Colln A. Vincent University of St. Andrews St. Andrews. Fife. Scotland
The Motion of Ions in Solution under the Influence of an Electric Field
Study of the conductance of electrolytic solutions has made a considerable contrihution to the development of physical chemistry, and conductance measurements are currently of great value both in advancing solution theory and as the basis of a number of important analytical methods. The teaching of the subject a t an elementary level, however, suffers from the use of parameters and units which are not easy to grasp, and from a picture of ionic motion that is far from reality. The purpose of this paper is 1) to clarify the understanding of the nature of the motion of ions in an electric field, and 2) to reappraise the units and nomenclature of the subject: A Picture of lonic Motion in an Electric Field
A common description of the movement of ions in solution under the influence of an electric field considers the solvent as a viscous, continuous and, ahove all, stationary medium. Ions are accelerated by an applied field towards anode or cathode and finally achieve a steady velocity due to the retarding action of a frictional force which is proportional to the ionic velocity. This approach, based on the application of Stokes' Law for macrmcopic bodies to a molecular system, has had some success, for example in predicting reasonable radii for solvated ions. Its fallacy lies in the assumption that the thermally generated motion of the ions can be regarded as a small and unimportant perturbation of a steady linear motion in the field, and this error, in quantitative terms, leads to a number of impossible conclusions. Despite the fact that this "ball-beariog-in-a-viscorn-fluid" or hydrodynamic model was effectively discredited almost 40 years ago ( I ) , there has been little attempt to introduce a more realistic treatment in textbooks of general or physical chemistry. Due to thermal energy, molecules of solvent and solvated ions are continually colliding, and thus move in a random manner with frequent changes of speed and direction. In the absence of an electric field, typical ions move over short distances at speeds of over 100 m s-' and experience in 1 s mean displacements of the order of m. If the motion of the ions could he examined more closely, one might perceive that they move by a series of jumps or activated transitions between equilibrium positions of relatively low free energy. An ion in such an equilibrium position or lattice site may be considered to vibrate several hundred times within a cage of solvent molecules under the influence of collisions with these molecules, until such a time as it has acquired sufficient activation energy to move through some transition state configuration to a new equilibrium position. I t can he shown from a detailed model that the frequency of the ion jumps between equilibrium positions is of the order of 1010-1011 s-1. When an electric field is applied to the solution, ions experience a force attracting them towards the oppositely charged electrode. This directional force, superimposed on the random collisional forces, causes the ions to have a net motion along the axis of the field. In terms of the transition state theory ( 2 4 )the field raises the energy of upfield cation "equilibrium positions" and lowers that of downfield positions (while having the opposite effect for anions). This results in a net drift of cations towards the cathode and anions towards the anode. I t is important to consider the relative magnitude of the effect of an applied field on the total motion of an ion. A typical experimental field of 1 Vm-' is found to produce mean net ionic velocities of the order of 50 nm s-'; the instantaneous 490 / Journal of Chemical Education
4
m
I
Apparatus.
velocities of ions in thermal motion are a factor of over logas large. In other words, if the total number of jumps performed by an ion in 1s is lo", a 1Vm-' field would produce a displacement during this interval equivalent to only 100jumps. Thus the effect of an electric field is to introduce a small anistropy into the violent but random Brownian motion of the ions. The net ionic velocity is due to the cumulative effect of this feeble though persistent perturbation. Mean Ionic Velocities and the Conductance of Soiutlons
For many purposes one is concerned only with the net motion of ions, and because of the enormous numher of -~ ions ~.-~present, even in dilute solution, one can consider mean velocities and mean displacements: i.e.. despite the nicture of ionic motinn on the molerular scale discussed ahove, nne can often ignore the "random walk" of the ions and concentrate attention solely on the drift or mean net velocity. Consider a solution of completely dissociated electrolyte in which each mole of electrolyte dissociates into u+ moles of cations of valency z+ and u- moles of anions of valencv z-. The maenitudes of the products z+u+ and z-u- must he the same, a n d a e may define ~
~
~
For example, for La? (SO& Further let 1dl of such a solution containing c moles of electrolyte contain N+ cations and N - anions, where N+ = v+eL
and L is the Avogadro constant. Consider a tube having length 1' and cross-section a, filled with the above solution and having a potential difference of E applied between electrodes across its ends (see figure). Under favorable conditions-when the electric impedance of -
' The convention that "physicalquantity = numericalvalue x S1
unit" ir mnintained, except where it has seemed advisable ro empha5ine the nature of the unit invulwd.
