NOTES
Oct., 1958 25" and 0.522 for 38". The constant B was assigned values of 1 and 2, corresponding to ionic diameters of about 3 and 6 PI., in order to test the possibility of two convergent extrapolation curves. The figures for p K given in the last line of each table were obtained from the graphs shown in Figs. 1, 2 and 3. The differences between the extrapolations indicate that the uncertainty is about 0.005 in pK1 or pK2 and 0.01 in pK8. Discussion A comparison of the ionization constants with those for citric acid may be of interest. Bates and Pinching' gave the p K values for citric acid as (7) R. G. Bates and G. D. Pinching, J. Am. Chem. Soc., 71, 1274 (1949).
1339
3.128, 4.761 and 6.396 a t 25" and 3.105, 4.750 and 6.429 a t 37". It appears that the first ionization of isocitric acid is definitely weaker, the second, slightly stronger, and the third, about the same. Increase in temperature increased the strength of all three stages in the ionization of isocitric acid, and of the first two for citric acid, but decreased the third ionization of citric acid. Acknowledgments.-The writer is indebted to Dr. H. B. Vickery for suggesting this problem and for supplying monopotassium isocitrate. The work was aided by a grant from the James Hudson Brown Memorial Fund of the Yale School of Medicine.
NOTES T H E KINETIC TERM I N ELECTROLYTIC CONDUCTANCE BY RAYMOND M. Fwossl A N D LARSONSAQER Contribution from the Istituto di Chimica-Fisica dell' Universitd degli Studi di Roma, Rome, Italy and the Sterling Chemwtry Laboratory of Yale Unzuerszty, New Haven, Conn. Received May 36,1968
I n a previous paper on the conductance of unassociated electrolytes,2 we mentioned that the conductance equation contains a small positive term, linear in concentration, which results from interionic collisions. As is well known, an external field produces an asymmetry in the ionic atmosphere of the reference ion. If the latter is, for example, a cation, then there will be a slight excess over the volume average concentration of anions behind it and a slight deficiency in front, compared t o the static equilibrium concentration. Mutual attraction of anions for cations favors the probability of mutual collisions over that corresponding to collisions of uncharged solute particles, and as a consequence of the asymmetry, the reference cation will be struck more often by anions from behind it than from in front of it. The result will be a slight component of velocity in its normal direction of travel. It is the purpose of this note to present a calculation of the corresponding increase of conductance. The effect is equivalent to a virtual force of kinetic nature which acts in the same direction as the applied field, or alternatively, it can be described as an asymmetry in the local osmotic pressure a t the location of the reference ion. The force AP is given by the directed component of the osmotic pressure II, integrated over the surface of the sphere of radius a which surrounds the reference ion. (This sphere is the one into which no other (1) On sabbatical leave from Yale University. Grateful acknowledgment is made for a Fulbright Grant. (2) R . M. Fuoss and L. Onsager, THIRJOURNAL, 61, 688 (1957). In order to save space, all symbols not explicitly defined in the present communication will have the meanings given in this reference.
charge can penetrate, if the ions are represented as point charges a t the centers of rigid spheres of radius 4 2 ; Le., a is the center-to-center distance at contact.) We have as the starting point for the calculation AI' = JII.dS = J I I
COS
0 dS
(1)
where dS = a2 sin 8 d8 d4. In the absence of an external field, the osmotic pressure due to anions (subscript 2 ) is IIo = n2kT
+ 0(x3a3)
(2)
Therefore the asymmetry term, which is a consequence of the fact that n12 # na12when an external field is acting, is given by TI
=
(n12 - n0&T
= (fiz
The distribution function terms f'iz
fiz
f12
- f%)kT/nl
(3)
is made up of three
+ + Fiz
(4)
QIZ
We shall neglect QIZ, because it would lead to a negligible term of order ca/l in the conductance. Hence TI
=
(5)
FlzkT/nl
and the force is given by
KT
AP = (kTa2/nl)
d+
Flz(a,8 ) cos 0 sin e dB (6)
where the asymmetry term in the distribution function is FIZ(T, e) = -(nlelX/4~~kTr2)[e-xr(l ~ r )
+
for 1
- 1electrolytes.
-Ae-yr(l
+ -tr)]cose
(7)
Abbreviating it as
FIP(v,e) = -Rlz(r)
COS
e
(8)
the above integral reduces to A P = -(4aaakT/3nl)Rl~(a)
(9)
Introducing r = a in R12 ( r ) and substituting the value of A obtained by solving the two unnumbered equations of ref. 2 which are located between equations 4.7 and 4.8, we find successively
NOTES
1340 Rl,(u) =
+
( n l e , X / 4 ~ ~ k T u 2 ) [ e - ~ a ( Kl U ) --Ae-ra(l
10' '1 = -(I
p-'[*
- g2)(KZU2/2b)(b -
+ ra)]
(10)
1) $- 0 ( K 3 U 3 ) (11)
and
+
AP = XeiK2U2(b- 1)/12b( 1
KU)
(1
+ +K2a2/6) (12) qKU
To order c3I2 in the final equation, the above reduces to AP
= Xelc[(b - l)/b3][~2u2b2/12~] (13)
for 1-1 electrolytes where q2 = 1/2. The ionic velocity then becomes vi = ( X
+ A X + A P ) ( e j w i - Aui)
(14)
and on following AP through t o the conductance equation, we find M P = (K2U2b2/12C)[(b - l ) / b 3 ] A o ~ -k
O(KSUs)
(15)
for the change in equivalent conductance resulting from the virtual osmotic force. The kinetic term has the same coefficient (K2a2b2/12C)as 81, which should therefore be replaced by a new function defined by 01
3
+
(K2U2b2/12C)[h(b)0.9074 f hl ( K U / C ' / , ) ]
where h(b) = (2b2
+ 2b - l)/bs
(16)
(17)
+
replaces the function g(b) = (1 2b)/b2 which appears in el. Typical numerical values are b
2
4
6
8
10
g(b) h(b)
0.750 0.875
0.562 0.609
0.361 0.384
0.266 0.280
0.210 0.219
The net numerical effect is small, in the sense that u1 differs only slightly from 81, because h is always less than the sum of the other two terms in the square brackets in the expression for (TI. Consistency however requires that this kinetic force be included in the catalog of linear terms.
