Affinity factors and transfer coefficients

Blcctrochimm

Acta. 1965. Vol 10. ~$8 107 to 109 Pergamon Press Ltd Prtnted tn Northern Ireland

SHORT COMMUNICATION

AFFINITY

FACTORS AND TRANSFER

COEFFICIENTS*

F GUTMAN* and P. VAN RYSSELBERGHE Departments of Chemistry and Chermcal Engmeermg Stanford Umverslty, Stanford, California, U S A THE

essential features of a treatment of electrode kmetrcs based upon the use of Marcehn-De Donder or MD formulas have been presented by one of us m a serves of commumcatronsl and more detailed consrderatrons and apphcatrons will be forthcoming Extensrve drscussron of the subject between the present authors has pointed to the desrrabrhty of putting greater stress on some fundamental aspects of the theory, two of whrch constitute the object of this paper. the concept of affinity factor and the sigmficance of transfer coefficients equal to i 1. THE

AFFINITY

FACTOR

The anodrc component Z, of the current densrty correspondmg to an electrode process for whrch step z is rate-determining IS grven by the expressron Z, = Z,, exp [(AT- _Q/RT],

(1)

m whrch ZlO1s the exchange current, A< 1s the forward electrochemrcal affimty, $, is the eqmhbnum value of A:, R IS the molar gas constant and T IS the absolute temperature. Srmrlarly the cathodrc component Z, of the current density IS grven by the expressron

(2)

IZ,] = IlO. exp [6? - $,YRTl~ A! and &

bemg now the reverse electrochemrcal affimtres z= z, - jZ& z* = % - $,

We have

S& = A,,.

When a Tafel behavrour is observed expenmentally

(3)

we have

m whrch the true transfer coefficrents cc,, and cc,, add up to one In terms of the storchrometnc number y. and of the affimty factor 6, of step z we have y,

2, = 6, . 3,

(5)

* Manuscnpt received 21 June 1964. t Permanent address Department of Physical Chemistry, Umverslty of New South Wales, Sydney, Austraha 107

F GUTMANNand P VAN RYSSELBERGHE

108

m which 2 = zFq

(6)

1s the

total electrochemical affinity of the electrode process, z being the overall charge number, F the faraday and 9 the overvoltage We then have a,, As = (a,,z UrJ FY, (7) a,, & = (a,,z UrJ

(8)

Q

The Tafel expressions bemg Z, = Z,, exp @,&IRT)

(9)

IZ,l = ZzO exp (--BAIRT),

(10)

and we have A, = a,,z &Iv,

A0 = a,,z UY,

(11)

a,, = AA%, + AC)

(13)

and

We easily verify that and that a 2a = B&A,

+ A,)

and

Since B,, and pze are experimentally obtained m a case of Tafel behavlour we see that the true transfer coefficients are likewise experimentally accessible The charge number z IS known from the way m which the overall electrode reaction has been written The ratio s,/y, 1s thus known, and, yz being a small integer whose value ~111 often be obvious, 6, will be obtained Normally 6, should be close to and smaller than one The fraction 6, of 2 residing m step z remains constant as Z and q vary, while the fractions 6,, all small and adding up to 1 - 6,, of the other steps do vary with Z and q since, m the steady state, we have Z, = Z = Z,, (A, + B,c) &IRT zz Z,, .&iRT = 120 (UY,)

(14)

APT,

and we see that a constant 6, would make Z proportional to q = Let us consider a hypothetical example It 1s experimentally found that @%a 0 72 and pEC= 1 08 This gives a,, = 0 72/l 80 = 0 4 and a,, = 1 08/l 80 = 0 6. With, for Instance, z = 2, we have 6,/y, = /?za/a,az = 0 9 If yz = 1 we have 6, = 0 9 and 1 - 6, = 0.1 For 7 = 0 1 V we have 2 = zFq = 4600 cal and (1 - 6,) 2 = 460 cal Since this last value 1s of the order of RT the mechanism steps other than zwill all follow formula (14), the 6, of each one of these steps being some function ofA”orq 2

SIGNIFICANCE

OF

TRANSFER

COEFFICIENTS

EQUAL

TO

&

In a previous paper on the MD method1 one of us has examined some implications of the equality of anodlc and cathodic overvoltage factors @,, = fiIC)in the case of hydrogen overvoltage The true transfer coefficients are then both equal to 4 We now present some general remarks for this type of case

Short

commumcatlon

109

Representing /l,,Fq/RT = @,,FqIRT by x we have, for the net current strength I, Z=2Z,

smhx=2Z,

(x+X3/3’+XS/5’+

. ),

05)

and we see that Z(x) = --I(-x)

(16)

for all values of x When p,, and p,C are not equal to each other the current Z 1s expressible as a senes mcludmg all powers of 7, odd and even, and we have Z(q) = -I(-$ only m the proportlonahty range of Z to q (see (14)), ze very near electrochemical eqmhbrmm The property Z(q) = --I(--r]) 1s then regarded as a mamfestatlon of rmcroscoplc reverslblhty Equation (16) somehow extends this mamfestatlon to large values of q when pza = &. Let us now consider a cell m which the anodlc and cathodic processes are the exact opposites of each other, as m a silver or a copper coulometer for instance The two electrodes function then with the same set of overvoltage factors btR, j3,, At sufficiently large current strengths the anodlc overvoltage IS given by ra = (RT/B,,F)

ln (Z/Z,)

(17)

ln (Z/I,)

(18)

and the cathodic overvoltage by l%l = (RTIBJ)

Subtracting the internal ohmic drop RI from the electric tenslon U of the cell we have U’= U-RZ=~,+(Y/J = (l/B,, + l/BJ

@T/F) . In U/&J7

(1%

(y,RT/z 63) . In (ZP,)

(20)

or u’ = (l/a,, + ll%)

We easily venfy that, at gven Z, U’ IS a mmlmum for an electrode process with %a = CI1C= l/2 (The function u = l/y + l/(1 - _JI)has a mmlmum equal to 4 when y = 1 - y = 4 ) The product U’Z 1s the power of polarlzatlon,2 equal to the entropy productlon per umt time multiplied by T We thus see that the property represented by formula (16), which holds when CI,, = u,, = $, also manifests Itself through a mmlmum entropy productlon or dlsslpatlon m this coulometer type of functlomng We then have q)7a= lrcl When u,, IS different from cc,, the sum q. + FCL;;t the entropy production are larger than when cc,, = CC,,= l/2, for given Z, z

E REFERENCES

1 P VAN RYSSELBERGHE, R C Accad Lmcez 31, 391 (1961), Electrochrm Acta 8, 583, 709 (1963), 9, 1547 (1964) 2 P VAN RYSSELBERGHE, Electrochemical Affinrty Hermann, Pans (1954, Thermodynamzcs of Zrreverszbfe Processes Hemann, Pam (1963)