Absolute ion hydration enthalpies from absolute hardness and some VBT relationships

Accepted Manuscript Research paper Absolute ion hydration enthalpies from absolute hardness and some VBT relationships Savaş Kaya, Robson Fernandes de Farias PII: DOI: Reference:

S0009-2614(17)31032-1 https://doi.org/10.1016/j.cplett.2017.11.015 CPLETT 35229

To appear in:

Chemical Physics Letters

Received Date: Revised Date: Accepted Date:

29 October 2017 8 November 2017 9 November 2017

Please cite this article as: S. Kaya, R. Fernandes de Farias, Absolute ion hydration enthalpies from absolute hardness and some VBT relationships, Chemical Physics Letters (2017), doi: https://doi.org/10.1016/j.cplett.2017.11.015

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Absolute ion hydration enthalpies from absolute hardness and some VBT relationships Savaş Kayaa, Robson Fernandes de Fariasb* a

Cumhuriyet

University,

Faculty

of

Science,

Department

of

Chemistry, 58140, Sivas, Turkey. b

Universidade Federal do Rio Grande do Norte, Cx. Postal 1664,

59078-970 Natal, RN, Brasil. *Corresponding author. [email protected]

2

Graphical abstract:

3

Highlights:

 In the present work, absolute hydration enthalpies are calculated from ion absolute hardness for a series of +1 and -1 ions.  It is shown that for d block monocations (Cu+, Ag+ and Au+), hydration enthalpy is closely related with Clementi effective nuclear charge by the equation:

ΔhydHo = -(9.645 η+ +

245.930) (Zeff/n-1), where n is the main quantum number.  Is shown that a typical VBT parameter (Vm-1/3) is related with η+ and η- values and so, with the energies of the frontier orbitals, that is, is stablished a direct relationship between a structural parameter available by x-ray data and the energy of atomic/molecular orbitals.

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Abstract In the present work, absolute hydration enthalpies are calculated from ion absolute hardness for a series of +1 and -1 ions. The calculated values are compared with those previously reported [2] and relationships between Vm-1/3

and absolute hardness are

stablished. The following empirical equations have been derived, for cations and anions, respectively: ΔhydHo = -(9.645 η+ + 245.930) and ΔhydHo = -(64.601 η- + 12.321). In such equations, η+ and η- are the absolute hardness. It is shown that for d block monocations (Cu+, Ag+ and Au+), hydration enthalpy is closely related with Clementi effective nuclear charge by the equation: 245.930) (Zeff/(n-1)),

ΔhydHo = -(9.645 η+ +

where n is the main quantum number.

Furthermore, is shown that a typical VBT parameter (Vm-1/3) is related with η+ and η- values and so, with the energies of the frontier orbitals, that is, is stablished a direct relationship between a structural parameter available by x-ray data and the energy of atomic/molecular orbitals. Keywords: Hydration enthalpy, volume based thermodynamics, absolute hardness, empirical equation

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The hydration enthalpy of an ion can be defined as the enthalpy change verified when one mole of the ideal gaseous ion is dissolved in an infinite volume of water at standard conditions [1]. In a recent paper [2], expanding the horizons of volume based thermodynamics (VBT), have been shown that there is a straightforward relation (r = 0.999) between hydration enthalpies and the inverse of the square root of the volume (Vm-1/3) for a series of cations. Based on hydration enthalpy data for group 1 cations and group 17 anions, the following empirical equations have been derived [2]:

ΔhydHo

=

-(48.2

Vm-1/3

+

154.6)

(1) ΔhydHo

=

-(214.71Vm-1/3

+

271.96)

(2)

Eq. (1) is valid for + 1 cations and Eq. (2) is valid for -1 anions. Both equations were then employed to calculate the hydration enthalpies for a series of another cations and anions.

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In the present work, absolute hydration enthalpies are calculated from ion absolute hardness for a series of +1 cations and -1 anions. The calculated values are compared with those previously reported [2] and and relationships between Vm-1/3

and absolute

hardness are stablished.

The hydration enthalpies for group 1 cations (from Li to Cs) previously reported [2] were plotted as a function of absolute hardness, and the following empirical equation (r= 0.996) was derived:

ΔhydHo

=

-(9.645

η+

+

245.930)

(3)

Were η+ = cation absolute hardness (eV). The absolute hardness for the considered cations were calculated as: Li+ (35.12), Na+ (21.08), K+ (13.64), Rb+ (11.56) and Cs+ (9.61), using ionization energy values [3,4]. Of course, such equation implies that the hardest cation will have the larger (most exothermic) hydration enthalpy.

