- Email: [email protected]
water boundaries
Chemical Physics Letters 429 (2006) 600–605 www.elsevier.com/locate/cplett
Dependence of ion–water distances on covalent radii, ionic radii in water and distances of oxygen and hydrogen of water from ion/water boundaries Raji Heyrovska Institute of Biophysics, Academy of Sciences of the Czech Republic, Kralovopolska 135, 612 65 Brno, Czech Republic Received 19 April 2006; in final form 10 August 2006 Available online 23 August 2006
Abstract This Letter shows that the ion–water (oxygen) distances for ions of monovalent, divalent and trivalent elements including lanthanides increase linearly with the covalent radii of atoms. The slopes and intercepts of the lines are found to depend on the Golden ratio. The ionic radii in water have been evaluated and ion/water boundaries located. The hydration distances of the oxygen of water to the peripheries of the cations are of four types and are constants. The distance of the hydrogen of water to the peripheries of halogen anions, which is the length of the ‘hydrogen bond’, is also a constant. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Recently, based on the finding [1] that the Golden ratio (/) plays an important role in atomic and ionic dimensions, the ionic radii in aqueous solutions, d(i), aq derived from ion–water(O) inter-nuclear distances [2], d(i . . . w) and molal volumes of solutions [3,4], were shown [5] to be linearly dependent on the crystal ionic radii of alkali metal and halogen ions evaluated [1] based on /. In [1], the cationic and anionic radii, d(A+) and d(A), respectively, of an atom (A) were shown [1] to be the Golden sections of the inter-atomic distance d(AA) between two atoms of the same kind, dðAþÞ ¼ dðAAÞ=/2 ¼ 0:382dðAAÞ ¼ 0:764dðAÞ
ð1aÞ
dðAÞ ¼ dðAAÞ=/ ¼ 0:618dðAAÞ ¼ 1:236dðAÞ dðAAÞ ¼ dðAþÞ þ dðAÞ ¼ 2dðAÞ
ð1bÞ ð1cÞ
where d(A) is defined as the covalent radius [6] and / = /2 1 = (1//) + 1 = (1 + 51/2)/2 = 1.618034. . ., is the Golden ratio [7]. The linear dependence found in [5] implies that d(i . . . w) must be linear with d(A+) and d(A). Since E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.08.073
the latter are Golden sections of d(AA), d(i . . . w) can be expected to be directly proportional to d(A). It is shown here that this is indeed the case, and the interesting results that emerge from it are summarized below. In this Letter, since the ionic as well as the covalent radii are apportioned distances, the letter ‘d’ is used instead of ‘R’. All distances in ˚. the text, figures and tables are in A 2. Ion–water distances and the covalent radii 2.1. Cationic and anionic radii in water For a unified treatment of the data, the covalent radii, d(A), given in column 2 in Table 1 (monovalent and divalent atoms) and column 2 in Table 2 (for trivalent atoms including lanthanides) have all been obtained using the same source of atomic crystal lattice parameters in [8]. In the case of the lanthanides, the d(A) values fall into two categories, La–Sm, nearer to the higher value of the two suggested radii in [6], which are a/2 and to the lower value, d(A) = 0.89a/2 for Gd–Lu and Y (excluding Eu and Yb), where a is the cell parameter [8]. It is interesting to note that 1//1/4 = 0.887. That lanthanides can be divided into two groups is known [6].
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
601
Table 1 Covalent radii d(A) [8], average ion–water distances d(i . . . w), av ˚ [2,9,10], ionic radii (mono and divalent) in aqueous solutions (±)0.08 A d(i), aq = kd(A): Eq. (3), where k is the slope (see Fig. 1) and the calculated radii d(i), / = f(/)d(A): Eq. (4), where k = f(/)
Table 2 Covalent radii d(A) [8], average ion–water distances d(i . . . w), av ˚ [2,9,10], ionic radii (trivalent ions) in aqueous solutions d(i), (±)0.08 A aq = kd(A): Eq. (3), where k is the slope (Fig. 2) and the calculated radii d(i),/ = f(/)d(A): Eq. (4), where k = f(/)
Atom (A)
Atom (A)
Li Na K Rb Cs Ag
d(A) 1.76 2.15 2.66 2.79 3.07 2.04
d(i . . . w), av
d(i), aq
2.04 2.39 2.80 2.91 3.13 2.33
k = 0.81 1.42 1.74 2.16 2.26 2.49 1.65
d(–O)
d(i), /
0.64 0.64 0.64 0.64 0.64 0.64
k = //2 1.42 1.74 2.16 2.26 2.48 1.65
1.60 1.60 1.60 1.60
k = 2//1/2 1.11 1.49 1.75 2.10
F Cl Br I
0.71 0.95 1.11 1.33
2.64 3.17 3.35 3.64
k = 1.57 1.11 1.48 1.74 2.08
Mg Ca Sr Ba
1.61 1.98 2.15 2.51
2.08 2.41 2.63 2.81
