DONALD G. MILLER
3588
ment of -5.0 eu at 25" for the hydrogen ion).24 From Table I, average values for Acp]26tand also values for ASt" can be obtained at 25, 60, 100, 150, and 200" for use with Criss and Cobble's estimates of ionic quantities a t these temperatures. From eq 1 for the dissociation equilibrium, the (assigned) entropy of the neutral species, I\IgS040, is equal to [S,O(Xg2+) 3," (S042-) 1 (from Criss and Cobble) minus AS,' (from Table I) where t refers to the temperature. The average heat capacities for 1\1gSO4"are obtained by a similar relationship. The calculated values for the two quantities are given in Table 11. They show that the entropy of ;\IgS040 is relatively small and, surprisingly, changes very little with temperature. The average heat capacity, of hIgSO4" rises with increasing temperature.
+
cp]25t,
Table 11: The Entropy and Average Heat Capacity of [ I I I ~ S O ~ ~ ]
T, O C
25 60
100 150 200
StO,
CpIm',
cal/mole deg
cal/mole
-1.6
- 50
-5.1 -6.1 -5.3 -4.5b
deg
-39"
- 25
- 10 -2
From smoothed Criss and Cobble valuesz6a t 60" for ?p]2560 for Mgz+ and S04Z-. * Calculated using smoothed value for S042- and e5trapolated value for Mg2- from Criss and Cobble24 t o 150".
Application of Irreversible Thermodynamics to Electrolyte Solutions. 111. Equations for Isothermal Vector Transport Processes in n-Component Systems'
by Donald G. Miller Lawrence Radiation Laboratory, University of California, Linermore, California 94550
(Receized April 3,1967)
Irreversible thermodynamics is applied t'o isothermal vector transport processes of an n-component system consisting of n - 1 electrolytes with a common anion dissolved in a neutral solvent. Rigorous expressions for the conductance A, transference numbers i f , and solvent-fixed thermodynamic diffusion coefficients (L& are given in terms of ionic transport coefficients, Zi,, and conversely. Rigorous expressions are also given .for ion flows J c in terms of ti, A, and either (L& or diffusion coefficients ( D f J 0 . Limiting expressions at infinite dilution are given for Lit and Di,in terms of limiting ionic conductances.
I. Introduction I n previous papers, parts I2 and II,3 isothermal vector transport processes in binary and ternary systems were treated in some detail, and comparisons with experiment were presented. I n view of increasing interest in higher order systems, particularly the accurate The Journal of Physical Chemistry
measurement of diffusion coefficients in quaternary it Seems worth Presenting briefly the results (1) This work was performed under the auspices of the U. S. Atomic Energy (2) D. G. Miller, J . p h y s . Chem., 70, 2639 (1966). (3) D. G. Miller, ibid., 71, 616 (1967).
ISOTHERMAL VECTORTRANSPORT PROCESSES IN ELECTROLYTE SYSTEMS
for the general case of an n-component system consisting of a neutral solvent and n - 1 binary electrolytes with a common anion. The common-cation case is obtained merely by relettering. Limiting expressions a t infinite dilution are also given for ordinary and thermodynamic diffusion coefficients. These may be of some practical use since, as far as is known, there are no complete experimental diffusion data on any quaternary or higher order system.
