molecules and ions. Such changes run contrary to the predictions of simple electrostatic theory,2 because the theory attributes them t o changes in electrostriction with ionic radius. Instead of this, the predominant effect is believed to arise from the difference in void volumes of the acid and its conjugate base. Because of the difference in packing density of the two species, this difference is not zero and it increases with increase in molecular size. Volume Changes for Ionization of Carboxylic Acids. The volume change for ionization of the glycine cation
+NH3CHzCOzH
+NHsCHzCOZ-
+ H+
can be completely accounted for by a combination of the partial molar volume of the hydrogen ion and the van der Waals volumes of the acid ( a ) and its conjugate base ( b ) , Vo(H+) Vw(b) V,.,(a) = -6.95 f 0.3 ml as compared with -6.80 ml mol-', the observed value of AV,O. All terms in the difference
+
A = AV,"
-
- Vo(H+) - VW(b)4- VW(a)
(27)
are known, and values of A are given in Table IV; their precision is limited to f0.3 ml mol-'. From eq 19 and 20 it follows that A can also be considered as a combination of void volumes and volume changes produced by (59) J. E. Desnoyers, R. E. Verrall, and B. E. Conway, J. Chem. P h y s . , 43, 243 (1965).
Volume 73, Number 6 May 1969
EDWARDJ. KING
1230 Table IV: Values of A for Carboxylic Acids Acid Glycine cation Formic acid Acetic acid +Butyric acid
A , ml mol-1
0.15 -1.5 -4.6 -7.3
interactions between particles of the acid and base and the solvent
+
A = VU(b>- V,(U>- Vs(b> V , ( U ) (28) The last two terms on the right-hand side may include the effect of electrostriction. For glycine the electrostriction has been estimated by comparing its partial molar volume with that of glycolamide, its uncharged isomer.60 The difference, after allowing for a small increase in van der Waals volume from HOCHzCONHz to -02CCHzNH3+, is - 12.7 ml mol-l, which is comparable with the following differences:Vo(NH4+) - Vo(NHs) - AV, = -12.8 f 0.5 ml mol-' and Vo(CHsNH9+)- Vo(CHaNH2)AV, = -10.2 f 0.2 ml mol-'. I n forming these differences, we expect the void volumes of ion and molecule largely to cancel, leaving an approximate estimate of the electrostriction. The cancellation will not be complete if the ion and molecule have widely different packing densities. There is a close similarity in packing densities of glycine, its cation, and the methylammonium ion, as recorded in Table V. The high values of these densities in comparison with those of acetate ion and acetic acid show that the ammonium group causes a greater electrostrictive effect than the carboxylate group. The slightly looser packing of glycine compared with its cation is probably due to overlapping, competitive effects of the two ionized groups of the dipolar ion on water. When the glycine cation ionizes, A is virtually aero because a small increase in void volume associated with the change to looser packing compensates for the contraction due to electrostriction caused by the newly created, negatively charged carboxylate group. Formic acid is unique among carboxylic acids because its molecule and ion have almost the same packing density: the values are 0.483 and 0.482, respectively. We can only speculate a t present about how formic acid and formate ion couId be accommodated in the structure of liquid water. The closeness of packing densities of the acid and ion to those of hydrocarbons indicates that they may be guest particles in cavities in the solvent. Another possibility, suggested by structural data for solid clathrate hydrates,8p is that the carboxyl group might replace several water molecules in a transient host lattice. I n neither mode of incorporation of formic acid into water would we expect any profound change in water structure, beyond a slight contraction, to result from ionization. The Journal of Physical Chemiatry
Table V: Some Packing Densities