k = 0.82 1.32 1.62 1.76 2.06
0.80 0.80 0.80 0.80
k = //2 1.30 1.60 1.74 2.03
Mn Fe Co Ni Cu Zn Cd Hg Sn
1.36 1.24 1.25 1.25 1.28 1.33 1.49 1.50 1.59
2.20 2.13 2.10 2.06 2.08 2.10 2.30 2.35 2.31
k = 0.79 1.08 0.98 0.99 0.98 1.01 1.05 1.18 1.19 1.26
1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10
k = //2 1.10 1.00 1.01 1.01 1.03 1.08 1.21 1.22 1.29
The distance of (O), water from the ion/water boundary at P(i/w) is d(–O). / = /2 1 = 1.618 is the Golden ratio, //2 = 0.809 and 2//1/2 = 1.572. ˚. All distances are in A
The d(i . . . w) data in [2,9,10] are all within about ˚ , and the average of these values, d(i . . . w), av, (±)0.08 A are given in column 3 of Tables 1 and 2. The linear increase of ion–water distances d(i . . . w) with the covalent radii d(A) are shown in Fig. 1 for monovalent and divalent ions and in Fig. 2 for trivalent ions of elements including lanthanides. The least square lines through the points are represented here by, dði . . . wÞ ¼ kdðAÞ þ dð–OÞ ¼ dðiÞ; aq þ dð–OÞ
ð2Þ
where k is the slope and d(–O) is the intercept (see column 5 in Tables 1 and 2) and d(i), aq is the ionic radii in aqueous solutions given by the equation, dðiÞ; aq ¼ kdðAÞ ¼ dði . . . wÞ–dð–OÞ
ð3Þ
The values of d(i), aq = kd(A) are presented in columns 4 in Tables 1 and 2. On examining the values of k for aqueous ions, it is found that k also depends on the Golden ratio, as found before [1] (see Eqs. (1a), (1b)). Thus, aqueous ionic radii, d(i), / based on the Golden ratio, could be calculated using the covalent radii d(A) thus, dðiÞ; / ¼ kdðAÞ ¼ f ð/ÞdðAÞ ¼ dði . . . wÞ–dð–OÞ
ð4Þ
d(A)
d(i . . . w), av
d(i), aq
d(–O)
d(i), /
1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40
k = 2//3 0.56 0.59 0.59 0.63 0.77 1.08 0.92 0.82
1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37
k = 1// 1.17 1.12 1.13 1.13 1.13 1.12 1.00 0.99 0.99 0.98 0.98 0.97 0.96 1.00
Al Cr Fe Rh In Eu Yb Tl
1.18 1.26 1.24 1.34 1.63 2.29 1.94 1.73
1.89 1.98 2.03 2.05 2.15 2.45 2.32 2.23
k = 0.47 0.55 0.59 0.58 0.63 0.76 1.08 0.91 0.81
La Ce Pr Nd Pm Sm Gd Tb Dy Ho Er Tm Lu Y
1.89 1.81 1.84 1.83 1.83 1.81 1.62 1.60 1.60 1.59 1.58 1.57 1.56 1.62
2.55 2.56 2.53 2.50 2.50 2.47 2.41 2.40 2.38 2.37 2.35 2.34 2.33 2.37
k = 0.62 1.17 1.13 1.14 1.14 1.14 1.13 1.01 1.00 1.00 0.99 0.99 0.98 0.97 1.01
The distance of the (O), water from the ion/water boundary at P(i/w) is d(–O). / = /2 1 = 1.618 is the Golden ratio, 2//3 = 0.472 and 1// ˚. = 0.618. All distances are in A
where (1) k = 1.57 = f(/) = 2//1/2 for halogen anions, (2) k = 0.80 f(1/) = 0.81 for cations of monovalent and divalent elements, (3) k = 0.62 = f(/) = 1// = 0.62 for cations of lanthanides (except Eu and Yb) and Y and (4) k = 0.47 = f(/) = 2//3 = 0.47 for cations of other trivalent elements and Eu and Yb. The linear dependence of d(i . . . w), av (column 3, Tables 1 and 2) on the calculated values of d(i), / (given in column 6, Tables 1 and 2) for all the ions is shown in Fig. 3. Moreover, the intercepts d(–O) obtained in Fig. 3 are the same as those obtained in Figs. 1 and 2. The calculated values of d(i . . . w),/ = d(i),/ + d(–O) are also plotted in Figs. 1 and 2, to show the validity of Eq. (4). Fig. 4. shows the sizes of various ions relative to d(A). 2.2. Oxygen (water)–cation (periphery) distances, d(–O) In Eq. (3), d(–O) obtained as the intercepts in Figs. 1 and 2 (see the values in column 5, Tables 1 and 2) represent the distance of the center of the oxygen (O) of water from the ion/water boundary, P(i/w) at the peripheries of the ions as shown: ð5Þ ˚ For cations, it is found that there are four values (in A) of d(–O): OðwaterÞ
dð–OÞ ! P ði=wÞ
dðiÞ; aq ! iðionÞ
602
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
Fig. 1. Linear dependence of ion–water distances, d(i . . . w), av for cations of mono and divalent elements on their covalent atomic radii, d(A). In the ordinate, d(i . . . w), / is the calculated value given by Eq. (4). For mono and divalent cations, slope = k = //2 = 0.81 and intercept d(– ˚ (monovalent), d(–O) = 0.80 (divalent, Mg, ) and 1.10 (±) 0.08 A ˚ (divalent, Zn, ). For halogen anions, k = 1.57 = 2//1/2 and O) = 0.64 (±) 0.03 A ˚ ˚ d(–O) = 1.60 (±) 0.04A = d(OH) + d(–H), where d(–H) = 1.60 0.97 = 0.63 A is the length of the ‘hydrogen bond’. In the legend of the figure, fi stands for the Golden ratio, / = 1.618.
