R. I. Holliday University of Westwn Australia Nedlands 6009 Australia
Electrolyte Theory and S.1. Units
The difficulties encountered in converting magnetochemistry equations from the c.g.s. e.m.u. system of units to S.I. have previously heen published (1, 21, hut changes in the basic equations used to develop the elementary theory of electrolytes when the c.g.s units are substituted by those of S.I. have not. The purpose of the present short article is to outline the essential differences between the basic equations of the two systems of units when these equations are used to develop Debye-Huckel and Onsager Theories of Electrolytic Conductance. It will then be shown how the derived equations of these theories can he applied to practical measurements of conductance. For easy reference, parallel equations in the text are given the same number but carry the suffix c.g.s. when they refer to that system. Debye Hiickel Theory
4?re2 L DhT 1000'~'~' There is a change in position of the constant 1000, and D, the dielectric constant, is used instead of permitivity 6 . Further application of Debye-Huckel theory develops an equation rflating ionic activity to the Debye-Huckel constant b. The change in chemical potential involved in forming the ion atmosphere is calculated and thence equated to the change in chemical potential due to the non ideal hehavior of the ion being considered. Thus if ji = jiO + kTlnci kTlnfi (symbols used are those in common usage) the last term viz kTlnfi can be equated to the change in chemical potential ~ i . 1 due to an ion atmosphere acting on an ion i (6). From Debye-Huckel theory the electrical work
+
In rationalized S.I. the force F between two electronic charges e is which on differentiation gives (e is in coulombs, r the distance between charges in meters, and 6 the permitivity in Faraday meter-' (or coulombs2 Ne~ton-~m-~).' Using the basic force equation, Poisson's equation is
per ion i. Thus
The corresponding c.g.s. equations are where p is the charge density of a sphere of radius r. The problem is to evaluate p in terms of an ion and its surrounding atmosphere. This is done by applying Boltzman statistics and integrating expression (2 S.I.). The integratdue to the ion and its ed expression gives the potential atmosphere as
*
jijel
=
aG,
-z2ezb - = ----
JN,
20
Ze * =4mr - - - Zeb 4xe where z = the valency of the ion and b = ((e2/ekT)ZNizi2)'I2 (Ni = NO. of ions i per cubic meter). The term zebl4s gives the reduction in potential due to the ion atmosphere and b as such has the dimensions of reciprocal length. Thus l l b is termed the Debye length or the thickness of the ion atmosphere. If lOOOci L is substituted for Ni
where L is Avogadro's number and ci is the concentration of i in mole (dm)-3. In the c.g.s. system the parallel equations to those above are
In eqns. (5)-(7) concentrations are expressed in mole (dm)3 and not mole kg-1. It is usual to express activities with a standard state based on the molal scale rather than the molar one as this enables activity coefficients obtained from colligative properties to be compared with electrochemical data. However, as Dehye-Huckel theory is only applicable to very dilute solutions no serious error is introduced if molarities are replaced by molalities in aqueous solution. Using molalities eqn. (7) becomes logy
z,2e2 e22000L ' I 2 8mkT ckt I/i
= ---
(
)
(8)
where yi is the activity coefficient of i when the standard state is unit molality and I is the ionic strength of solution. Both eqns. (8 S.L) and (8 C.G.S.) reduce to
'
log Y,
may also be written as D m where D is the dielectric constant of the medium and a the permitivity in free space with a value of 8.854 X 10-l2 C2Nm-2.
=
-A=,
0
e
where A, a constant, is 0.509 kg1l2 mole-If2 for aqueous solutions at 25'C whichever system of units is used. Volume 53.Number 1, January 1976 / 21
Onsager Llrniting Law
Using Dehye-Hiickel theory Onsager developed expressions relating conductance to concentration of electrolyte. Although many more sophisticated expressions have been developed, Onsager theory is still the most widely taught in undergraduate courses. In the original derivation, Onsager considered the effect of relaxation on the applied electric field and the change in velocity of the ion caused hy the electrophoretic force acting on it (3).Glasstone (4) equates the forces acting on an ion, when it reaches terminal velocity. Both derivations give an identical result which for uniunivalent electrolytes is
Table 1. Valuer of Physical and Electrical Constants Useful in Electrolyte Theory Calculations conrtant
z:;:;t;2;.rj'e
C.Q.I.rywem
,4i:yz
stant k
Famay F Oie'ectr'c (water 2 5 ' ~ ) Permitivity (Water 2 5 ' ~ )
vircority
(water 2 5 ' ~ ) Potential
2.892 78'5
x
8.95 X 1e.r.u. =
S.I.system
~$
;'$iz:Ki.,
izIi:E;;:eg.,
l
10" e.5.u.