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For anions, the following equation (r= 0.999) was derived:

ΔhydHo

=

-(64.601

η-

+

12.321)

(4)

Were η- = anion absolute hardness (eV). The absolute hardness for the considered anions were calculated as: F- (7.01), Cl(4.70), Br- (4.24) and I- (3.70), using ionization energy values [3,4]. For another anions (shown in Table 2) values from literature [5] were employed. Of course, such equation implies that the hardest anion will have the larger (most exothermic) hydration enthalpy. The experimental ΔhydHo values shown in Table 1, were calculated using tabulated data [3,4], and a thermochemical cycle to the reaction M(s) + H+(aq) → M+(aq) + ½ H2(g). The enthalpy of such reaction is the formation enthalpy of the M+(aq) ion.

M(s) → M(g)

ΔH1

M(g) → M+(g)

ΔH2

M+(g) → M+(aq)

ΔH3

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H+(aq) → H+(g)

ΔH4

H+(g) → H(g)

ΔH5

H(g) → ½ H2(g)

ΔH6

Hence, ΔH3 = ΔhydHo = ΔHf M+(aq) - ΔH1 - ΔH2 - ΔH4 - ΔH5 - ΔH6. In such cycle, ΔH1 = ΔHf M(g) - ΔHf M(s); ΔH2 = the first ionization energy of M; ΔH4 = the negative of the hydration enthalpy of H+ = 1094.2 kJmol-1; ΔH5 = the negative of the first ionization energy of hydrogen = -1302.1 kJmol-1; and ΔH5 is the negative of half the dissociation energy of H2 = -217.9 kJmol-1. Two points must be clarified here: the experimental ΔhydHo values calculated in the present work are, sometimes, a very few kJmol-1 different from those previously calculated by Smith [6] since we have employed more precise tabulated values [3,4] to calculate the hydration enthalpy for H+ as -1094.2 kJmol-1, whereas Smith [6] reported a value of -1091 kJmol-1. The ΔhydHo

value for H+

calculated in the present work is a little (55.9 kJmol-1) different from those previously used by Jenkins [2] = -1150.1 kJmol-1. Equation (3) was applied to a series of +1 cations (with exception of group 1 cations, for which it works very well, of course),

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and the results are summarized in Table 1.

Equation (4) was

applied to a series of -1 anions (with exception of group 17 anions, for which it works very well, of course), and the results are summarized in Table 2. In Table 1, the values between parenthesis in the fourth column are the

Vm

-1/3

values recalculated, subtracting, from the

previously calculated values [2], 55.9 kJmol-1 (the difference between the hydration enthalpy for H+ of -1094.2 kJmol-1 calculated in the present work and the value of -1150.1 kJmol-1 used by Jenkins [2]). For Au+ the value obtained using Eq.(3) was submitted to a relativistic correction, because is necessary to remember that for gold, (Z = 79), relativistic contributions matters [7], and that gold is the

element

with

the

(proportionally)

higher

relativistic

contraction/effects. The relativistic and non-relativistic equations can be related by using γ = 1/[1-(v2/c2)], where v is the velocity of the considered body (in our case, an electron). The velocity of the 1s electron is ≈ Z/137, where Z is the atomic number. Hence, γ = 1/[1-((Z/137)2/c2)]1/2. For gold (Z = 79), γ = 1.224.

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Multiplying the hydration enthalpy value calculated using Eq. (3), by γ, a “corrected” hydration energy value is calculated for Au+, and is shown between parenthesis in Table 1. As can be verified from Table 1 data, Eq (3) works very well for Tl+ and not well for the group 11 monocations. Smith [6] have previously derived an empirical equation to calculate the hydration enthalpies for ions and have noted that for dn cations we have a more complicated situation than for cations with a noble gas configuration. Hence, some kind of “correction” in Eq (3) is necessary in order to enhance the reliability of such equation. As is well known, the crystal field stabilization energy (CFSE) plays a crucial role in hydration enthalpy values [8]. However, Cu+, Ag+ and Au+ are ions with d10 configuration and so, zero CFSE. Hence, the difference between the calculated hydration enthalpy values (Eq. 3) and the experimental values cannot be attributed to the CFES. Hence, to Cu+, Ag+ and Au+ the following empirical equation (a modification of Eq. 3) was derived:

ΔhydHo (5)

=

-(9.645

η+

+

245.930)

(Zeff/(n-1))