Formate ion exerts an electrostrictive effect on water that is equal to A if V , for the molecular acid can be neglected. The difference in void volumes, V,(b) - V , ( u ) , is virtually zero because the ion and molecule have equal packing densities. The value of A, -1.5 ml mol-l, is far smaller than estimates of the electrostriction due to ammonium groups, which were between -10 and -13 ml mol-l. The electrostriction can also be estimated as the difference between the partial molar volume of the formate ion and that of a hypothetical HCO2 molecule with the same van der Waals volume. The packing densities of hydrocarbons, water, primary alcohols, and molecular carboxylic acids can be fitted to a smooth curve with a standard deviation of only f0.0063.61~62 From this curve the packing density of HC02, ,for a van der Waals volume of 15.28 ml mol-', is 0.461, and its partial molar volume is therefore 33.17 ml mol-l, as compared with 31.7 for the ion. The electrostriction is thus -1.5 f 0.4 ml m o F , which is precisely the v a l u e 2 A. R e can also make a theoretical estimate of the electrostriction. Let the formate ion be represented by two conducting spheres separated by a V-shaped insulating rod. Each sphere bears one-half of the unit charge e or -2.40 X 1O-Io esu. Suppose that the average number of water molecules coordinated to each sphere over a period of time is 2 and that the distance r from the center of a sphere to the center of one of these water molecules is 2.58 A, the 0-H---0 internuclear distance in crystalline formic wideB3 T h e (60) E". T. Gucker, Jr., I. M. Klotz, and T. W. Allen, Chem. Rev.. 3 0 , 181 (1942). (61) It is assumed that there are two hydrogen bonds between water molecules and the oxygen atom of a carbonyl group and two more, one a donor and the other an acceptor bond, between water molecules and a hydroxyl group. If one donor and two acceptor bonds are formed by the hydroxyl group, the packing densities of alcohols and acid6 are displaced by a constant amount, 0.018, below those of hydrocarbons. It would doubtless be more realistic to assume some distribution either in the number of hydrogen bonds per molecule of acid or alcohol or in the 0-H---0 bond lengths. but the correct distributions are unknown. (62) The largest molecules of hydrocarbons, alcohols, and acids show some slight divergence of behavior, so that there appear to be somewhat different plateau densities d-(RCOz-)
> do(RCHs) > da(R0H) > do(RC0zII)
DifPerent polar or charged groups in solute particles, as we have noted before, evidently stabilize different water structures or distributions of structures. (83) F. Holtzberg, B. Post, and I. Rrankuchen, Acta Cryst., 6, 127 (1953).
VOLUMECHANGES FOR IONIZATION OF FORMIC, ACETIC,AND %-BUTYRIC ACIDS electric field intensity E is given by
E
=
e/2er2
(29)
Grahame’s empirical equatiod4 relates the differential dielectric constant E to the bulk dielectric constant of water e,, the square of its index of refraction no2,the field intensity, and a constant b which has the value 1.08 X esu2 d y r 2 t
=
(E,
- n*2)/(1
+ bE2) + n.2
(30)
From these two equations we find the value of E to be 1.52 X lo4 dyn esu-l. The relationship between field intensity and electrostriction has been worked out by Desnoyers, Verrall, and C o n ~ a y . If~ ~only the first layer of water molecules about the formate ion is affected by the weak field, we estimate the electrostriction to be -1.4 ml mol-’, in excellent agreement with -1.5 ml mol-’ for A. An error of f0.09 A in r propagates to one of f0.3 ml mol-’ in the electrostriction. The values of AV,” for the glycine cation and formic acid have now been completely accounted for. In the preceding section the difference between the AV,” values of n-butyric acid and acetic acid was predicted, and the behavior of acids with still longer alkyl chains can presumably be accounted for in the same way. It is more difficult to explain the remarkable difference between formic and acetic acids: the volume change for ionization of acetic acid is more negative than that for formic acid by 3.1 ml mol-l; the packing density of the acetate ion is considerably larger than that of the molecular acid instead of being equal to it. The difference in packing densities of the molecule and ion of acetic