CONDUCTIVITY OF STRONG ELECTROLYTES
477
ON THE CONDUCTIVITY OF STRONG ELECTROLYTES S. G. CHAUDHURY Department of Chemistry, University College of Science and Technology, 92 Upper Circular Road, Calcutta, India Received A p r i l 2.4, 19.46 I . INTRODUCTION
The theoretical treatment of the variation of the conductivity of an electrolyte with its concentration is due to Milner (8), Ghosh (3), Debye and Huckel (2), and Onsager (11). An equation showing the relation between conductivity and concentration-derived by Onsager-includes all experimental results. Its recent modification by Shedlovsky (15), who has introduced an additional empirical constant in the Onsager equation, agrees better with experimental results. I n all these investigations, the effect of the change in the concentrations of ions near the electrode surface (during the time the current is on) from those in the bulk on the conductivities or mobilities of ions has not been considered. It is the purpose of this paper to take this effect into consideration and to deduce an equation for the conductivity of electrolytes. 11. THEORETICAL
Fundamental consideralions Consider two electrode surfaces in contact with a certain concentration of an electrolyte. They will be charged similarly, owing to the adsorption of ions of one sign (Mukherjee (9), Stern (16), Chaudhury (1)). Owing t o this charge, there would be an unequal distribution of positive and negative ions over a distance d (= thickness of the double layer) from each electrode surface. Thus, within this element of volume (thickness d ; area = electrode surface; cf. 4, 9, 16) there would be more ions of opposite charge and fewer ions of similar charge t o that of the electrode surface. Thus the distribution of ions within this element of volume would be different from that in the bulk of the solution. By the application of an external alternating field, the two electrodes are alternately charged oppositely. At the moment the current is just on, these ions would be subjected to interionic attractions and repulsions and the corresponding electrophoretic effects which would differ from the similar quantities in the bulk of the solution (the field near the surface of the electrode being already non-symmetrical) . Near the positively charged electrode when the current is, first just on (if the electrode was originally negatively charged owing to the adsorption of the ions), the velocity of the positive and negative ions under unit potential gradient would be increased, one owing to repulsion and the other t o attraction. Thus, layers of escess positive ions would move away and layers of negative ions would move towards the electrode surface and if the time (t) for which the positive charge is on the electrode is sufficient, there would accumulate an excess of negative ions near the electrode surface, while layers of excess positive ions will have moved into ihe bulk of the sohtion. The gradual
478
S. G. CHAUDHURY
decrease in the concentration gradient of positive ions from layer t u layer as they move away is exactly equal t o the gradual increase in the concentration gradient of negative ions from layer t o layer as they move towards the electrode.. The charge given to the electrode is sufficient to effect this in a very short time. Thus it would be seen that during the latter part of this process, when increasing concentrations of negative ions are accumulating near the positively charged electrode surface, the velocities of the positive and negative ions would be retarded. It must be noted that this accumulation of excess charge near the electrodes would be greater and also spread over a greater distance, the stronger the field applied. Thus the acceleration of the velociiies during the time t / 2 would be exactly neutralized by the retarding e$ect produced afterwards, also in the time t/2. So, also, the electrophoretic e$ects on each of the ions near the electrodes are cancelled out. This involves the assumption that the reversal of the charge of an electrode is instantaneous, so that the ions already oriented near a charged surface do not mix, owing to thermal agitation, before a charge of another sign is on the electrode. The effect of thermal agitation, insofar as it affects the average distribution or orientation of ions near the electrode surface when the field is on, Le., when the ions are moving with a certain velocity and the electrode is charged, may be considered nil. Its effect (effect of Brownian motion) on the velocities of ions in bulk has been included in the electrophoretic and the interionic effects considered by Onsager (11). After a time t, this electrode would be charged negatively and an exactly reverse process would occur. Similar phenomena would occur near the other electrode. Thus, a t any one instant interionic and electrophoretic effects would be nil over a certain volume A d , where A is the surface of the electrode and d is the distance from the electrode t o a point in the bulk of the solution where the outermost layer of the double layer has, for a constant frequency of the current, reached a given voltage of the field applied and a definite concentration of the electrolyte. Had the double layer not been considered a t all from the very beginning, similar concentration gradients of layers of positive and negative ions would have been created near the charged surface by the application of an alternating field. The distribution of ions is given by the distribution law of Boltzmann.’ The net result of the applied field would thus be to annul the e$ect of the interionic attraction over a certain volume near the electrodes.
Modification of the Onsager equation Let n, be the number of positive and negative ions in the volume A d near one electrode where interionic and electrophoretic effects are nil. Of n, numbers of ions, let x = number of positive ions and y = number of negative ions. The corresponding quantities near the other electrode are n:, x’, and y‘. Hence if in Onsager’s (11) equation for the mobility or equivalent conductance of an ion
1
For the number of ions see reference 1.
’
CONDUCTIVITY OF STRONG ELECTROLYTES
me put C
- C8 instead of
479
C , where
C, = either Ct
x
+
2'
= - or
NAd
Y + Y' G=NAd
(considering the total number of either positive or negative ions near both the electrodes in volume A d ) , we have
where Xo
=
the mobility a t infinite dilution,
D = the dielectric constant of the solvent, T = the absolute temperature, N = Avogadro's number, ? I = the viscosity of the solvent, the charge carried by the ion (absolute value), c = the equivalent concentration, x1 and zz = the charges carried by the anions and cations, X!andXi = the mobilities a t infinite dilution of anions and cations, and
z=
For the equivalent conductance, A, of an ion constituent in a solution of a zcni-univalent electrolyte, expression 1 becomes
in which Xo is the equivalent conductance at zero concentration, and this can be put in the simpler form =
A0
-
(ax0
+ 0)de
(4)
1vhel.e
aud
and therefore equation 2 is reduced t o h
=A0
-(ah0
+0 ) d c - E
(5)
For the equivalent conductance, A, of a uni-univalent electrolyte, the relation is A =
no -
(&I
4-0 ) d n -
(ab
P ) d C - C,
(6)
480
G. CHAUDHURY
S.
where XOI = equivalent conductance of the positive ion constituent and XW = equivalent conductance of the negative ion constituent at zero concentration. But
c:
cz =
=
c,
and therefore equation 6 is reduced to A = no
- (ah0 + 2 / 3 ) 4 G - G
(7)
Therefore A from this equation would be greater than nonsager by an amount dA. The corresponding Onsager equation is = AO
A~nsager
-.( ~ A 4o 2P)&
(8)
Subtracting equation 8 from equation 7, we have dh =
(~uno+ 2/3) c, -
C* 2
+ -81 -c, + -1 c: +-5 ct + ......} C 16 C2 128Cs
(9)
+---,+ 1 c3 5cf ...,.,}
(10)
+
16 C
= no -
128 Ca
A ~ +EA ct C S
(11)
+
where A = an0 2p and C,/C is negligible in comparison to unity, as is usually the case. Equation 11 can be put in a somewhat different form,-equation 12, 13.1, or 13.2.
= Ao - A
