Radii of Redox Components from Absolute Redox Potentials

Short Communication Short Communication

Radii of Redox Components from Absolute Redox Potentials Compared with Covalent and Aqueous Ionic Radii Raji Heyrovska Institute of Biophysics, Academy of Sciences of the Czech Republic, 135 Kralovopolska, 612 65 Brno, Czech Republic *e-mail: [email protected] Received: October 14, 2009 Accepted: December 23, 2009 Abstract Aqueous standard potentials, referred to that of the SHE as zero, were recently shown to vary linearly with gaseous ionization potentials, with the absolute potential of SHE as the intercept. This enabled arriving at the absolute redox potentials of elements. Here, the distances between the oxidized and reduced forms in aqueous solutions have been evaluated. From the linear dependence of these distances on the covalent radii of atoms, the radii of the redox components have been obtained. The latter also vary linearly with the aqueous ionic radii estimated earlier from ionwater distances, and indicate the presence of aqueous molecular anions. Keywords: Electrochemistry, Absolute redox potentials, Radii of redox components, Covalent radii, Aqueous ionic radii, Molecular anions DOI: 10.1002/elan.200900503

Aqueous standard potentials, E8 of redox equilibria, O þ ne ¼ R, between the oxidized (O) and reduced forms (R), are conventionally referred to that of the standard hydrogen electrode (SHE), E8 (SHE) as zero [1]. It was recently shown [2] for the first time that E8 varies linearly with the respective gaseous ionization potentials, I (the sum of the first n ionization potentials for n > 1) for many groups of elements. Using the data for E8 and I in [3], it was found that the linear relation, E8 ¼ kaqI E8I¼0

(1a)

tures of the elements in Table 1. The E8abs values have been used here to evaluate the sizes of the oxidized and reduced species in aqueous solutions as described below. In an earlier publication [6] many different radii of atoms (A) including the covalent radii [7 – 9], d(A)cov (radii are symbolized by d instead of r, since they are apportioned distances), defined [7] as half the interatomic distances d(AA), were shown to be directly proportional to the Bohr radii, aB, evaluated [6] from their first ionization potentials, I ¼ e/(8p eaB). On replacing I in Equation 2b by 0.719982/aB (for I in eV and aB in nm), one obtains

E8abs(SHE) ¼ E8I¼0

(1b)

E8abs ¼ kaq(0.719982/aB) ¼ 0.719982/d(O R)aq

(3)

d(O R)aq ¼ aB/kaq ¼ 0.719982/E8abs

(4)

holds, where kaq is the slope of the E8 vs. I straight lines and E8I¼0, the intercept at I ¼ 0, is the absolute potential, E8abs (SHE) of the SHE. E8I¼0 ¼ 4.20 ( 0.03) V, was found to be the common intercept for all except for group VIIA. As pointed out in [2], this value coincides with that obtained [4] by electron capture by gas-phase nanodrops, and is also close to the earlier theoretical value, 4.44 ( 0.02) V, suggested [5] for the Hþ/(1/2)H2 equilibrium. For group VIIA elements, E8I¼0 ¼ 2.87 ( 0.01) V) and it refers [2] to that of the (1/2)H2/H redox. A detailed introduction to the literature on absolute potentials can be found in [4, 5]. Thus, the absolute redox potentials in aqueous solutions, E8abs, were obtained as, E8abs ¼ E8 þ E8abs(SHE) ¼ kaqI

where d(O R)aq denotes the distance between O and R in the aqueous medium. Their values obtained from E8abs using Equation 4 are given in Table 2. It can be seen that these distances in aqueous solutions are comparable with the corresponding covalent radii [7 – 9], d(A)cov given in the next column. The graphs in Figure 1 show that, in fact, linear relations exist between the two (data for those elements with at least three points on the lines are shown), revealing thereby, for the first time, a direct link between sizes of the redox species in aqueous solutions and their covalent radii. The straight lines in Figure 1 are represented by the equation,

(2a, b) d(O R)aq ¼ maqd(A)cov þ d(w)

(5)

for elements of many groups of the Periodic Table, The values are tabulated here along with the electronic strucElectroanalysis 2010, 22, No. 9, 903 – 907

2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

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R. Heyrovska

Fig. 1. Linear dependence of the distance, d(O R)aq, obtained from the absolute redox potentials [2] (E8abs), on covalent radii [7 – 9] d(A)cov, for elements of several Groups of the Periodic Table (data are in Table 2). The regressed slopes (maq) are given in the legend for the Figure and the intercepts, d(w) are at the headings in column 4, Table 2. [Note: The straight lines for elements of groups IA and IIA have the same slope.]