11. General Equations and Notation Equations from papers I2and 113will be distinguished by the prefixes I and 11, respectively. We introduce the following notation for our n-component system. Let the neutral solvent be denoted by subscript 0, the cations of the n - 1 binary electrolytes by subscripts running from 1 to n - 1, and the common anion by subscript n. Each individual electrolyte as a whole can also be denoted by the subscript of its cation. Each of these binary electrolytes ionizes as (ci)r&Tia
+
ricCtZi riaAzn
+
(1)
where C , denotes the cation, A the common anion, x t the signed valence of the ions, and ric and r i a the stoichiometric coefficients for the ionization. If p,, p,, and p i n are the chemical potentials in joules per mole for the cation i, anion, and electrolyte as a whole, respectively, then ptn
r w
=
+
i
Ttap,
= 1,
. . ., n - 1
(2)
Moreover, by charge conservation rtczt
+
riaZ,
i
= 0
= 1,
. . ., n - 1
(3)
If c i (i = 1, . , ., n - 1) are the concentrations in moles per liter of the electrolytes as a whole, then the number of equivalents per liter of a cation is given by
N,
= riczfci
i
= 1,
. . ., n
-1
(4)
The total number of equivalents per liter, N , is the same as the number of equivalents per liter of the anion and is n-1
N
=
n- 1
xridici = -Z,CriaCt = Nn
(5)
i-1
i=l
3589
fi
is the gas constant in joules per mole degree, and a function of T only. This can be written in terms of the moles per liter mean activity coefficient y t of the binary electrolyte i as5
puz$is
ai = cic~iCcrari8ytTi
(8)
where rt, the total number of ions of electrolyte i on ionization, is ri =
Tic
+ ria
(9)
For a common anion system n-1 cic
=
rlcCt;
cia
=
CrlaCl
(10)
1-1
so that
Chemical potential derivatives will be useful. Therefore, substituting eq 11 into 7, differentiating, and making use of eq 5 and 6 , we obtain upon relettering
common anion system, k , j = 1, . . ., n
-
1
(12)
where the notation p k j is defined by eq 12, and 6 k j is the Kronecker 6. We now turn to the irreversible thermodynamic description of this system. There are n 1 diffusion constituents, i.e., n ions and the solvent. However only n of them are independent, because of a choice of reference frame and because of the Gibbs-Duhem equation.2 As b e f ~ r e ,it~ ,is~convenient to choose the solvent-fixed reference frame and choose the n ion flows as the independent set. I n this situation we may write
+
n
Jz = C l t j X , j=1
i
= 1,
*
~
a ,
n
(13)
where Ji are the solvent-fixed ion flows in moles/cm2 see, li, the solvent-fixed ionic transport coefficients in mole2/joule cm see, and X I the thermodynamic forces (in one dimension) given by
The equivalent fraction is defined as =
Ni/N
(6)
andx, = 1. The expression for ptn is given in terms of the activity a t of the electrolyte as a whole by pin = pino
+ $T In ai
i = 1, . . ., 72
-1
(7)
where T is the absolute temperature in degrees Kelvin,
where 5 is the Faraday in coulombs per equivalent, 4 the electrical potential in volts, and x the distance in (4) H. Kim and L. J. Gosting, private communication. (5) D. G. Miller, J. Phys. Chem., 63, 570 (1959); corrections, ibid., 63, 2089 (1959).
Volume 71, Number I1 Odober 1967
DONALD G. MILLER
3590
centimeters. The solvent flow J o is zero on this reference frame. Because the J iand X ihave been properly chosen,2g6 the Onsager reciprocal relations (ORR) lij
i , j = 1, . . ., n
= 111
+
(15) l)/2 of the n2
are valid. Consequently, only n(n 1l5 are independent. We now turn t o the general equations for pure diffusion, where the flows are those of electrolytes as a whole. The generalized Fick law expressions in terms of solvent-fixed flows and solvent-fixed diffusion coefficients (Dij)oare
and in terms of solvent-fixed thermodynamic difin niole2/joule cm sec are fusion coefficients
where the flows of electrolyte as a whole Ji, in moles/ cm2 sec are J i
i = l , . . ., n - 1
Jtn=-
TlC
(18)
and where ( f i t j ) 0 (in liters/cm sec) are given in terms of the usual diffusion coefficients (in cm2/sec) by