Fig. 2. Linear dependence of ion–water distances, d(i . . . w), av for trivalent cations of elements including of lanthanides (upper line) on their covalent atomic radii, d(A). In the ordinate, d(i . . . w), / has the same significance as in Fig. 1. For the trivalent cations of Al, , Eu and Yb, ˚ . For the trivalent cations of lanthanides (except Eu and Yb) and Y, slope = k = 0.62 = 1// slope = k = 0.47 = 2//3 and intercept d(–O) = 1.40 (±) 0.06 A ˚. and intercept = d(–O) = 1.37 (±) 0.05 A
dð–OÞ ¼ 1:39ðÞ0:07½ 0:64ð2=/3 Þ ¼ 1:36;
dð–OÞ ¼ 0:64ðÞ0:03½¼ dð–OÞ; univalent cationðLiþ ; ; Csþ ; Agþ Þ
ð6Þ
dð–OÞ ¼ 0:80ðÞ0:05½ 0:64/1=2 ¼ 0:81 ðMgþ2 ; ; Baþ2 Þ dð–OÞ ¼ 1:10ðÞ0:05½ 0:64/ ¼ 1:04 ðMnþ2 ; ; Snþ2 Þ
ð7aÞ ð7bÞ
ðAlþ3 ; ; Tlþ3 and Laþ3 ; ; Yþ3 Þ
ð8Þ
˚ for where d(–O) in Eq. (8) is the average of 1.40 (±) 0.05 A ˚ the cations of lanthanides and Y and 1.37 (±) 0.06 A for cations of other trivalent elements and the lanthanides, Eu and Yb. In the above Eqs. (6)–(8), it is shown inside the square brackets that the values of d(–O) are expressible
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
603
Fig. 3. Linear dependence of ion–water distances, d(i . . . w), av on calculated aqueous ionic radii d(i),/ = kd(A), where k = f(/) is based on the Golden ratio.
Fig. 4. Ionic radii in aqueous solutions, d(i), aq = kd(A), relative to the covalent radii, d(A) = 1, where k > 1 for anions and k < 1 for cations.
in terms of the Golden ratio. Note that d(–O) < R(w) in ˚, Eqs. (6), (7a,7b), where R(w) = d(O) + 2d(H) = 1.34 A ˚ are the covalent radii of d(O) = 0.60 and d(H) = 0.37 A O and H, and that in Eq. (8), d(–O) R(w). Therefore cations are likely to compete with water–water interactions as concluded recently from molecular dynamic simulation studies [11]. ˚ is subtracted In [2], one uniform value of R(w) = 1.39 A from d(i . . . w) for all ions to get the aqueous ionic radii, and they are found to be close to Pauling’s ionic radii. This implies that all ions having about the same d(i . . . w) like ˚ and Hg+2 the pairs Ag+ and Lu+3: d(i . . . w) = 2.33 A +3 ˚ and Er : d(i . . . w) = 2.35 A will have the same ionic radii in aqueous solutions. It was shown in [3] using the data in [2] that the alkali metal cations and halogen anions have different values of R(w). In this work, since d(–O) is shown to have four different values, the aqueous ionic radii d(i), aq given by Eq. (3) are different from those in [2]. For exam-
ple, for the above mentioned ions (see Tables 1 and 2) d(i), ˚ ): Ag+ = 1.65, Lu+3 = 0.97, Hg+2 = 1.19 and Er+3 aq (in A 0.99. In [10], the authors subtract values of crystal ionic radii from d(i . . . w) and find that the radius of the water ˚ , depending on molecule varies between 1.34 and 1.43 A the charge on the ion. The latter conclusion is correct, but the values of d(–O) found here in Eqs. (6)–(8) are ˚ , and only trivalent ions have 0.64, 0.80, 1.10 and 1.39 A ˚ d(–O) 1.39 A as found in [2,10]. Fig. 5 (I, III, IV and V) (drawn to scale) shows the sizes of various ions in aqueous solutions, the water–cation points of contact P(i/w) and hydration lengths d(–O). 2.3. Distances of O(water) and H(water) from the peripheries of anions ˚ of the oxygen of The distance, d(–O) = 1.60 (±) 0.04 A the water molecule (column 5, Table 1) from P(i/w) at the peripheries of the anions is the same for all the anions ˚ ) is larger (F, Cl, Br and I). The value, d(–O) (=1.60 A than those for the cations given by Eqs. (6)–(8), since the hydrogen of water is bound to the anions by ‘hydrogen bond’ as shown, OðwaterÞ dðOHÞ ! H dð–HÞ ! P ði=wÞ dðiÞ; aq ! iðanionÞ dð–OÞ ¼ dðOHÞ þ dð–HÞ ¼ 1:60
ð9Þ ð10Þ