Vol. 62
as in ref. 2) with absolute perchloric acid. The perchloric acid was prepared by the method of Smith.3 The solid that resulted was dried under vacuum until it was powdery. This powder was washed with anhydrous nitromethane and redried under vacuum. All transfers of the material were made under an atmosphere of dry nitrogen in a dry-box. The samples (1.5 9.) used in the thermodynamic measurements were placed in thin-walled glass bulbs which were quickly removed from the dry-box and sealed with a torch. During this last step, care was taken not to decompose any of the sample by excess heating. The bulbs were then transferred to the calorimeter where they could be broken under water at the appropriate time. The temperature rises in the calorimeter were measured with a temperature sensitive resistance bridge. The standard heat source was a resistance operatsingunder a measured voltage and a measured current. The precision was roughly f 2 cal., and the temperature rise was of the order of 0.1". No attempt was made to correct the data from 24.7 to
25.0'.
The acid equivalent of the nitronium perchlorate was obtained by dissolving the solid in standard, carbonate-free sodium hydroxide. Excess base was used and back titrated with standard potassium acid phthalate to a phenolphthalein end-point. The observed a.e.w. was 73.8 f 0.2 and the calculated value is 72.7. The discrepancy is probably due to some pirk up of water during handling and to the escape of some NO2 that formed during dissolution in the base, although this gas was confined in the titrating vessel and eventually redissolved.
Calculation of the Heat of Formation of the Gaseous Nitronium Ion.-The heat of formation of an ionic solid can be given in terms of the beats of formation of the gaseous ions and the lattice energy U of the solid. For a salt containing a monovalent anion the equation is A H o f w w p )= AH'frM+n(,) nAH0tx-cg) - u - ( n 1)nT
+
+
(A)
The lattice energy can be approximated by the formula of K a p u ~ t i n s k y . ~For salts of a univalent anion this equation is In the equation, n is the charge on the cation and are, respectively, the cationic and anionic radii expressed in fL. U is then given in kcal./mole. This formula has the advantage of not requiring detailed knowledge of the crystal structure, but will correspondingly introduce some error into the estimation of U . When this formula was applied to the ionic azides the results were found to be uniformly low by about 3% as compared to more reliable estimates.6 If equations A and B are to be applied to the determination of the heat of formation of the gaseous nitronium ion, then the radii of the ions must be known along with the heats of formation of the anion and of the salt. The heats of formation of two salts are known: for nitrogen pentoxide' and the present value for nitronium perchlorate. The heats of formation of perchlorate and nitrate ions and their radii can be obtained by applying equations A and B to a series of perchlorate and a series of nitrate salts, for which the heats of formation of the solids and of the gaseous cations are known. I n addition, the ionic radii of these cations need to r + and r-
THE IIEAT OF FORMATION OF NITRONIUM PERCHLORATE AND OF THE GASEOUS NITRONIUM ION BY H. F. CORDESAND N. R. FETTER U.S. Naval Ordnance Test Station, China Lake, California Reoeived May $0, I068
The aqueous enthalpy of solution of nitronium perchlorate has been determined a t 24.7' with an isothermal calorimeter. The value for this enthalpy of solution was found to be -20.4 f 0.4 kcal./mole for a lo4to 1 mole ratio of water to salt. Taking the reaction to be NOaC104(s) HzO(1) + HNOo(aq) HC104(aq) and using the appropriate values for the enthalpies of formation of the species in this equation,' the enthalpy of formation of nitronium perchlorate is derived as 8.0 i 0.4 kcal./mole.
+
+
Experimental The nitronium perchlorate was prepared by mixing, under anhydrous conditions, an excess of nitryl chloride (prepared (1) F. D. Rossini. "Selected Values of Chemical Thermodynamic Properties," Circular of the National Bureau of Standards 500, U. S. Government Printing O.ffice. 1952.
(2) M. Volpe and H. 8. Johnston, J . Am. Chem. Soc., 78, 3903 (1956). (3) G. F. Smith, $bid., 75, 184 (1953). (4) A. F. Kapustinsky, Acta Phy8icochim. U.R.S.S.. 18, 370 (1943). (5) P. Gray and T. C. Waddington, Proc. Roy. Soe. (London), 285, 481 (1956).