Fig. 2. Linear dependences of d(O)aq, d(R)aq and their sum obtained from the absolute redox potentials (E8abs) on aqueous ionic radii, d(i)aq evaluated [17] from ion – water distances (data are in Table 2). Slopes are given in the legend for the Figure and the intercepts are zero in all cases. [Note: the lines for Groups IA and IIA elements have the same slopes and the line for the radii sum for Group VIIA coincides with the d(O)aq vs. d(i)aq lines for Group IA and IIA elements.]

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Electroanalysis 2010, 22, No. 9, 903 – 907

Radii of Redox Components Table 1. Absolute aqueous redox potentials [2], E8abs ¼ E8SHE¼0 þ E8abs ( SHE ) (in V ) for the redox equilibria, O þ ne ¼ R. E8abs ( SHE ) ¼ 2.87 ( 0.01) V for gp. VIIA and 4.20 ( 0.03) V for the others. E8SHE¼0 are from [3]. The values of n are given in each column either at the head or in brackets next to the E8abs. The list of elements in the Periodic Table are from: http://www.webelements.com/ Elec.struc. Gp.IA: s1 1s1 [ He]2s1 [ Ne]3s1 [ Ar]4s1 [ Kr]5s1 [ Xe]6s1 [ Rn]7s1 Gp.IIIA: s2p1 [ Be]2p1 [ Mg]3p1 [ Zn]4p1 [ Cd]5p1 [ Hg]6p1 [ Cn]7p1 Gp.IVA: s2p2 [ Be]2p2 [ Mg]3p2 [ Zn]4p2 [ Cd]5p2 [ Hg]6p2 [ Cn]7p2 Gp.VA: s2p3 [ Be]2p3 [ Mg]3p3 [ Zn]4p3 [ Cd]5p3 [ Hg]6p3 [ Cn]7p3 Gp.VIA: s2p4 [ Be]p4 [ Mg]3p4 [ Zn]4p4 [ Cd]5p4 [ Hg]6p4 [ Cn]7p4 Gp.VIIIA: s2p6 1s2 [ Be]2p6 [ Mg]3p6 [ Zn]4p6 [ Cd]5p6 [ Hg]6p6 [ Cn]7p6

Atom H Li Na K Rb Cs Fr B Al Ga In Tl Uut C Si Ge Sn Pb Uuq N P As Sb Bi Uup O S Se Te Po Uuh

E8abs

Elec.struc.

n¼1 4.20 1.16 1.49 1.28 1.28 1.28

Gp.IB: s1

n¼3 3.35 2.53 3.67 3.86 5.45 n¼2 4.72 3.39 4.45 4.06 4.08 n¼3 5.68 3.70 4.44 4.4 4.52 n¼2 5.42 3.75 4.09 3.06 2.8

He Ne Ar Kr Xe Rn Uuo

[K]3d10 [Rb]4d10 [Cs]4f145d10 [Fr]5f146d10 Gp.IIIB: s2d1

Atom

Cu Ag Au Rg

E8abs

Elec.struc.