the electrode/solution interfaces has been made negligibly small-a homogeneous field, of magnitude P = El1 and direction xy, then acts at all points within the solution. Under its influence, cations and anions experience a net force which results in their moving along the axis of the tube with and i-,respectively. mean net velocities of i+ Let there be a planepl normal to the axis of the tube at any ~ositionalong the axis, and a second plane, p2, parallel t o p , i n d at a distance u+ (in meters) to the cathodicside of p, (see figure). Then in 1 s, a "mean cation" originally situated in plane pl will have moved to plane p2. Further, during this time interval, effectively all the cations originally situated in the volume between p l andpz will have moved t h r o u g h p ~Now . the volume of solution between the two planes is au+ (in units of m3) or 1000 au+ in units of dl, and thus the numher of cations in this volume is lOOON+au+.Hence the numher of cations moving through any planep in 1s must he lOOON+au+. Further, since each cation carries z+e coulombs of charge, the total flux of charge through any cross-section of the tuhe due to the motion of cations in the electric field is ;+ =
+
+
and for a simple electrolyte t+ t- = 1where t + and t- are the transport numbers of cation and anion,. respectively. . By theabove definition
1000z+eN+o++
(where li+l has units of C s-I or A). Similarly one can show that the numher of anions moving through any cross-section (in the opposite direction to cation flow) is 1000 N-av-, so that the flux of charge due to anion movement is Then the total charge flux along the axis of the tuhe in direction xy is =
known as the cell constant for a conveniently sized cell by measuring R when the cell contains a solution of known conductance. If n is found for a solution of known concentration, then (u+ u-) for the salt (in units of ms-l) can be readily determined from eqn. (2)-i.e. conductance measurements provide the sum of ionic mobilities of the constituent cation and anion of the salt. In order t o consider individual ion mohilities one must divide the total current in the cell into current due m the motion of cations and current due to the motionof anions. The electric transport numher, ti, of an ionic species i is defined as the fractionofthe totalcurrent carried hy that species. Obviously
lOWr+enN++++ 1OW z-eaN-+= lWOz'Fea(++ - t-1
Now*+ = lu+I.x a n d i - = Iv-14-x), whereir is theunitvelocity vedor. Hence and therefore
Define ui, the mobility of an ion as magnitude of its mean velocity in unit field strength, i.e. in a field of 1Vm-': then
u+ = t+(u++ U-) (3) u- = t-(u+ + u-) The t r a n s ~ o rnumher t of an ion can be measured hv a number f or the of techniiues (5)such as the classical ~ i t t o rmethod, generally more accurate moving houndary method, and thus the sum of the ionic mobilities can he split into the constituent individual ion velocities. The independence of movement of ions in an electric field was recognized by Kohlrausch who showed that a t infinite dilution (see helow) the velocitv of the cation was comoletelv independent of the nature of the accompanying anion in solution, and vice versa. Values of the sum of the ionic mohilities for a series of potassium and sodium salts in water at 25°C are given in Table 1.Subtraction of the values corresponding t o a pair of salts with a common anion then gives the difference in velocity between the potassium and sodium ions. For the examples given, this is seen to have a mean value of 24.2 nm s-' and to be inde~endentof anion, with a variance of 0.1 nm s-1.