dð–HÞ ¼ dð–OÞ dðOHÞ ¼ 1:60 0:97 ¼ 0:63 ð11Þ ˚ In Eq. (11), d(–H) = 0.63 A, is the length of the ‘hydrogen ˚, bond’. This is close to the sum, d(H) + d(H+) = 0.65 A + + where d(H ) is a distance equivalent to d(H ) = d(HH)//2
604
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
Fig. 5. (drawn to scale) The oxygen (O) and two hydrogen (H) atoms of water are shown as circles with covalent radii, d(O) and d(H). The circles on the right represent ions with aqueous ionic radii, d(i), aq = kd(A). The length of the hydration bond, d(–O) is the distance between the center of O(water) and ˚ . d(i), aq (in A ˚ ): 1.42 (Li+) < 1.65 (Ag+ circle with P(i/w) (shown as dotted lines in Figs. I–V): (I) Li+, , Cs+: k = 0.81 = //2, d(–O) = 0.64 (±) 0.03 A + + + + 1/2 ˚ . d(i), aq (in A ˚ ): 1.11 (F) < 1.48 dashes) < 1.74 (Na ) < 2.16 (K ) < 2.26 (Rb ) < 2.49 (Cs ). (II) F , , I : k = 1.57 = 2// , d(–O) = 1.60 (±) 0.04 A ˚ . (III): Ni+2, , Ba+2: k = 0.80 //2 = 0.81, d(– (Cl) < 1.74 (Br) < 2.08 (I). The ‘hydrogen bond’ length d(–H) = d(–O) d(OH) = 0.63 A ˚ . d(i), aq (in A ˚ ) (only some are shown to avoid overcrowding): 0.98 (Ni+2) < 1.19 (Hg+2) < 1.32 (Mg+2) < 2.06 (Ba+2). (IV): O) = 1.06 (±) 0.08 A ˚ . d(i), aq (in A ˚ ) (only some are shown): 0.55 (Al+3) < 0.63 (Rh+3) < 0.81 (Tl+3) < 0.91 (Yb+3, Al+3, , Eu+3: k = 0.47 = (2//3), d(–O) = 1.40 (±) 0.06 A +3 +3 +3 ˚ . d(i), aq in increasing order (in A ˚ ) (only some dashed line) < 1.08 (Eu , dash and dot line). (V) Lu , , La : k = 0.62 = //2, d(–O) = 1.37 (±) 0.05 A are shown): 0.97 (Lu+3) < 1.01 (Y+3, dashed line) < 1.13 (Sm+3) < 1.17 (La+3).
˚ [1], the value suggested for the radius of hydrogen = 0.28 A in hydrogen halides in [6]. Fig. 5II shows the water–anion points of contact P(i/w) and hydration length d(–O). 3. The H . . . X distance for halogen (X) anions and the length of the hydrogen bond Since anions form hydrogen bonds with the hydrogen of water [12], the distances, d(H . . . X) obtained as the sums, dðH . . . XÞ ¼ dð–HÞ þ dðiÞ; aq
ð12Þ
˚ ], are with d(–H) = 0.63 (±) 0.04 [d(H) + d(H+) = 0.65 A ˚ ˚ d(H . . . F) = d(–H) + d(F) = 0.63 + 1.11 = 1.74 A (1.74 A ˚ ˚ [13]), d(H . . . Cl) = 0.63 + 1.48 = 2.11 A (2.15 A [13]), ˚ (2.40 A ˚ [13]) and d(H . . . Br) = 0.63 + 1.74 = 2.37 A ˚ ˚ d(H . . . I) = 0.63 + 2.08 = 2.71 A (2.72 A [13]). The comparable values in parentheses are from Table 3.25 in [13]. Acknowledgements This research was supported by grants of the Academy of Sciences of the Czech Republic AVOZ50040507 and of
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
the Ministry of Education, Youth and Sports of the Czech Republic, LC06035. References [1] [2] [3] [4] [5]
R. Heyrovska, Mol. Phys. 103 (2005) 877. Y. Marcus, Chem. Rev. 88 (1988) 1475. R. Heyrovska, Chem. Phys. Lett. 163 (1989) 207. R. Heyrovska, Marine Chem. 70 (2000) 49. R. Heyrovska, 6th Working meeting of physical chemists and electrochemists, Masaryk University, Brno, February 2006, Book of abstracts, p. 38.
605
[6] L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, NY, 1960. [7] M. Livio, The Golden ratio, the story of Phi, the World’s most astonishing number, Broadway Books, NY, 2003. [8] http://www.webelements.com/webelements/elements/text/La/ radii.html. [9] H. Ohtaki, T. Radnai, Chem. Rev. 93 (1993) 1157. [10] F. David, V. Volkhmin, G. Ionova, J. Mol. Liquids 90 (2001) 45. [11] A. Grossfield, J. Chem. Phys. 122 (2005) 024506. [12] M. Chaplin, http://www.lsbu.ac.uk/water/hbond.html. [13] R. Desiraju, T. Steiner, The weak hydrogen bond in structural chemistry and biology, International monographs on crystallography, Oxford Science Publications, 1999.