n¼1

Gp.IIA: s2

4.72 5.00 6.03 n¼3

[Ca]3d1 [Sr]4d1 [Ba]5d1 [Ra]6d1 Gp.IVB: s2d2

Sc Y La Ac

2.17 1.83 1.82 2.07 n¼4

[Ca]3d2 [Sr]4d2 [Ba]4f145d2 [Ra]5f146d2 Gp.VB: s2d3

Ti Zr Hf Rf

3.01 2.50 2.64

[Ca]3d3 [Sr]4d3 [Ba]4f145d3 [Ra]5f146d3 Gp.VIB: s2d4

V Nb Ta Db

n¼3

Cr Mo W Sg

[Ca]3d6 [Sr]4d6 [Ba]4f145d6 [Ra]5f146d6

Fe Ru Os Hs

3.44 4.00 4.08(4) n¼2

3.76 5.24(8) 4.89(4)

where maq (< 2) is the slope (values are given in the legend for the figure) and d(w), the intercept, is the distance between O and R in the aqueous (w) medium. The values of the latter are given in column 4, Table 2. In Equation 5, the product, maqd(A)cov ¼ d(O)aq þ d(R)aq

(6)

can be taken as the sum of the radii, d(O)aq and d(R)aq of O and R, respectively (see column 6, Table 2). These radii sums were then divided into the component radii, d(O)aq and d(R)aq as described below. Electroanalysis 2010, 22, No. 9, 903 – 907

Be* Mg Ca Sr Ba Ra La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

E8abs

Elec.struc.

n¼2

Gp.IIB: s2

2.23 1.84 1.36 1.31 1.28 1.28 n¼3 1.82 1.86 1.85 1.88 1.91 1.90 2.21 1.92 1.89 1.91 1.87 1.88 1.88 1.98 1.90

[ Ca]3d10 [ Sr]4d10 [ Ba]4f145d10 [ Ra]5f146d10 Gp.IIIB: s2fxd1 [ Ra]6d1 [ Ra]6d2 [ Ra]6d15f2 [ Ra]6d15f3 [ Ra]6d15f4 [ Ra]6d05f6 [ Ra]6d05f7 [ Ra]6d15f7 [ Ra]6d05f9 [ Ra]6d05f10 [ Ra]6d05f11 [ Ra]6d05f12 [ Ra]6d05f13 [ Ra]6d05f14 [ Ra]6d15f14

Atom

E8abs n¼2

Zn Cd Hg Cn Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

3.44 3.80 5.05 n¼3 2.07 2.37(4) 2.74(4) 2.54 2.41 2.20 2.13 2.14 2.19 2.27 2.20 2.24 2.50 3.00 2.20

3.95 3.10

n¼3

[Ca]3d4 [Sr]4d4 [Ba]4f145d4 [Ra]5f146d4 Gp.VIIIB: s2d6

[ He]2s2 [ Ne]3s2 [ Ar]4s2 [ Kr]5s2 [ Xe]6s2 [ Rn]7s2 Gp.IIIB: s2fxd1 [ Ba]5d1 [ Ba]5d04f2 [ Ba]5d04f3 [ Ba]5d04f4 [ Ba]5d04f5 [ Ba]5d04f6 [ Ba]5d04f7 [ Ba]5d04f8 [ Ba]5d04f9 [ Ba]5d04f10 [ Ba]5d04f11 [ Ba]5d04f12 [ Ba]5d04f13 [ Ba]5d04f14 [ Ba]5d14f14

Atom

Gp.VIIA: s2p5 [ Be]2p5 [ Mg]3p5 [ Zn]4p5 [ Cd]5p5 [ Hg]6p5 [ Cn]7p5 Gp.VIIIB: s2d7

F Cl Br I At Uus

[ Ca]3d7 [ Sr]4d7 [ Ba]4f145d7 [ Ra]5f146d7

Co Rh Ir Mt

n¼1 5.74 4.23 3.94 3.41 3.07* n¼2

3.92 4.96(3) 5.36(3)

Gp.VIIB: s2d5

n¼2

[ Ca]3d5 [ Sr]4d5 [ Ba]4f145d5 [ Ra]5f146d5 Gp.VIIIB: s2d8

Mn Tc Re Bh

[ Ca]3d8 [ Sr]4d8 [ Ba]4f145d8 [ Ra]5f146d8

Ni Pd Pt Ds

3.02 4.60

n¼2

3.94 5.12 5.39

The covalent bond lengths, d(AA) between two atoms of the same kind in solids and gases were shown [11, 12] to be quantitatively related to their cationic and anionic radii, d(A þ ) and d(A ), through the Golden ratio [13], f. This ratio, f ¼ (1 þ 51/2)/2 ¼ 1.618 ¼ a/b, is the positive root of the equation, (a/b)2 (a/b) 1 ¼ 0 obtained when a and b are such that a/b ¼ (a þ b)/a. It was shown in [11, 12] that d(AA) ¼ 2d(A)cov ¼ d(A þ ) þ d(A )