acid may be due to the effect of the methyl group on the structure of water. Alkyl groups may stabilize icelike regions about them.e6 The electrostatic field of the ion might cause some “melting” of these “icebergs,” so that water molecules could pack more tightly about the ion than about the molecule. Some reservations must be expressed about this explanation. Molecular acetic acid appears t o be a normal solute. Its packing density is only 0.6% higher than that of a hypothetical hydrocarbon or alcohol with the same van der Waals volume. If we compare methyl alcohol with molecular formic acid, which has no methyl group, their packing densities, 0.492 and 0.483, respectively, do not differ by more than we would expect from the difference in the van der Waals volumes. Thus, if “icebergs” exist about the methyl groups in neutral solute molecules, their presence apparently does not affect the packing density. Grunwald and Ralph68also failed to find evidence in their kinetic data of multilayer water structures about alkyl groups. The value of A for the ionization of acetic acid cannot be used as a measure of the electrostriction of the acetate ion. If the methyl group does stabilize an
1231
icelike structure in water, A will include a contribution V , from the molecular acid. Moreover, there should be a decrease in void volume as a result of ionization because of the tighter packing of the acetate ion. The electrostriction itself may be the same as that of formate ion-about -1.5 ml mol-1-or it may be larger if the methyl group enhances the field of the carboxylate group. Humphreys and HammetP suggested that the methyl group, behaving as a cavity of low dielectric constant, would repel the negative charge. Because this charge would be forced closer to the periphery of the carboxylate group and nearer the water, it could cause greater electostriction than that observed for the formate ion. I n terms of our simple model of the carboxylate group, we would expect r to be smaller for acetate than for formate. Even if we attribute all of A to electrostriction, the theory does not predict such a decrease in r . The model is so primitive that it is perhaps more surprising that the theory succeeds for formate ion than that it fails for acetate. Entropy-Volume Relationships. A discussion of correlations between entropy and volume changes must be postponed until additional accurate values for AV,“ of carboxylic acids become available. The following observations evolve from the preceding discussion. We have seen that the volume change for ionization of the glycine cation can be entirely accounted for by combination of the partial molar volume of the hydrogen ion and the difference in van der Waals volumes of glycine and its cation. Likewise, the entropy change for ionization of the glycine cation, namely67 -7.4 f 0.1 cal deg-1 mol-’ a t 25”, can be accounted for by the partial molar entropy of the hydrogen ion (-5.0 to -5.5)68 and internal contributions to the entropy change due to translational, rotational, and vibrational motions of the cation and dipolar ion (about -2.2 cal deg-l mol-’). Solvation effects for the acid and its conjugate base cancel in the entropy change just as they did in the volume change. For formic acid the estimated entropy change would likewise include contributions of -5.0 to -5.5 from H+ ion and -2.39 from internal motionsBg plus another contribution, AS,, from polarization of water by the ionic charge. A rough estimate of AS, can be obtained by using the Born model2
(64) D. C. Grahame, J. Chem. Phys., 21, 1054 (1951). (65) See, for example, H. 9. Frank and M. W. Evans, ibid., 13, 507 (1945), and G. NBmethy and €1. A. Scheraga, ibid., 36, 3401 (1962). (66) H. M. Humphreys and L. P. Hammett, J. Amer. Chem. SOC. 78, 521 (1956). (67) E. J. King, i b i d . , 73, 155 (1951): J. Sturtevant ibid., 63, 88 (1941); B. B. Owen, i b t d . , 56, 24 (1934). (68) 0. M. Criss and J. W. Cobble, ibid., 8 6 , 5385 (1964); B. E.
Conway, R. E. Verrall, and J. E. Desnoyers, Z . Phys. Chem. (Leipzig). 230, 157 (1965). (69) Reference 2, p 145.