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In Eq. (5) Zeff is the effective nuclear charge for the valence electron of the neutral atom (4s, 5s and 6s, respectively for Cu, Ag and Au): 5.54, 6.76 and 10.94 [9,10], respectively, and n is the main quantum number for the valence electron (4, 5 and 6, for Cu, Ag and Au, respectively). The hydration enthalpy values calculated by using Eq. (5) are in very good agreement with the experimental ones, and are more accurate than those calculated simply by a VBT approach [2]. As can be verified from Table 2 data, Eq (4) works well for another (other than group 17) anions, and provides, specially for SH, a calculated value in much better agreement with the experimental value than the one provided simply by VBT approach [2]. Since Eq (1) and (3) works really very well for group 1 cations, equating both equations, we have:

Vm-1/3 (6)

=

(9.645

η+

+

89.53)/48.2

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Such equation, of course, stablish a relation between V m-1/3 and η+, in agreement with the fact that, for cations with the same charge, minor cations will be the hardest ones. Since Eq. (2) and (4) works really very well for group 17 anions, equating both equations, we have:

Vm-1/3

=

(64.601

η-

-

259.64)/214.7

(7)

Such equation, of course, stablish a relation between V m-1/3 and η-, in agreement with the fact that, for anions with the same charge, minor anions will be the hardest ones. Furthermore, since η+ and η- are related with the energies of homo and lumo orbitals, Eq. (6) and (7) stablish a relationship between a typical VBT parameter (Vm-1/3) and the energies of the frontier orbitals, that is, a direct relationship between a structural parameter

available

by

atomic/molecular orbitals.

x-ray

data

and

the

energy

of

13

References

[1] W.E. Dasent, Inorganic energetics, 2nd Ed., Cambridge University Press, Cambridge, 1982. [2] C.E. Housecroft, H.D.B. Jenkins, RSC Adv., 7 (2017) 27881. [3] http://www.rsc.org/periodic-table. [4] CRC Handbook of Chemistry and Physics 96th ed., Taylor and Francis, Boca Raton, 2016. [5] R.G. Parr, R.G. Pearson, J. Am. Chem. Soc., 105 (26) (1983) 7512. [6] D.W. Smith, J. Chem. Edu., 54 (9) (1977) 540. [7] J. Leszczynski, Relativistic methods for chemists, Springer, New York, 2010. [8] B.W. Pfennig, Principles of inorganic chemistry, Wiley, New Jersey, 2015. [9] E. Clementi, D. L. Raimondi, J. Chem. Phys., 38 (1963) 2686. [10] E. Clementi, D. L. Raimondi, W. P. Reinhardt, J. Chem. Phys., 47 (1967) 1300.

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.Captions of figures

Figure 1. The negative of hydration enthalpy as a function of absolute hardness for group 1 monocations (from Li to Cs). Figure 2. The negative of hydration enthalpy as a function of absolute hardness for group 17 monoanions (from F to I).

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Table 1. Hydration enthalpies (kJmol-1) for +1 cations, calculated by using Eq (3). Cation

Cu+

Ag+

Au+

η+ (eV)

6.3

7.0

5.6

- ΔhydHo(Eq.

- ΔhydHo(from Vm-

3)

1/3 a )

306.7

466.1

(566.4)d

(410.2)b

313.4

419.2

(529.7)d

(363.3)b

299.9

372.9

(367.1)c

(317.0)b

- ΔhydHo(exp)

593

474

615

(656.3)d Tl+

7.2

315.8

355.3

326

(299.4)b a

Ref. 2. bThe values between parenthesis are the

Vm

-1/3

values

recalculated, subtracting, from the values calculated in Ref.2, 55.9 kJmol-1 (the difference between the hydration enthalpy for H+ of 1094.2 kJmol-1 calculated in the present work and the value of 1150.1 kJmol-1 used in Ref.2). multiplied by γ; dUsing Eq. (5).

c

Calculated by using Eq. (3)

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Table 2. Hydration enthalpies (kJmol-1) for -1 anions, calculated by using Eq (4). η-

- ΔhydHo(Eq.

- ΔhydHo(from Vm-1/3)a

- ΔhydHo(exp)b

(eV)

4)

OH-

5.67

378.6

404.3

460

CN-

5.10

341.8

_

324

NO2-

4.50

303.0

_

405

SH-

4.10

277.2

672.6

333

Cation

a

Ref. 2; bRef. 6.

17

600

Hydration enthalpy/kJmol

-1

550

500

450

400

350

300 10

15

20

25

Absolute hardness/eV

Figure 1

30

35

18

500

Hydration enthalpy/kJmol

450

400

350

300

250

3.5

4.0

4.5

5.0

5.5

6.0

Absolute hardness/eV

Figure 2

6.5

7.0

7.5