I
Ion-Ion lnteractlons Now Ohm's Law applies to electrolytic solutions a t low field strengths, so that i = PlIR
where R is the resistance of the electrolytic solution. Also R = plIA = llxa where p is the resistivity of the solution and nits conduciiuity. (The units of x are n-lm-', now often given as Sm-' where the reciprocal ohm takes the name siemens and symbol S.) Suhstituting in eqn. (1) above, we have From eqn. (2) i t is easy to see that the conductance of an electrolytic solution becomes larger 1) the greater the numher of units of charge belonging to the salt, 2) the greater the concentration of the salt, and 3) the greater the velocities of the individual ions in unit field. K can he determined by measuring R , usually by means of an ac bridge, if 1and a are known. Normally one determines the quantity
Long Range
So far it has been assumed that ion velocities are independent of salt concentration. Equation (2) states that the conductance of a salt solution increases with concentration as the number of charge carriers per unit volume increases, hut does not take into account the fact that a t higher concentrations an ion is surrounded hy ions of opposite charge (the "ionic atmosnhere") which mav .imoede . its movement. Kohlrausch observed that a t low concentrations the sum of the ionic mohilities for uni-univalent salts obeyed the relationship ~~~
~~
where B is a constant. The parameter up is known as the ionic Table 1. Limiting Ionic Velocities of Some Potassium and Sodium Salts ,,OK+
~nion 1x1
+ "OX
lnm *-'I
uDNa+ + uOx lnm r-'1
u°K+ lnm 8
1
Volume 53.Number 8, August 1976 / 491
mobility "at infinite dilution." For dilute solutions (60.001 M for 1:l salts) the value of (u?, + u)! can be found by measurement of K and hence (u+ u-) a t a series of values of c and extrapolating the resulting linear (u+ u-) versus 6plot. More complex theoretically based relationships allow ionic atmosphere retardation effects to he allowed for up to about 0.1 M in favorable cases. but these eenerallv reauire the computer processing of d a k in order to'kxtrapiate infmite dilution. T r a n s ~ o rnumbers t are also concentration denendent ' and infinite diiution values must he found. Then
+
Table 2. LimitingVelocities of Ions in Unit Field at 25-C in Water
+
tb
uQ + - t0 + (u$
+ u)!
Ion velocities (Table 2) are almost always compared a t infinite dilution-i.e. in a situation where ions are too far apart in the solution to affect each other's motion in any way. Short Range
Up till now we have discussed only the situation of the "classical strone electrolvte" and have not considered the existence in sol&ion of a&ciated species. If part of a solute exists in molecular or associated form, then there will he fewer charge carriers in solution than if thesolute werecompletely dissociated. Such a situation is well known in the case of "weak electrolytes" (mainly organic acids and bases). However, in addition, salts, especially those of high valency or when dissolved in solvents of low dielectric constant, form associated "ion pairs!' Usually these are short-lived and held together by electrostatic forces. The ion pairs dissociate and reform continuously so that there is an equilibrium concentration of uncharged or charge-reduced species in solution. For the acetic acid system
+
HAc = H+ (aq) Ac- (aq)
define a as the fraction of acetic acid molecules which have ionized in solution (i.e. as the degree of dissociation). As only H+ (aq) and Ac- (aq) species are able to conduct electricity, we have r
+
= 1000F(uc)(u~+UA,-)
(4)
since z' = 1. For very weak electrolytes, the concentration of ions is so small that their velocities can be taken as approximately equal to the velocities at infinite dilution. That is r
= lWOF(ac)(uR++ u&)