(Dtj)0/1000.027 (19) The ( f i i j ) 0 are related to the (Ltj)0and conversely by the equations’ (blj)o
=
n-1 (fiij)O
= C(Lik)Opkj k=l
i, j
=
1, . . . I n - 1
(20)
111. Relations among the Transport Quantities The analysis of conductance leading to eq 1-15 and 11-12 is straightforwardly generalized for the n-com-
11-23 to 11-25, except that sums run from 1 to n. The bracketed term of eq 11-25 can be written in general in terms of electrolytes as a whole as
zz- ax
BCL 1
23’ B X
=
(1 - 65,)-
&jn
--
ric Bx
(1 - 6,,)-
zj
&zn
-
rzc ax
(23)
When j = n, the first term on the right side vanishes; when 1 = n, the second term vanishes; and when j = I, the whole thing vanishes. When eq 23 is substituted into the generalized eq 11-25, with the above conditions applied and a dummy index appropriately relettered, one obtains the generalization of eq 11-27 with i, j = 1, . . . , n - 1 and 12, 1 = 1, . . . , n. Identification of terms yields the result analogous to eq 11-28
i, j
= 1, . . ., n
- 1 (24)
Because the ORR apply to the Zij, eq 24 has the consequence that there are (n - l)(n - 2)/2 ORR among the (L&. Therefore, only n(n - 1)/2 of the (Llj)oare independent. Consequently, eq 20 implies that only n(n - l ) / 2 of the (DiJ0 are independent as well. The independent quantities are the conductance, n - 1 transference numbers, and n(n - 1)/2 diffusion coefficients. These add up to n(n 1)/2 independent experimental quantities, which is precisely the number of independent lij. Consequently, eq 21, 22, and 24 can be solved simultaneously for the lij. The technique is as follows. Fori, j = 1, . . . , n - 1, form the sum of (a2tit5/ztz1) aricrjc(Llj)o.After appropriate cancellation, one obtains
+
+
ponent system, yielding To obtain It,, eq 25 is substituted into eq 22, which after rearrangement yields where X is the specific conductance in (ohm cm)-l, A is the equivalent conductance in cm2/ohm equiv, and cr is defined by eq 21. Similarly, the analyses leading to eqs 1-22 and 11-14 for the Hittorf transference number yield directly the general result
The pure diffusion case is more complex. The argument is everywhere analogous to that leading from eq The Journal of Physical Chemistry
atit,
lin
= Xi&
n- 1
+ ricCr1a(Li~)0
i
= 1,
. . n - 1 (26) - 1
z=1
Finally, I,, is obtained by substituting eq 25 and 26 into eq 22 for t,, yielding eq 27. (6) S, R. DeGroot and P. Maeur, “Non-Equilibrium Thermodynamics,” Interscience Publishers, Inc., New York, N. Y., 1962, pp 64-69. (7) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, and G. Kegeles, J. Chem. Phye., 33, 1505 (1960).
ISOTHERMAL VECTOR TRANSPORT PROCESSES IN ELECTROLYTE SYSTEMS
1nn
=
atn2
+
7 C zn
?l--l
k=l
n-l
E Tkaria(Ltt)o 2=1
(27)
With eq 3, eq 25-27 can be combined into the single equation
(6,~ -
6jn)Tkarla(~tdo
i, j = 1,
.
J
3591
n
-1
PI/Ioreover, because the right sides of eq 16 and 17 are equal, eq 30 and 33 can be written in terms of diffusion coefficients as
(28)
Equations 21, 22, 24, and 28 are the desired relations between the I f , and the ordinarily measured transport quantities when ~ i N / l O ~is5substituted ~ for a.
IV. Flaws in Terms of Experimental Quantities We now turn our attention to expressing ion flows
J i in terms of the usual experimental quantities ti, A, and (Dfi)o. If we substitute eq 28 into eq 13, we obtain
Equations 30, 33, 34, and 35 are the desired general expressions for J r (i = 1, . . ., n) in terms of experimental quantitiess and represent the general statement of superposition of diffusion and electrical flows.
V. Limiting Equations at Infinite Dilution i
= 1,
. .., n (29)
where { } is a triple summation term involving X, and the ( L f J O term of eq 28. If the Kronecker 6 terms of the { ] are multiplied out and the four 6 products are applied separately, the triple sums become double sums. By appropriately relettering a dummy index, adding up the four double sums, factoring, and canceling, we find that the dqb/bx terms cancel and the dp,/dx terms combine to form gradients of electrolytes as a whole. When the result is substituted into eq 29, we obtain
1.