Volume 78, Number 6 May 1080
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DAVIDG. WILLIAMSON AND KYLE D. BAYES
For water the ratio of differential quotients70is -2.358 cal deg-;-' ml-l a t 25' and, as we have seen, AV8 is -1.5 ml mol-l, so that AS, is roughly -3.5 cal deg-l mol-l. M7e can thus account for only -11.4 of the observed entropy change of -17.2 cal deg-l mol-' for the ionization of formic acid. The entropy and volume evidently respond somewhat differently to ion-water interactions.6
Summary The concept of packing density has been shown to be useful for examination of the volume changes associated with ionization reactions. Explicit account can be taken of hydrogen bonding between solute and solvent. Because the packing density is a dimensionless variable with only a modest dependence on molecular size, it is convenient for making comparisons of solutes of different structures and molecular volumes. We have noted, for example, the close similarity of packing densities of hydrocarbons, primary alcohols, the packing densities and acids' of most alkyl-substituted ammonium ions fall on one
curve regardless of the degree of substitution. Above a certain minimum size, all molecules or ions of a given type have the same packing density, and these limiting values are close to those for the random packing of spheres. The concept of plateau density can be used to account for the effect of chain length on the volume changes associated with ionization reactions. Packing of water about small ions is strongly affected by electrostriction, but for large ions the dead space or void volume next to the ion may be more significant. The concept of packing density should be useful in developing a detailed molecular interpretation of solute-solvent interactions. Acknowledgments. I gratefully acknowledge the support of the University of New England by my appointment as Visiting Reader in Physical Chemistry. I thank also Professor R. H. Stokes and Dr. L. A. Dunn for advice about the measurements and Dr. G. W. King for help with the analyses. (70) B. B. Owen, R . C . Miller, C. E. hlilner, and H . L. Cogan, J. p h y s . Chem., 65,2065 (lesi).
Reactions of Oxygen Atoms with Acetylene by David G. Williamson and Kyle D. Bayes Contribution No. 2845 from the Department of Chemistry, University of California, Los Angeles, California g0O.W (Received July 3 , 1 9 6 8 )
Ground-state oxygen atoms, generated by the mercury-sensitized decomposition of N20, have been reacted with C2Hz. Products include CO, CaH4, Hz, and a polymer. The formation of C3H4 demonstrates that CHz is generated in this system. However, the maximum CO yield accounts for only half of the oxygen atoms, so that the reaction 0 C2Hz+ CH2 CO cannot be the exclusive primary step. The dependence of the hydrogen yield on acetylene pressure suggests that most of the H2 comes from hydrogen atoms. Under conditions where most of the hydrogen atoms are being scavenged, a small H2 yield remains. Experiments with mixtures of CzHt and C2Dz demonstrate that the reaction 0 C2H2 + H2 C20 occurs to a small extent. The possibility of a relative long-lived (C2Hz0)* complex is suggested to account for the large amount of oxygen lost to the polymer. It is concluded that all of the reactions which are energetically possible do occur.
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Introduction The oxidation of acetylene is of primary importance in understanding combustion processes, since acetylene is commonly formed as an intermediate during the combustion of other hydrocarbons.'-a One postulated step4b6in the low-temperature oxidation is the addition of an oxygen atom to the acetylene triple bond to form an unstable complex, (C2H20)*, in analogy to the known addition of O(aP) to the olefins.6 I n a solid matrix, this initial complex is stabilized to ketene,' HzCCO, but in the gas phase a t low pressures various primary The Journal of Phyaical Chemistry
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fragmentations are possible. Fenimore and Jones4originally suggested the fomiation of CH2 and CO, which (1) D. J. Cole and G. J. Minkoff, Proc. Roy. SOC.,A239, 280 (1957). (2) K. H. Homann, M. Mochieuki, and H. G. Wagner. Z . Phys. Chem. (Frankfurt am Main), 37, 299 (1963). (3) I. D. Gay, G. P. Glass, R . D. Kern, and G. B . Kistiakowsky. J . Chem. Phys., 47, 313 (1967). (4) C. P. Fenimore and G. W. Jones, i b i d . , 39, 1514 (1963). (5) 0.A. Arrington, W. Brennen, G. P. Glass, J. V. Michael, and H. Niki, i b f d . , 43, 525 (1965). (6) R. J. CvetanoviC, Adwan. Photochem., 1, 115 (1963). (7) I. Haller and G. 0. Pimentel, J . Amer. Chem. SOC.,84, 2855 (1962).