Dependence of ion–water distances on covalent radii, ionic radii in water and distances of oxygen and hydrogen of water from ion/water boundaries Raji Heyrovska Institute of Biophysics, Academy of Sciences of the Czech Republic, Kralovopolska 135, 612 65 Brno, Czech Republic Received 19 April 2006; in final form 10 August 2006 Available online 23 August 2006
Abstract This Letter shows that the ion–water (oxygen) distances for ions of monovalent, divalent and trivalent elements including lanthanides increase linearly with the covalent radii of atoms. The slopes and intercepts of the lines are found to depend on the Golden ratio. The ionic radii in water have been evaluated and ion/water boundaries located. The hydration distances of the oxygen of water to the peripheries of the cations are of four types and are constants. The distance of the hydrogen of water to the peripheries of halogen anions, which is the length of the ‘hydrogen bond’, is also a constant. Ó 2006 Elsevier B.V. All rights reserved.
1. Introduction Recently, based on the finding [1] that the Golden ratio (/) plays an important role in atomic and ionic dimensions, the ionic radii in aqueous solutions, d(i), aq derived from ion–water(O) inter-nuclear distances [2], d(i . . . w) and molal volumes of solutions [3,4], were shown [5] to be linearly dependent on the crystal ionic radii of alkali metal and halogen ions evaluated [1] based on /. In [1], the cationic and anionic radii, d(A+) and d(A), respectively, of an atom (A) were shown [1] to be the Golden sections of the inter-atomic distance d(AA) between two atoms of the same kind, dðAþÞ ¼ dðAAÞ=/2 ¼ 0:382dðAAÞ ¼ 0:764dðAÞ
ð1aÞ
dðAÞ ¼ dðAAÞ=/ ¼ 0:618dðAAÞ ¼ 1:236dðAÞ dðAAÞ ¼ dðAþÞ þ dðAÞ ¼ 2dðAÞ
ð1bÞ ð1cÞ
where d(A) is defined as the covalent radius [6] and / = /2 1 = (1//) + 1 = (1 + 51/2)/2 = 1.618034. . ., is the Golden ratio [7]. The linear dependence found in [5] implies that d(i . . . w) must be linear with d(A+) and d(A). Since E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2006.08.073
the latter are Golden sections of d(AA), d(i . . . w) can be expected to be directly proportional to d(A). It is shown here that this is indeed the case, and the interesting results that emerge from it are summarized below. In this Letter, since the ionic as well as the covalent radii are apportioned distances, the letter ‘d’ is used instead of ‘R’. All distances in ˚. the text, figures and tables are in A 2. Ion–water distances and the covalent radii 2.1. Cationic and anionic radii in water For a unified treatment of the data, the covalent radii, d(A), given in column 2 in Table 1 (monovalent and divalent atoms) and column 2 in Table 2 (for trivalent atoms including lanthanides) have all been obtained using the same source of atomic crystal lattice parameters in [8]. In the case of the lanthanides, the d(A) values fall into two categories, La–Sm, nearer to the higher value of the two suggested radii in [6], which are a/2 and to the lower value, d(A) = 0.89a/2 for Gd–Lu and Y (excluding Eu and Yb), where a is the cell parameter [8]. It is interesting to note that 1//1/4 = 0.887. That lanthanides can be divided into two groups is known [6].
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
601
Table 1 Covalent radii d(A) [8], average ion–water distances d(i . . . w), av ˚ [2,9,10], ionic radii (mono and divalent) in aqueous solutions (±)0.08 A d(i), aq = kd(A): Eq. (3), where k is the slope (see Fig. 1) and the calculated radii d(i), / = f(/)d(A): Eq. (4), where k = f(/)
Table 2 Covalent radii d(A) [8], average ion–water distances d(i . . . w), av ˚ [2,9,10], ionic radii (trivalent ions) in aqueous solutions d(i), (±)0.08 A aq = kd(A): Eq. (3), where k is the slope (Fig. 2) and the calculated radii d(i),/ = f(/)d(A): Eq. (4), where k = f(/)
Atom (A)
Atom (A)
Li Na K Rb Cs Ag
d(A) 1.76 2.15 2.66 2.79 3.07 2.04
d(i . . . w), av
d(i), aq
2.04 2.39 2.80 2.91 3.13 2.33
k = 0.81 1.42 1.74 2.16 2.26 2.49 1.65
d(–O)
d(i), /
0.64 0.64 0.64 0.64 0.64 0.64
k = //2 1.42 1.74 2.16 2.26 2.48 1.65
1.60 1.60 1.60 1.60
k = 2//1/2 1.11 1.49 1.75 2.10
F Cl Br I
0.71 0.95 1.11 1.33
2.64 3.17 3.35 3.64
k = 1.57 1.11 1.48 1.74 2.08
Mg Ca Sr Ba
1.61 1.98 2.15 2.51
2.08 2.41 2.63 2.81
k = 0.82 1.32 1.62 1.76 2.06
0.80 0.80 0.80 0.80
