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On tlze Hall Effect in a Binary Electrolyte. 465

XL VI I. Tl1emy of the II all Effect in a Binar.IJ Electrolyte. By F. G. DoNNAN, .M.A., Ph.D.*

I N 1883 Roiti t investigated the subject of a possible Hall . _effect in electrolytic solutions, but failed Jo obtain any

positive result. Uecently, however, the question has been taken llp by Bngard :t, who has obtained very considerable effects in aqueous solutions of zinc ami copper snlphates. On the other hand, negati\e resuHs have been obtnined by Florio, and as both Barrard and Florio maintain the correctness of thei1 experimental work, a 1wlemic on the subject has arisen between them. In this condition of affairs it seemed worth while to examine what effect micrbt be expected theoretically. :With this purpose in view I made the calculation contained

~~ the following pages. Subsequently I discovered that a Similar theory had been given by Van Everdingen,jun.jj; but as I do not arrive at quite the same results, and ha \'e considered the snl~ect somewhat more generally, it seemed to me not to be needless repetition to communicate this note.

Tho basis of the followinrr calculation may be stated briefly as follows. The diagram slwws the directions of the primary .current and the marrnetic field. The lines of flow of the primary current are ~upposed to be straight and the magnetic

Communicated by the Physical Society: read June 24th, 180~. t Jown. de Pl1y.ique, 1883. . ~ ~ ... _ _ t Comptes Rendus, vol. cxxi1. pp. 71-191 and cxxm. pp. 1210-12,3 (1896). Nuoro Oimento, r4] vol. iv. pp. 106-111 (189G). II J.;Ietingen over het Verschijnsel van Hall en ~e Toenan~e vau den

Weerstand in het Magnetisch Veld, p. 102 et seq. (Leiden, 1891).

466 Dr. F. G. Donnan on tiw field everywhere uniform. The effect of the ponderomotive forces on the moving ions in the magnetic field is to urge both positive and negative ions in the positive direction of z. The moving ions will thus acquire component velocities in this direction, and these velocities being in general different for each sort of ion, the result is the separation of positive

~nd negt~tive charges, whereby a potential-gradient is set up m the duection of the z-axis which reduces the oriainally !ln~qual velocities to equality. we thus obtain a flow of

~omc matter in the positive z-direction. This produces in 1ts turn a reverse osmotic gradient or concentration-fall both

~or dissociate~ and undissociated salt. The stationary Rtate Is finally aHamed when the net flux of ionic matter in the

posi~ive direction of z is balanced by the flux of undissociated salt In the opposite direction.

z

X

Let J = current-de?sity. of primary current, c= col!-centratwn (m mols.) of positive plus negative

Ions, P = corresponding osmotic pressure C= concentrati?n (in mols.) of undi~sociated salt, P= correspondmO' osmotic pressure

u, v= veloc~ties in "centim. per seco~d acquired under umt !ore~ ~y one gram-mol. of positive and

_ ne~ahve w.mc matter respectively, G- veloCity acqmred under unit force by one gram-

d7r mol. of undissociated salt d.r: = potential-gradient of primacy current, de dz '"" potential-gradient of Hall-effect

'

q " 1 1 I

Hall Effect in a Binary Elect1olyte. 461 e = quantity of electricity travelling per gram-equi-

valent. of ionic matter, ro = valency of each ion, H= magnetic field-strength, t = temperature (absolute).

We shall suppose the laws p=cRt and P=CRt to hold for the solution in question, although this is not essential. We have furthermore

CrJ22C d7r J=---y(u+v)dx' (i.)

Now an ion with a positive charge roe moving with a velocity V in the posithe direction of axis of :c will he acted on by a force in the positive z direction equal to roe VH; so that the force in this direction on a positive gram-ion is given by -ro2e2uH ~;, whence it has a component velocity in the same d. t' t' 2 -2 2Hd7r B . 'I . 1rec IOn amoun mg to -ro ~:u dx. y sum ar reasomng the velocity of a negative gram-ion in the positive z direction d7r IS - ro2e2v2H dx.

Expressing quantities of matter in mols., we can now draw up the following list, which takes into account all the fluxes of matter occurring in the solution. 1. Quantity of positive ionic matter tra-

versing per unit time unit section perpendicular to axis of z in positive direction ofthis axis, due to pondero-motive forces arising from the mag-netic field.

