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Analytical expressions for the reversible Randles-Sevcik function
J. Electroanal. Chem., 105 (1979) 373--375
373
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
Short communication ANALYTICAL EXPRESSIONS FOR THE REVERSIBLE RANDLES-SEVCIK FUNCTION KEITH B. OLDHAM
Trent University, Peterborough (Canada) (Received 4th May 1979)
When the potential ramp signal E = Eo - - vt is applied to a working electrode immersed in a solution containing, in addition to excess supporting electrolyte, a concentration c of a species that is reducible by a reversible n-electron reaction, then a plot of the faradaic current i versus the time t or the potential E is a characteristic curve that displays an asymmetric peak. Randles [ 1] and Sevcik [2] described the mathematical form of this curve, which is usually expressed in terms of a transcendental function X. They showed that, provided E0 -- Eh is sufficiently large compared with R T / n F , then (i/nAFc)(RT/nFDv)'/2
= ~,/2 X(X)
(I)
where R is the gas constant, T is the thermodynamic temperature, F is Faraday's constant, A is the constant electrode area, D is the diffusion coefficient of the electroreducible species, Eh is its polarographic half-wave potential, and x is an abbreviation for n F [ E h - - E ] / R T . The ~r1:2 X(X) function has been tabulated by several authors, most notably by Nicholson and Shain [3], and a similar tabulation accompanies this note. For some purposes tabular data are satisfactory; for other purposes however, particularly where experimental values are to be processed by computer, tabulations are markedly inferior to analytical representations of the function. Reinmuth [4] showed that 7r'/2 X(x) = -- ~
(__)jjl/2 exp(jx)
(2)
j=l
and this expression provides a convenient m e t h o d for the calculation of ×(x) for negative values of x. However, the summation diverges for x > 0 which, as Table 1 shows, includes the important peak region. The purpose of this note is to present alternative expressions for ×(x). A summation equivalent to (2) is 1/2
k=l
fl= [ ( 2 k - - 1)27r2 + x 2 ],/2 which converges for all values of x, though the convergence is sometimes rather
374 slow. T o h a s t e n the process, o n e m a y develop f o r m u l a (3) i n t o
Irl/2X(x)=L+Mx--Nx2
+(2) 1/2
(~ - x),,= (~ + ~ : ) k=l
8b 2 + 12bx--
~3
15x 2
(4)
8b 7/2
where L = 0 . 3 8 0 1 0 4 8 1 3 , M = 0 . 1 1 8 6 8 0 8 7 1 , N = 0 . 0 4 3 9 2 0 5 6 0 and b = (2k -- 1)~r. F o r m u l a (4) proves t o be an efficient m e t h o d o f calculating Ir ' n X(x) for all x and was, in fact, e m p l o y e d in the c o n s t r u c t i o n o f Table 1. T h e d a t a in this t a b u l a t i o n are c o n s i s t e n t with, b u t m o r e precise than, previously published values o f the f u n c t i o n [3]. A f o u r t h analytical expression f o r the X f u n c t i o n , n a m e l y 1 { ~_~_2 497r4 341~6 } ~'/~ X(X) - (~x),/~ 1 + 8x 2 + 34 8 -4 x + 1024x6~ + O(x -s)
(5)
is valid f o r large positive values o f x, c o r r e s p o n d i n g t o the " t a i l " o f the voltammogram. TABLE 1 Values and features of the function X(X) X
~l/2~X)
--~
exp(x)
--9.0000 --8.0000 --7.0000 --6.0000 --5.0000 --4.0000 --3.0000 --2.0000 --1.0934 --1.0000 --0.7315 0.0000 +1.0000 1.1090 2.0000 2.5950 3.0000 4.0000 5.0000 6.0000 6.8400 7.0000 8.0000 9.0000 10.0000
0.00012 0.00034 0.00091 0.00247 0.00667 0.01785 0.04648 0.11314 0.22315 0.23681 0.27719 0.38010 0.44572 0.44629 0.41815 0.38362 0.35951 0.30747 0.26886 0.24093 0.22315 0.22020 0.20427 0.19146 0.18093
~
t ~ x ~-xn
Limitingexpression
Halfpeak Inflection point Peak Inflection point
Half peak
Limiting expression
375
The derivations of expansions (3), (4) and (5) are being reported in the mathematical literature [ 5]. REFERENCES 1 2 3 4 5
J.E.B. Randles, Trans. F a r a d a y Soc., 4 4 ( 1 9 4 8 ) 3 2 7 . A. Sevcik, Collect. Czech. Chem. C o m m u n . , 13 ( 1 9 4 8 ) 349. R.S. N i c h o l s o n a n d I. Shain, Anal. Chem., 36 ( 1 9 6 4 ) 706. W.H. R e i n m u t h , Anal. Chem., 34 ( 1 9 6 2 ) 1 4 4 6 . K.B. O l d h a m , S . T . A . M . J . Math. Phys., s u b m i t t e d .