k = //2 1.30 1.60 1.74 2.03
Mn Fe Co Ni Cu Zn Cd Hg Sn
1.36 1.24 1.25 1.25 1.28 1.33 1.49 1.50 1.59
2.20 2.13 2.10 2.06 2.08 2.10 2.30 2.35 2.31
k = 0.79 1.08 0.98 0.99 0.98 1.01 1.05 1.18 1.19 1.26
1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10 1.10
k = //2 1.10 1.00 1.01 1.01 1.03 1.08 1.21 1.22 1.29
The distance of (O), water from the ion/water boundary at P(i/w) is d(–O). / = /2 1 = 1.618 is the Golden ratio, //2 = 0.809 and 2//1/2 = 1.572. ˚. All distances are in A
The d(i . . . w) data in [2,9,10] are all within about ˚ , and the average of these values, d(i . . . w), av, (±)0.08 A are given in column 3 of Tables 1 and 2. The linear increase of ion–water distances d(i . . . w) with the covalent radii d(A) are shown in Fig. 1 for monovalent and divalent ions and in Fig. 2 for trivalent ions of elements including lanthanides. The least square lines through the points are represented here by, dði . . . wÞ ¼ kdðAÞ þ dð–OÞ ¼ dðiÞ; aq þ dð–OÞ
ð2Þ
where k is the slope and d(–O) is the intercept (see column 5 in Tables 1 and 2) and d(i), aq is the ionic radii in aqueous solutions given by the equation, dðiÞ; aq ¼ kdðAÞ ¼ dði . . . wÞ–dð–OÞ
ð3Þ
The values of d(i), aq = kd(A) are presented in columns 4 in Tables 1 and 2. On examining the values of k for aqueous ions, it is found that k also depends on the Golden ratio, as found before [1] (see Eqs. (1a), (1b)). Thus, aqueous ionic radii, d(i), / based on the Golden ratio, could be calculated using the covalent radii d(A) thus, dðiÞ; / ¼ kdðAÞ ¼ f ð/ÞdðAÞ ¼ dði . . . wÞ–dð–OÞ
ð4Þ
d(A)
d(i . . . w), av
d(i), aq
d(–O)
d(i), /
1.40 1.40 1.40 1.40 1.40 1.40 1.40 1.40
k = 2//3 0.56 0.59 0.59 0.63 0.77 1.08 0.92 0.82
1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37 1.37
k = 1// 1.17 1.12 1.13 1.13 1.13 1.12 1.00 0.99 0.99 0.98 0.98 0.97 0.96 1.00
Al Cr Fe Rh In Eu Yb Tl
1.18 1.26 1.24 1.34 1.63 2.29 1.94 1.73
1.89 1.98 2.03 2.05 2.15 2.45 2.32 2.23
k = 0.47 0.55 0.59 0.58 0.63 0.76 1.08 0.91 0.81
La Ce Pr Nd Pm Sm Gd Tb Dy Ho Er Tm Lu Y
1.89 1.81 1.84 1.83 1.83 1.81 1.62 1.60 1.60 1.59 1.58 1.57 1.56 1.62
2.55 2.56 2.53 2.50 2.50 2.47 2.41 2.40 2.38 2.37 2.35 2.34 2.33 2.37
k = 0.62 1.17 1.13 1.14 1.14 1.14 1.13 1.01 1.00 1.00 0.99 0.99 0.98 0.97 1.01
The distance of the (O), water from the ion/water boundary at P(i/w) is d(–O). / = /2 1 = 1.618 is the Golden ratio, 2//3 = 0.472 and 1// ˚. = 0.618. All distances are in A
where (1) k = 1.57 = f(/) = 2//1/2 for halogen anions, (2) k = 0.80 f(1/) = 0.81 for cations of monovalent and divalent elements, (3) k = 0.62 = f(/) = 1// = 0.62 for cations of lanthanides (except Eu and Yb) and Y and (4) k = 0.47 = f(/) = 2//3 = 0.47 for cations of other trivalent elements and Eu and Yb. The linear dependence of d(i . . . w), av (column 3, Tables 1 and 2) on the calculated values of d(i), / (given in column 6, Tables 1 and 2) for all the ions is shown in Fig. 3. Moreover, the intercepts d(–O) obtained in Fig. 3 are the same as those obtained in Figs. 1 and 2. The calculated values of d(i . . . w),/ = d(i),/ + d(–O) are also plotted in Figs. 1 and 2, to show the validity of Eq. (4). Fig. 4. shows the sizes of various ions relative to d(A). 2.2. Oxygen (water)–cation (periphery) distances, d(–O) In Eq. (3), d(–O) obtained as the intercepts in Figs. 1 and 2 (see the values in column 5, Tables 1 and 2) represent the distance of the center of the oxygen (O) of water from the ion/water boundary, P(i/w) at the peripheries of the ions as shown: ð5Þ ˚ For cations, it is found that there are four values (in A) of d(–O): OðwaterÞ
dð–OÞ ! P ði=wÞ
dðiÞ; aq ! iðionÞ
602
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
Fig. 1. Linear dependence of ion–water distances, d(i . . . w), av for cations of mono and divalent elements on their covalent atomic radii, d(A). In the ordinate, d(i . . . w), / is the calculated value given by Eq. (4). For mono and divalent cations, slope = k = //2 = 0.81 and intercept d(– ˚ (monovalent), d(–O) = 0.80 (divalent, Mg, ) and 1.10 (±) 0.08 A ˚ (divalent, Zn, ). For halogen anions, k = 1.57 = 2//1/2 and O) = 0.64 (±) 0.03 A ˚ ˚ d(–O) = 1.60 (±) 0.04A = d(OH) + d(–H), where d(–H) = 1.60 0.97 = 0.63 A is the length of the ‘hydrogen bond’. In the legend of the figure, fi stands for the Golden ratio, / = 1.618.