Corresponding' quantity of negative ionic matter.

Flux of positive ionic matter in same direction due to potential-gradient along z-axis.

Corresponding flux for negative ionic matter.

5. Flux of positive ionic matter in same direction due to oslllotic gradient.

6. Corresponding flux of negative ionic matter.

7. Flu.'\: of undissociated salt in positive z direction due to osmotic gradient.

CrJEC de =+yv([z

Rt de =-u2 dz'

Rtdc =-v 2dz'

dC =-GRt-1 CZ

468 Dr. F. G. Donnan on tlte The stationary condition then gives the equations:-

ro2e2c 2Hdr. roec de uRt de +GRt dC -O ~u dx + T u dz + 2 dz dz - ' ro

2e2e v2H dr. _ roec v de + vRt de + G Rt dC = O. 2 dx 2 dz 2 dz dz

(ii.)

(iii.)

dr. dC h . (' ) (" ' d Elimination of- and - from t e equatiOns 1. , n.;, an dx dz (iii.) leads to the result

de_ 2 u-v HJ Rt u-v de (iv.) dz - roee u + v - roec u + v ' dz '

It is to be observed that equation (iv.) holds for the v.ariable as well as the stationary state, because although durmg the Yariable state the left-hand members of equations (ii.) and (iii.) are not zero, they are always equal to each other.

Accordingly for the initial phenomenon, before any appre-ciable concentration-gradient occurs, we obtain, putting c=co

de and dz =0,

de 2 u-v dr. dz = roec

0 u +vHJ =roe(v-u)H dx' (v.)

or, integrating,

s.J cdle dy dz= 2H ~s.rJdy dz ; z roee0 u+v whence

2 U-t Hi e=--- . (vi.)

roec0 u+v d'

wh~r? e=tota.l difference of potential measured in direction of pos1hve z-ax1s, d = thickness measured in direction of mag-

neti~ field, and i =primary current-strength. So that we obtam for the constant of the initial Hall-effect the nlue

2 u-v const. = - --.

roec0 u+v In order to furthe: _in~estigate the stationary state, we shall

suppose t~at !he e9u~hb:mm-equation required by the laws of electrolyt~c dissocwbon IS everywhere satisfied. 'l'his ai~o~nts tQ supposmg that the processes which adjust this equihbrmm

procee~ ve_rymuch more rapidly than any of the other changes occurrmg m the system. In the uncertain state of knowledge concerning the equilibrium-equation it may be written for the

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Hall Effect in a Binar.IJ Electrolyte. 469

h dO I ) de EI' . r dC present C=cp(c); so tat dz =cp (e dz' umna mg dz between this equation and (ii.) and between this equation and (iii.), we obtain the equations

de ~-u-HJ-Rt {2Gcp'(e) +1 }de, (vii.) 'dz = roee u + v roec u dz

de = _ ~ _v_HJ + ~ { 2Gcp' (el + 1} de. . (viii.) dz roee u+v roec v dz WriiinO' 2G'(e) + 1 = L, 2G' (c) + 1 = l\I, we obtain from

.., u v

(vii.) and (viii.)

or

de=_!___ _1_ Jiu-i~v HJ, J dz roer: L + .1\f u + v I

~ de roe dr. I dz = L+l\1 (Lv-.1\Iu)H d.'C _j

(ix.)

For a completely dissociated electrolyte L = 1, l\1 = 1, and we get instead of equations (ix.)

de =-l_u-vHJ=roe(v-u)Hdr.. (ix.a) ~ ro~u+v ~ (~

Comparing equations (v.) and (ix. ~), ":e see that the fin~! potentiul-gradient for a completely dissociated electrolyte IS just one-half the initial gradient.

de!dr. V I I t . d Let- -= D and denote by U, t 1e ve oc1 ws acqmre dz dx '

under unit potential O'radicnt by a gram-mol. of positive and negative ionic matt~r respecti,ely. Equation (ix. a) may then be written

D=-k(V- U)H. (x.) Van Everdingen arrived at the equation D= (V- U)H

for the stationary state in an electrolyte supposed to be .''ell slightly dissociated; but be assumes in this case that t~w wmc concentration docs not vary throughout the. solub~n, un -assumption which is inconsistent with the equation.:; ( \'lL) and {viii.); and t.herefore appears to me to he erroneous. .