where the last term on the right side is the { The first term of the right side of eq 30 can be written in terms of the current density I as
we considered a number of approximaI n paper tions for ternary 1(, in strong electrolyte solutions, which in turn led to estimates of A, tt, (Ltj)0,and (DtJ0. The success of these estimates depended on how many binary data were available. However, fair estimates of (LfJ,and (D,,)ocould be obtained from the infinite dilution approximation, although A and ti were unsatisfactory a t finite concentrations. Therefore it seems reasonable that infinite dilution equations should yield a fair estimate of (L,)o and (D& for higher order systems, and this approximation has the advantage that only the limiting ionic conductances are required. The infinite dilution approxiniation analogous to eq 11-68 is
where bhe superscript zero refers to infinite dilution, A$ is the limiting ionic conductance, and all ionic cross terms I,O/N are zero. Substitution of eq 36 into eq 21 and 22 yields the trivial results n
(31)
Ao =
where { ] is again the last term of the right side of eq 29 or 30. The sum over the zt { ] is found to be identically zero and Zti = 1. Hence
ti = x t h j / h Q
--I = $a4- + E -t 5-dr, sa ax j p l x j ax If eq 32 is substituted into 29, we obtain
(32)
EX& i=l i
= 1,
(37)
. . ., n
(35)
If we substitute eq 36 into eq 24, factor out the unsummed i quantities, apply the various Kronecker a's, (8) The p i of eq 30 and 34 are of course not really "experimental." However, in the analysis of any experiment either the [ ] vanishes or the Ji's are coupled with external electrodes such that the g j ' s combine to form expressions involving only electrolytes as a whole and the 4 term is absorbed in a measurable emf.
Volume 71,Number 11
October 1967
H.L. RETCOFSKY AND R. A. FRIEDEL
3592
use eq 37, and use the fact that 6 t j / z i = 6 , j / x j , we obtain the nontrivial result
L$ _ -- X& [ 6 i p i 0 - x ~ A P ] N 10%ic~5,ziz552h0
out and apply the various Kronecker 6 terms, cancel and rearrange, and use 6j,/x, = 61j/zt and eq 19, we obtain
i , j = 1, . . ., n - 1 (39) which is analogous to eq 11-72, 11-73, and 11-74. To get a diffusion coefficient expression, we need the limiting expression for p R j ,namely pk5 =
i?T -b"kC6kj
c5
i , j = 1, . . ., n
-1
(41)
which is analogous to eq 11-76 to 11-79. Equations 41 are the wary analogs of the Nernst-Hartley equation.
Vi. Remarks f rkaxJ1
(40)
This is obtained from eq 12 by noting that c j b In yk/bcj goes to zero as as a result of the Debye-Huckel theory, where S is the ionic strength. If we substitute eq 39 and 40 into eq 20, multiply
The quaternary system equations are easily obtained by specializing the general eq 21, 22, 24, 28, 30, 33, 34, 35, 39, or 41. Better estimates for diffusion coefficients can be obtained by means of better approximations to l i j and by data or better approximations for p k j .
Carbon-13 Nuclear Magnetic Resonance Studies of 4-Substituted Pyridines
by H. L. Retcofsky and R. A. Friedel U.S. Department of the Interior, Bureau of Mines, Pittsburgh Coal Research Center, Pittsburgh, Pennsylvania 16819 (Received April 3,1967)
Carbon-13 magnetic resonance spectra of nine 4-substituted pyridines have been obtained and analyzed. Both electron-releasing and -withdrawing substituents were investigated. Substituent effects were generally within *3 ppm of those found in the corresponding monosubstituted benzenes and suggest similar shielding mechanisms for the two classes of compounds. The 6-vinyl carbon shielding in 4-vinylpyridine supports the common proposal that the nitrogen atom in pyridine resembles electronically the CNOz group in nitrobenzene.
introduction During the course of an extensive investigation of the structure of coal using spectroscopic techniques, the need for a collection of carbon-13 magnetic resonance spectra of nitrogen-containing heterocyclic molecules became evident since these materials are used extensively as extracting agents for coal and have been shown to form molecular complexes with coal. Heterocyclic The Journal of Physical Chemistry
nitrogen is generally assumed to be present in coal although its existence has never been confirmed. C13nuclear magnetic resonance parameters obtained for nine 4-substituted pyridines are reported here as the first phase of an investigation of carbon shieldings and spin-spin couplings in monosubstituted pyridines. Data for the parent compound and 4-picoline have been previously published.' The 4-substituted compounds