96.494 x 103c
poirle 300 volts
10.'
~
6.95 x 10-" 6.95 x lo-'' c'N-'~' 8.95 X 10-'Nrm-'
volt
,
Table 2. Valuer af Some Electrochemical Constants for Aqueous Solutions for Uni-Univalent Electrolvtes st 2S°C Constant
4&kT or A=Ao-
2454 X lo-*
+ 2.16(c XTYI2lo-''
In the c.g.s. system allowance must he made for the conversion of e.s.u to practical units (i.e., e.s.u of potential is 300 V). Thus the parallel equations are
e2(2 - lh) (-)Aofi (8re2L) lOOODkT lOOODkT 82.4
8 . 2 ~ 1 0 ~
A0)fi
(9c.g.s.) (10 c.g.s.1
At constant temperature for a given solvent hoth equations can he written as A = Ao -(A
+ BAo) 6
(llc.g.s and S.1.)
where A and B are constants. Thus for water a t 25°C A (S.I.) is 6.02 X Sm2 (dm)3" mole-'I2 and A (c.g.s.) is 60.2 ohm-' cm2 1'f2 mole-If2 whereas B(S.1.) is 0.229 (dm)3f2mole-'I2 and B(c.g.s.) is 0.229 1'f2 r n ~ l e - ~ f ~ . Practical Units and Calcuiatlons
The unit of conductance has the same magnitude in hoth systems of units hut is called the Siemen in S.I. and ohm-' in c.g.s. Conductivity r is defined as the conductance of a cube of unit volume and is thus Sm-' in S.I. and ohm cm-' in c.g.s. Molar conductivities A are of more interest to chemists and are defined as the conductance between two plates unit distance apart and of such an area as to include 1 mole of electrolyte. Thus A (S.I.) has the units Sm2 mole-' and A (c.g.s.) has the units of ohm-' cm2 mole-'. Because the concentration scale is the same A(S.L)
=
and
22 1 Journal of Chemical Educatbn
lRMc
(12)
D. H. constant b ( c = 0.1 MI Debye length l l b (c = 0.1 MI onsager conrtant A Onsager constant B
C.Q.S. sy~tem 1.04
X ~o'cm-'
S.I.ryrtem 1.04 X
109m-'
9.6 X 10-'cm 9.6 X 10-"M 60.2'ohm-' cm'~'" mole1* 6.02 x 1 0 ~ ' S m ~ , , 2 (dm)'" mole 0,2291r* mole-m 0.229 ldml'" moiei'"' 0.509 kg'" mole-In Oebye ~ i m i t i n g~ a w0.509 kg1* mole'* Constant A
I n order to calculate electrochemical quantities one ohviously needs different values of physical constants for hoth systems of units. Table 1 gives these values and Table 2 some electrochemical constants calculated from them. Although S.I. is more rational than the c.g.s. system it still contains many irrationalities. These are basically due to keeping the definition of the mole the same as in the c.e.s. svstem. Thus if the mole was defined in terms of 12 kg of =arb& a i d Avogadro's eonstant became 6.023 X 1026 newmole-' (5) the factor 1000 would disappear from all the S.I. formula quoted above. Concentrations would be in newmole m-3 and would have the same numerical value as the present m~le(dm)-~ (molarity). In the ideal gas law R would have the value 8.314 kJ K-I mole-' and molar volumes would be a thousand times Lareer. Values of thermodynamic functions and many other physical quantities would be 103greater than at present but would not be troublesome to handle. Literature Cited (1) Quiekenden,T.I.,and Maraha1.R. C..J. CHEM.EDUC..49.114 (19721 (2) Oavies.N.H.,Chem.Brit.. 7.331 (1971). (31 Onsapr. L.. Tuns. Forodoy Soc., 23.341 (19271. (0Glasstone, S.."lntroduction to Elcctrachernistry:' 1954,rr. 87. (51 McMsnur.F.R.. Chem.Brif.. 7.909 (19711. (61 Hunter. R. I.,J.CHEM. EOUC.43.550 (19661.
2The suggestions in the appendix are contrary to current IUPAC policy. The "newmole" is not an accepted concept at this time and should not he used.
~
~