Equation (x.) may now be applied to the_ data obtamed by Bagard. He finds, for example, the followm~gresults.

.Loc. cit. p. 207.

470 Dr. F. G. Donnan on tlze CuS04 solution. Concentration = 1Iu- grm. cquiv. per litre.

Temperature 21-26 C.

H. D. D il" ---------- _-__ ------

385 C.G.S. units. 0018

0024

0034

46xl0-'

34xiO-'

35xlO-'

707

962

To test these results by means of the theory, it is only neces~ary to calculate the value of !(V-U), where it must be noticed that V and U are the velocities of the gram-mols. in centim. per second under a potential-gradient of one C.G.S. electromagnetic unit of potential per centim. From Ostwald's Lehrbuch, vol. ii. p. 770, the molecular conductivity of CuS04 solutions for the highest dilution at 18 C. is 217 in Siemens units, and therefore 230 in mho's. As the calculation is only very approximate, we may put consequently

230 (U + V)tso= 96540 X 2 X 108'

From the same source, p. 612, we get U 'l:T = 36, a value U+~ ~vhich may be regarded as fairly correct for temperatures Ill the neighbourhood of :20 C. (accordina to Hittorf and Bein, Zoe. cit.). Hence we obtain finally '"'

t(V-U)I8o= 16 X 10-13, The value of t(V-U) for temperatures 21-26 C. will probably be somewhat smaller. Of course the solution of concentration.=--fs grm.-equiv. per litre is not by any m~ans _ completely dissociated; and by using the proper equahon, namely,

D Lv-Mu H =we L+M '

d h Lv-1\Iu v-u smce v > u an t erefore > -- we should obtam a L+l\1 2 '

higher value for~ Nevertheless this would never account for the difference between 39 x I0-5 as observed by Bagard and 16 x 10-13 as deduced theoreticaliy for the case of complete dissociation.

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Hall Effect in a Bina1y Electrolyte. 471 So far as I can see, the theory here given is wholly in

favour of the negative results obtained by Roiti and Florio; and it would therefore seem that Bagard has measured a phe-nomenon not contemplated in the foregoing theory. Van Ever-dingen in the paper refened to above supports Bagard; but this is owing to his having accidentally omitted the factor 10-8 in his numerical work.

The ionic concentration-fall can be readily calculated for a c?mpletely dissociated electrolyte. From equations (iv.) and (u. a) we get at once

Rtt!!!_ =HJ dz '

or

whence

or

2Rt log~ =H(U + V)(z2-z1) 71"2-71"1 me c3 .x2-.r1

Thus, callina E the P.D. between two transverse electrodes (of the same 1~etal as the kation) due only to the differences in concent.ration set up (a case which could be realized by taking an electrolyte such as silver nitrate, for which U and V are very nearly equal), it follows that

71"2-71"1 E=tH(U + V)(z2-zt) --.

.'1:2-.xl Taking the case of copper sulphate, we get, as before,

u + v =00119.

Putting z2 -z1 = 1 centim., and 71"2-71"1

= 1 (volt per centim.), T2-T1

and H=20,000 C.G.S. units, we obtain E=l2x 10-8 volts.

Renee it would appear that in this, or in any other similar experimental arranaement it would be neces~ary not only to e~ploy an extremely strodg magnetic field, but also a very high primary potential-gradient, i. e. of the orde.r of 10,00.0 volts per centim. This might perhaps be reahzed exper1-

472 Mr. J. Walker on tlte Admi8sible Width of mentally by employing very dilute solutions and a large accumulator-battery, such as that recently employed by Pro-fessors Trowbridge and Richards*

20th ~lay, 1898.

Note added J1Iay 27tlt.-Dr. Van Everdingen informs me that he has discovered the slip in his calculation, and agrees with. the remarks made above concerning the experiments of Flono and Bagard.

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Phil. )lag. February (1897). l t Communicated bv the Author t Ph}l. .Mag. [.'>J. xxviii. p. 80 (IS89). . . I find tha.t thrs result of the progreE~ive widemng of the Elit has

already been grven by i\I. Ch. Fabrv in his thesis for the de!!Tee of Doctor of Science, published at Marseilles in I89'l and recrret that I was not aware of this interesting work until my pap7r was in type.

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