373
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
Short communication ANALYTICAL EXPRESSIONS FOR THE REVERSIBLE RANDLES-SEVCIK FUNCTION KEITH B. OLDHAM
Trent University, Peterborough (Canada) (Received 4th May 1979)
When the potential ramp signal E = Eo - - vt is applied to a working electrode immersed in a solution containing, in addition to excess supporting electrolyte, a concentration c of a species that is reducible by a reversible n-electron reaction, then a plot of the faradaic current i versus the time t or the potential E is a characteristic curve that displays an asymmetric peak. Randles [ 1] and Sevcik [2] described the mathematical form of this curve, which is usually expressed in terms of a transcendental function X. They showed that, provided E0 -- Eh is sufficiently large compared with R T / n F , then (i/nAFc)(RT/nFDv)'/2
= ~,/2 X(X)
(I)
where R is the gas constant, T is the thermodynamic temperature, F is Faraday's constant, A is the constant electrode area, D is the diffusion coefficient of the electroreducible species, Eh is its polarographic half-wave potential, and x is an abbreviation for n F [ E h - - E ] / R T . The ~r1:2 X(X) function has been tabulated by several authors, most notably by Nicholson and Shain [3], and a similar tabulation accompanies this note. For some purposes tabular data are satisfactory; for other purposes however, particularly where experimental values are to be processed by computer, tabulations are markedly inferior to analytical representations of the function. Reinmuth [4] showed that 7r'/2 X(x) = -- ~
(__)jjl/2 exp(jx)
(2)
j=l
and this expression provides a convenient m e t h o d for the calculation of ×(x) for negative values of x. However, the summation diverges for x > 0 which, as Table 1 shows, includes the important peak region. The purpose of this note is to present alternative expressions for ×(x). A summation equivalent to (2) is 1/2
k=l
fl= [ ( 2 k - - 1)27r2 + x 2 ],/2 which converges for all values of x, though the convergence is sometimes rather
374 slow. T o h a s t e n the process, o n e m a y develop f o r m u l a (3) i n t o
Irl/2X(x)=L+Mx--Nx2
+(2) 1/2
(~ - x),,= (~ + ~ : ) k=l
8b 2 + 12bx--
~3
15x 2
(4)
8b 7/2
where L = 0 . 3 8 0 1 0 4 8 1 3 , M = 0 . 1 1 8 6 8 0 8 7 1 , N = 0 . 0 4 3 9 2 0 5 6 0 and b = (2k -- 1)~r. F o r m u l a (4) proves t o be an efficient m e t h o d o f calculating Ir ' n X(x) for all x and was, in fact, e m p l o y e d in the c o n s t r u c t i o n o f Table 1. T h e d a t a in this t a b u l a t i o n are c o n s i s t e n t with, b u t m o r e precise than, previously published values o f the f u n c t i o n [3]. A f o u r t h analytical expression f o r the X f u n c t i o n , n a m e l y 1 { ~_~_2 497r4 341~6 } ~'/~ X(X) - (~x),/~ 1 + 8x 2 + 34 8 -4 x + 1024x6~ + O(x -s)
(5)
is valid f o r large positive values o f x, c o r r e s p o n d i n g t o the " t a i l " o f the voltammogram. TABLE 1 Values and features of the function X(X) X
~l/2~X)
--~
exp(x)
--9.0000 --8.0000 --7.0000 --6.0000 --5.0000 --4.0000 --3.0000 --2.0000 --1.0934 --1.0000 --0.7315 0.0000 +1.0000 1.1090 2.0000 2.5950 3.0000 4.0000 5.0000 6.0000 6.8400 7.0000 8.0000 9.0000 10.0000
0.00012 0.00034 0.00091 0.00247 0.00667 0.01785 0.04648 0.11314 0.22315 0.23681 0.27719 0.38010 0.44572 0.44629 0.41815 0.38362 0.35951 0.30747 0.26886 0.24093 0.22315 0.22020 0.20427 0.19146 0.18093
~
t ~ x ~-xn
Limitingexpression
Halfpeak Inflection point Peak Inflection point
Half peak
Limiting expression
375
The derivations of expansions (3), (4) and (5) are being reported in the mathematical literature [ 5]. REFERENCES 1 2 3 4 5
J.E.B. Randles, Trans. F a r a d a y Soc., 4 4 ( 1 9 4 8 ) 3 2 7 . A. Sevcik, Collect. Czech. Chem. C o m m u n . , 13 ( 1 9 4 8 ) 349. R.S. N i c h o l s o n a n d I. Shain, Anal. Chem., 36 ( 1 9 6 4 ) 706. W.H. R e i n m u t h , Anal. Chem., 34 ( 1 9 6 2 ) 1 4 4 6 . K.B. O l d h a m , S . T . A . M . J . Math. Phys., s u b m i t t e d .