Fig. 2. Linear dependence of ion–water distances, d(i . . . w), av for trivalent cations of elements including of lanthanides (upper line) on their covalent atomic radii, d(A). In the ordinate, d(i . . . w), / has the same significance as in Fig. 1. For the trivalent cations of Al, , Eu and Yb, ˚ . For the trivalent cations of lanthanides (except Eu and Yb) and Y, slope = k = 0.62 = 1// slope = k = 0.47 = 2//3 and intercept d(–O) = 1.40 (±) 0.06 A ˚. and intercept = d(–O) = 1.37 (±) 0.05 A
dð–OÞ ¼ 1:39ðÞ0:07½ 0:64ð2=/3 Þ ¼ 1:36;
dð–OÞ ¼ 0:64ðÞ0:03½¼ dð–OÞ; univalent cationðLiþ ; ; Csþ ; Agþ Þ
ð6Þ
dð–OÞ ¼ 0:80ðÞ0:05½ 0:64/1=2 ¼ 0:81 ðMgþ2 ; ; Baþ2 Þ dð–OÞ ¼ 1:10ðÞ0:05½ 0:64/ ¼ 1:04 ðMnþ2 ; ; Snþ2 Þ
ð7aÞ ð7bÞ
ðAlþ3 ; ; Tlþ3 and Laþ3 ; ; Yþ3 Þ
ð8Þ
˚ for where d(–O) in Eq. (8) is the average of 1.40 (±) 0.05 A ˚ the cations of lanthanides and Y and 1.37 (±) 0.06 A for cations of other trivalent elements and the lanthanides, Eu and Yb. In the above Eqs. (6)–(8), it is shown inside the square brackets that the values of d(–O) are expressible
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
603
Fig. 3. Linear dependence of ion–water distances, d(i . . . w), av on calculated aqueous ionic radii d(i),/ = kd(A), where k = f(/) is based on the Golden ratio.
Fig. 4. Ionic radii in aqueous solutions, d(i), aq = kd(A), relative to the covalent radii, d(A) = 1, where k > 1 for anions and k < 1 for cations.
in terms of the Golden ratio. Note that d(–O) < R(w) in ˚, Eqs. (6), (7a,7b), where R(w) = d(O) + 2d(H) = 1.34 A ˚ are the covalent radii of d(O) = 0.60 and d(H) = 0.37 A O and H, and that in Eq. (8), d(–O) R(w). Therefore cations are likely to compete with water–water interactions as concluded recently from molecular dynamic simulation studies [11]. ˚ is subtracted In [2], one uniform value of R(w) = 1.39 A from d(i . . . w) for all ions to get the aqueous ionic radii, and they are found to be close to Pauling’s ionic radii. This implies that all ions having about the same d(i . . . w) like ˚ and Hg+2 the pairs Ag+ and Lu+3: d(i . . . w) = 2.33 A +3 ˚ and Er : d(i . . . w) = 2.35 A will have the same ionic radii in aqueous solutions. It was shown in [3] using the data in [2] that the alkali metal cations and halogen anions have different values of R(w). In this work, since d(–O) is shown to have four different values, the aqueous ionic radii d(i), aq given by Eq. (3) are different from those in [2]. For exam-
ple, for the above mentioned ions (see Tables 1 and 2) d(i), ˚ ): Ag+ = 1.65, Lu+3 = 0.97, Hg+2 = 1.19 and Er+3 aq (in A 0.99. In [10], the authors subtract values of crystal ionic radii from d(i . . . w) and find that the radius of the water ˚ , depending on molecule varies between 1.34 and 1.43 A the charge on the ion. The latter conclusion is correct, but the values of d(–O) found here in Eqs. (6)–(8) are ˚ , and only trivalent ions have 0.64, 0.80, 1.10 and 1.39 A ˚ d(–O) 1.39 A as found in [2,10]. Fig. 5 (I, III, IV and V) (drawn to scale) shows the sizes of various ions in aqueous solutions, the water–cation points of contact P(i/w) and hydration lengths d(–O). 2.3. Distances of O(water) and H(water) from the peripheries of anions ˚ of the oxygen of The distance, d(–O) = 1.60 (±) 0.04 A the water molecule (column 5, Table 1) from P(i/w) at the peripheries of the anions is the same for all the anions ˚ ) is larger (F, Cl, Br and I). The value, d(–O) (=1.60 A than those for the cations given by Eqs. (6)–(8), since the hydrogen of water is bound to the anions by ‘hydrogen bond’ as shown, OðwaterÞ dðOHÞ ! H dð–HÞ ! P ði=wÞ dðiÞ; aq ! iðanionÞ dð–OÞ ¼ dðOHÞ þ dð–HÞ ¼ 1:60
ð9Þ ð10Þ