(7a)

d(A þ ) ¼ d(AA)/f2 ¼ 0.764d(A)cov

(7b)

d(A ) ¼ d(AA)/f ¼ 1.236d(A)cov

(7c)



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Table 2. Absolute redox potentials [2], E8abs (in V ), distance d( O R )aq ¼ 0.72/E8abs ¼ maqd( A )cov þ d(w), the distance d(w) between O and R, covalent radii [7 – 9], d( A )cov, radii sum maqd( A )cov ¼ d( O R )aq d(w) ¼ d( O )aq þ d( R )aq, radii d( O )aq and d( R )aq, and aqueous ionic radii [17], d(i)aq from ion-water distances, for the elements of the Periodic Table. All distances are in nm. 1 Group

2 Atom (A)

IA IA IA IA IA IA

H Li [a] Na K Rb Cs

IIA IIA IIA IIA IIA

Be Mg Ca Sr Ba

IIIA IIIA IIIB

B Al Sc

IB IB IB IIIA

Cu Ag Au Tl [a]

VA VA VIB

N P Cr

VIA VIA VIA

O S Te

VIIA VIIA VIIA VIIA

F Cl Br I

3 E8abs O þ ne ¼ R

4 d( O R )aq

n¼1 4.20 1.67 1.59 1.34 1.29 1.20 n¼2 2.23 1.83 1.46 1.35 1.23 n¼3 3.35 2.50 2.07 n¼1 5.07 4.98 6.06 4.02 n¼3 5.68 3.77 3.44 n¼2 5.42 3.75 3.10 n¼1 5.75 4.26 3.88 3.45

d(w) ¼ 0.111 0.171 0.431 0.453 0.537 0.559 0.599 d(w) ¼ 0.178 0.323 0.393 0.494 0.533 0.584 d(w) ¼ 0.091 0.215 0.288 0.347 d(w) ¼ 0.035 0.142 0.145 0.119 0.179 d(w) ¼ 0.062 0.127 0.191 0.209 d(w) ¼ 0.063 0.133 0.192 0.232 d(w) ¼ 0.032 0.125 0.169 0.186 0.209

5 d( A )cov

6 maqd( A )cov

7 d( O )aq ¼ 0.43maqd( A )cov

8 d( R )aq ¼ 0.57maqd( A)cov

0.037 0.200 0.211 0.261 0.279 0.302

0.060 0.320 0.342 0.426 0.448 0.488

0.026 0.139 0.148 0.184 0.194 0.211

0.034 0.182 0.194 0.241 0.254 0.277

0.090 0.131 0.197 0.215 0.251

0.145 0.215 0.316 0.355 0.406

0.063 0.093 0.137 0.154 0.176

0.082 0.122 0.179 0.201 0.230

0.090 0.143 0.187

0.124 0.197 0.256

0.054 0.085 0.111

0.070 0.111 0.145

0.128 0.145 0.102 0.173

0.107 0.110 0.084 0.144

0.046 0.047 0.036 0.062

0.061 0.062 0.047 0.082

0.055 0.111 0.125

0.065 0.129 0.147

0.060 0.103 0.143

0.070 0.129 0.169

0.028 0.056 0.064 0.45maqd( A )cov 0.031 0.058 0.076

0.037 0.073 0.084 0.55maqd( A)cov 0.039 0.071 0.093

0.071 0.099 0.115 0.133

0.093 0.137 0.154 0.177

0.042 0.061 0.069 0.079

0.052 0.076 0.085 0.098

9 d(i)aq ¼ ki, aqd( A )cov ki, aq ¼ 0.81 0.030 0.162 0.171 0.211 0.226 0.245 ki, aq ¼ 0.81 0.073 0.106 0.160 0.174 0.203

ki, aq ¼ 1.57 0.112 0.156 0.181 0.209

[a] E1/2 from [10] was used for obtaining Eabs8.