above and for accurate determination of a or K, i t is necessary to allow for these. Again, this is often done using an iterative computer program. Ionic Velocities, Equivalent and Molar Conductance Until recently the results of conductance measurements have been given in terms of the Darameten A. the "eauivalent conductan>en of the salt, and Xi the equivalent conductance of an ion, where v i. , , , = A?"" + A?"'" = F(u+ + u-) From eqn. (2) i t can be seen that n = 1000z'c Aeq~". The equivalent conductance of an ion, while not so simple a concept to grasp as the mean velocity, is a useful quantity since, as with ui values, tables of ionic equivalent conductances can he drawn up which allow AquiVfor a salt to be determined (and hence n calculated) simply by adding the equivalent conductances of the cation and anion involved. Further, since the velocities of ions of any charge can he compared directly by inspection of such a tahle. The currently favored parameter, the molar conductance of a salt, Amolar, is defined as
or
= J~WOFC(UR+
+ ug,)
(5)
Note that i t is not practicable to find
and is related to the "molar conductances" of the constituent ions, XP""'" by the expression
(uR++ uL-) hy measuring (UH+
+ UA,-)
as a function of concentration of HAc, since i t would be necessary to know a in advance in order to to this. Instead
so that
(uR++ u L ) may be found using such an identity as (uR++ uic-) = (uR++ &)+
(uL++ ue-) - (&+
+ u&)
The approximation of taking ui = up in the case of weak electrolytes produces the same equation for the dissociation constant as used by Ostwald, Arrhenius, and others and known as "Ostwald's Dilution Law." In the case of acetic acid, derivation of ol from eqn. (5) gives a K , value constant to within 1% up to concentrations of 0.03 M. However the ions are subject to the long range ionic atmosphere forces discussed 492 / Journal of Chemical Education
and hence it is seen that ui/ui= p ' z . / pJ l J-
Zi
Thus the velocities of ions are not directly comparable in tables of molar ionic conductance. In addition, the stoichiometric numbers, ui, must he taken into account when deriving Am"'ar from a table of Xmo" values. While the velocities of ions can be readily determined from the equivalent conductance of a salt a t any concentration (if transport numbers are known a t that concentration) this is
not generally done since ionic velocities at c # 0 are coupled quanritien-i.e.. the velocityofthecation is dependent on the nature of the anion and vice versa. On the &her hand, we suggest that values of UP be used in preference to either hPsqu" or-$""',' a t least in elementary &aching, since the foimer quantity is much easier to visualize. Further an equation such as (2) makes i t easy to separate mentally the contributing effects to the conductance, r , of number of ions, ionic charges, and ionic velocities. Such a separation may he masked in equations using XequiV or Xmol'. In Table 2, the limiting mean ionic velocities are calculated for a number of common ions from the tables of Robinson and Stokes (14). Because the value of the Faraday constant is close t o 105Cmol-1 the numerical value of the mean ionic velocitv in units of nm s-I in a unit field of 1 Vm-' is rather similar w t h e eauivaient condudance of the ion in units of I)-' cdequiv-1 0;s cm2equiv-1, the units of most published tables. Fbr example, the limitina (i.e. infinite dilution) equivalent conductance of the sodium ion is 50.10 S cm2 equiv-' while its
mean velocity in unit field is 51.92 nm s-'. While such a &ifference in numerical value is by no means insignificant (considering the accuracy of conductance data) it is seen that equivalent conductance tables can be used as a quick guide to up values. Literature Cited
. ..,. ... ... ..,...., . ... C., and Gordon. A. R.. J. Chom. Phya.. 13.413 (1945). ewm, R. E., Bduir, D. R., Butler. J. P., and Gordon, A. R.. J. Amer. Chem. Soc., 7.5.
( 6 ) Benson, G.
.,AOK,> -""\.""",.
(81 Monk. C. B., J . Amer Chom. Sm.. 70,3281WW. (9) P m n a , R.. "Handbook of ElBnmehmehiealConsUnW Buttewortha. Londrm. L9S9. p. 82.
(10) Owena, B. B., andZeldes,H..J C h m . Phyr.. 1%1083i19~). ill) L a d l e , P. A,, and Aston. J. G., J. A m r . Chpm Sm.. 56,3067i1933l. (12) S i u e r t z , V . , R e i t m e r , R . E . , a n d T ~ , HV. . , J A m e r . Chm.Sae., 68,1379(1940). (13) Jenkins, J. L., and Monk, C. B., J A m , Ckem Soe, 72,269511950). 114) Robinson. R A . and Stokes.R. H..'"Eleetrolvte Solutions," 2nd Ed., Butlemrlhs.
Volume 53,Number 8, August 1976 / 493