dð–HÞ ¼ dð–OÞ dðOHÞ ¼ 1:60 0:97 ¼ 0:63 ð11Þ ˚ In Eq. (11), d(–H) = 0.63 A, is the length of the ‘hydrogen ˚, bond’. This is close to the sum, d(H) + d(H+) = 0.65 A + + where d(H ) is a distance equivalent to d(H ) = d(HH)//2
604
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
Fig. 5. (drawn to scale) The oxygen (O) and two hydrogen (H) atoms of water are shown as circles with covalent radii, d(O) and d(H). The circles on the right represent ions with aqueous ionic radii, d(i), aq = kd(A). The length of the hydration bond, d(–O) is the distance between the center of O(water) and ˚ . d(i), aq (in A ˚ ): 1.42 (Li+) < 1.65 (Ag+ circle with P(i/w) (shown as dotted lines in Figs. I–V): (I) Li+, , Cs+: k = 0.81 = //2, d(–O) = 0.64 (±) 0.03 A + + + + 1/2 ˚ . d(i), aq (in A ˚ ): 1.11 (F) < 1.48 dashes) < 1.74 (Na ) < 2.16 (K ) < 2.26 (Rb ) < 2.49 (Cs ). (II) F , , I : k = 1.57 = 2// , d(–O) = 1.60 (±) 0.04 A ˚ . (III): Ni+2, , Ba+2: k = 0.80 //2 = 0.81, d(– (Cl) < 1.74 (Br) < 2.08 (I). The ‘hydrogen bond’ length d(–H) = d(–O) d(OH) = 0.63 A ˚ . d(i), aq (in A ˚ ) (only some are shown to avoid overcrowding): 0.98 (Ni+2) < 1.19 (Hg+2) < 1.32 (Mg+2) < 2.06 (Ba+2). (IV): O) = 1.06 (±) 0.08 A ˚ . d(i), aq (in A ˚ ) (only some are shown): 0.55 (Al+3) < 0.63 (Rh+3) < 0.81 (Tl+3) < 0.91 (Yb+3, Al+3, , Eu+3: k = 0.47 = (2//3), d(–O) = 1.40 (±) 0.06 A +3 +3 +3 ˚ . d(i), aq in increasing order (in A ˚ ) (only some dashed line) < 1.08 (Eu , dash and dot line). (V) Lu , , La : k = 0.62 = //2, d(–O) = 1.37 (±) 0.05 A are shown): 0.97 (Lu+3) < 1.01 (Y+3, dashed line) < 1.13 (Sm+3) < 1.17 (La+3).
˚ [1], the value suggested for the radius of hydrogen = 0.28 A in hydrogen halides in [6]. Fig. 5II shows the water–anion points of contact P(i/w) and hydration length d(–O). 3. The H . . . X distance for halogen (X) anions and the length of the hydrogen bond Since anions form hydrogen bonds with the hydrogen of water [12], the distances, d(H . . . X) obtained as the sums, dðH . . . XÞ ¼ dð–HÞ þ dðiÞ; aq
ð12Þ
˚ ], are with d(–H) = 0.63 (±) 0.04 [d(H) + d(H+) = 0.65 A ˚ ˚ d(H . . . F) = d(–H) + d(F) = 0.63 + 1.11 = 1.74 A (1.74 A ˚ ˚ [13]), d(H . . . Cl) = 0.63 + 1.48 = 2.11 A (2.15 A [13]), ˚ (2.40 A ˚ [13]) and d(H . . . Br) = 0.63 + 1.74 = 2.37 A ˚ ˚ d(H . . . I) = 0.63 + 2.08 = 2.71 A (2.72 A [13]). The comparable values in parentheses are from Table 3.25 in [13]. Acknowledgements This research was supported by grants of the Academy of Sciences of the Czech Republic AVOZ50040507 and of
R. Heyrovska / Chemical Physics Letters 429 (2006) 600–605
the Ministry of Education, Youth and Sports of the Czech Republic, LC06035. References [1] [2] [3] [4] [5]
R. Heyrovska, Mol. Phys. 103 (2005) 877. Y. Marcus, Chem. Rev. 88 (1988) 1475. R. Heyrovska, Chem. Phys. Lett. 163 (1989) 207. R. Heyrovska, Marine Chem. 70 (2000) 49. R. Heyrovska, 6th Working meeting of physical chemists and electrochemists, Masaryk University, Brno, February 2006, Book of abstracts, p. 38.
605
[6] L. Pauling, The Nature of the Chemical Bond, Cornell Univ. Press, Ithaca, NY, 1960. [7] M. Livio, The Golden ratio, the story of Phi, the World’s most astonishing number, Broadway Books, NY, 2003. [8] http://www.webelements.com/webelements/elements/text/La/ radii.html. [9] H. Ohtaki, T. Radnai, Chem. Rev. 93 (1993) 1157. [10] F. David, V. Volkhmin, G. Ionova, J. Mol. Liquids 90 (2001) 45. [11] A. Grossfield, J. Chem. Phys. 122 (2005) 024506. [12] M. Chaplin, http://www.lsbu.ac.uk/water/hbond.html. [13] R. Desiraju, T. Steiner, The weak hydrogen bond in structural chemistry and biology, International monographs on crystallography, Oxford Science Publications, 1999.