where d(A þ ) and d(A ) are respectively, the cationic and anionic radii of A, and d(A þ ) < d(A)cov < d(A ) as is generally observed [7]. For hydrogen, the inter-atomic distance, d(HH) ¼ 0.074 nm [7], and from Equation 7b, d(H þ ) ¼ 0.074/f2 ¼ 0.028 nm. This is the value suggested [7] for the radius of hydrogen in the partially ionic bonds in hydrogen halides, and it holds for many hydrides [11, 14, 15] and also for several hydrogen bonds [16]. Many more bond lengths [11 – 17] including the crystal ionic distances in all alkali halides [11] were shown to be exact sums of the covalent and or Golden ratio-based ionic radii given by Equations 7. From Equations 7, one gets the following ratios for the cationic and covalent radii,

and the following for the anionic and covalent radii, d(A)cov/[d(A)cov þ d(A )] ¼ 1/(1 þ 2/f) ¼ 0.447

(9a)

d(A )/[d(A)cov þ d(A )] ¼ 1/(1 þ f/2) ¼ 0.553

(9b)

By using the ratios given by Equations 8 and 9, the values of d(O)aq and d(R)aq in aqueous medium were evaluated (see columns 7 and 8, Table 2) from their sums (see Equation 6), maqd(A)cov (given in column 6). The above radii were then compared with the aqueous ionic radii [17], d(i)aq (see column 9, Table 2), obtained earlier from ion-water distances, d(i)aq ¼ ki, aqd(A)cov

d(A þ )/[d(A)cov þ d(A þ )] ¼ 1/(1 þ f2/2) ¼ 0.433

(8a)

d(A)cov/[d(A)cov þ d(A þ )] ¼ 1/(1 þ 2/f2) ¼ 0.567

(8b)

906


(10)

where ki, aq ¼ f/2 ¼ 0.809 for cations of groups IA and IIA and ki, aq ¼ 2/f1/2 ¼ 1.572 for anions of group VIIA. The



Radii of Redox Components

graphs of d(O)aq, d(R)aq and their sum, maqd(A)cov, versus d(i)aq in Figure 2 show that all are straight lines with zero intercepts with slopes as given in the legend for Figure 2. It follows from Fig. 2 and the data in Table 2 that, d(i)aq ¼ 0.809d(A)cov ¼ 0.5[d(O)aq þ d(R)aq] ¼ 1.16d(O)aq (gps. IA & IIA) (11a) d(i)aq ¼ 1.572d(A)cov ¼ 1.17[d(O)aq þ d(R)aq] ¼ (f3/2)d(R)aq (gp. VIIA) (11b) Since from Equation 6, d(O)aq þ d(R)aq ¼ maqd(A)cov one gets from Equation 11a, maq ¼ 0.809/0.5 ¼ f(¼ 1.62, the slope of the lines for gps IA and IIA, in Fig. 1). In Equation 11b for gp. VIIA, 1.17 ¼ f/(1 þ 1/f2) and hence maq ¼ 1.57/ 1.17 ¼ 1.34 (¼ the slope of the line for group VIIA in Fig. 1). Also, since in Equation 11b, (f3/2) ¼ 2.12, d(i)aq ¼ 2.12d(R)aq > 2d(R)aq and d(i)aq probably pertains to the molecular halogen anions d(A2)aq in aqueous solutions. Thus, presented here of fundamental importance to electrochemistry and electroanalyses are: 1) the absolute redox potentials, E8abs of elements of the Periodic Table, 2) the distances between the oxidized and reduced forms in aqueous solutions, which are inversely proportional to E8abs and 3) the radii of the oxidized and reduced forms in aqueous solutions. Acknowledgement The author thanks the Institute of Biophysics for institutional grant Nos. AV0Z50040507 and AV0Z50040702 of the Academy of Sciences of the Czech Republic.


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