New insights from the relation between lattice energy and bond stretching force constant in simple ionic compounds

Accepted Manuscript New Insights from the Relation between Lattice Energy and Bond Stretching Force Constant in Simple Ionic Compounds Savaş Kaya, Eduardo Chamorro, Dimitar Petrov, Cemal Kaya PII: DOI: Reference:

S0277-5387(16)30679-9 http://dx.doi.org/10.1016/j.poly.2016.12.028 POLY 12386

To appear in:

Polyhedron

Received Date: Revised Date: Accepted Date:

8 September 2016 12 December 2016 18 December 2016

Please cite this article as: S. Kaya, E. Chamorro, D. Petrov, C. Kaya, New Insights from the Relation between Lattice Energy and Bond Stretching Force Constant in Simple Ionic Compounds, Polyhedron (2016), doi: http://dx.doi.org/ 10.1016/j.poly.2016.12.028

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New Insights from the Relation between Lattice Energy and Bond Stretching Force Constant in Simple Ionic Compounds Savaş Kaya,1,* Eduardo Chamorro,2 Dimitar Petrov,3 Cemal Kaya1 1

Department of Chemistry, Faculty of Science, Cumhuriyet University, Sivas, 58140, Turkey Universidad Andres Bello, Departamento de Ciencias Quimicas. Facultad de Ciencias Exactas. Millennium Nucleus Chemical Processes and Catalysis (CPC), Avenida República 275, 8370146 Santiago, Chile 3 Department of Physical Chemistry, University of Plovdiv “Paisii Hilendarski”, 24, Tsar Asen Str., 4000 Plovdiv, Bulgaria * corresponding author e-mail: [email protected] 2

ABSTRACT: Lattice energy and bond stretching force constant are two quantities considered in many fields of chemistry and physics. Both quantities can be used to describe the stability or reactivity of a simple system. In the present study, by inspecting the correlation between lattice energies and bond stretching force constants for some sets of simple ionic molecules having similar bonding type, we present simple and useful equations to estimate the bond stretching force constants of simple ionic compounds. In order to determine lattice energies taken into account in this study, Born-Haber-Fajans (BHF) thermochemical cycle and lattice energy equations proposed by Jenkins and Kaya were used. It is shown that bond force constant of a simple ionic compound can be easily calculated in agreement with its spectroscopic counterpart through its lattice energy value. Percentage deviation values determined considering calculated and observed bond stretching force constants for the tested ionic compounds are pretty small except of some molecules like cesium halides (CsF, CsCl, CsBr). Such small values of percentage deviation probe the reliability and validity of the correlation between lattice energy and bond stretching force constant. Keywords: Bond stretching force constant, Lattice energy, Chemical Hardness, Molar Volume, Maximum Hardness Principle, Minimum Polarizability Principle 1. INTRODUCTION In the description of different characteristics of chemical species related to their chemical bonding, stability and reactivity, popular chemical concepts like hardness (η) and molar volume (Vm) are commonly used [1-4]. Maximum Hardness (MHP) [5-6] and Minimum Polarizability (MPP)[7] Principles are two useful electronic structure rationalizing tools in chemistry. Maximum Hardness Principle proposed by Pearson [6] states “there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible.” The formal proofs regarding the validity of this electronic structure principle was provided by Parr and Chattaraj [5]. Some studies also showed that the validity of this electronic structure principle may require certain conditions including both temperature and chemical potential remain constant. As an extension of MHP, Sengupta and Chattaraj [7] argued a Minimum Polarizability Principle (MPP) which states that “the natural direction of evolution of any system is toward a state of minimum polarizability.” It can be thus noted on the light of these

electronic structure principles that chemical systems having high chemical hardness and low polarizability are energetically favored (i.e., become more stabilized). Lattice energy (U) is a key parameter for the prediction of the stability of ionic compounds [8]. It provides an estimate of the strength of the bonds in an ionic compound and corresponds to the enthalpy of the reaction: A x By (c) → xA y + + yB x − Lattice energy is not a

quantity that can be experimentally determined directly because in practice, the crystalline solid dissociates into atoms and not into gaseous ions. Lattice energy values obtained via Born-Haber-Fajans (BHF) thermochemical cycle [9] for ionic materials are indeed considered experimental data. Although many theoretical methodologies [8-15] proposed for the estimation of lattice energies of ionic materials are available in the literature, only some [8,9,14,15] of these become quite useful and provide extremely consistent results with experimental data. Recently, Jenkins [15] reported the following alternative lattice energy estimation method that can be used for lattice energy values less than 5000 kJ/mol, i.e.

U (kJ/ mol) = 2I α (Vm −1/3 ) + β 

(1)

where Vm represents the molar volume of the ionic compound, being α and β appropriate fitted coefficients chosen taking into account the stoichiometry of the salt. In Eq. (1), I represent the ionic strength of the lattice which is simply calculated in terms of the number and type of ions with charge zi entering the unit formula, namely, I = 1/ 2∑ ni zi 2 For the estimation of lattice energies, the advantage of the use of molar volume values of ionic compounds is that it can be determined from the chemical formula mass (Mm) and density (ρ m) values ( ρm = M m / Vm ). Density is a readily accessible, easily measurable and widely reported property for ionic compounds. It should be noted that the Jenkins equation (1) supports the validity of Minimum Polarizability Principle aforementioned because the Lorentz-Lorenz [16,17] correlation states that polarizability (PB)[18,19] of any compound is closely related to its molar volume, namely, molar volume can be considered as a measure of the polarizability. Chemical hardness has been identified as the resistance of an atomic or molecular system to deformation [20]. Recently, in analogy to Eq. (1) Kaya et.al [14] have proposed a new formula that incorporates both hardness and molar volume to calculate the lattice energy of inorganic ionic compounds, namely,

U (kJ/ mol) = 2I  aηM (Vm −1/3 ) + b 

(2)

In this equation, ηM is the chemical hardness of the ionic compound. a and b are coefficients that takes different numerical values for ionic salts having different stoichiometry. In the application of this method, molecular hardness values of ionic crystals are calculated using Eq. (3) [21].



N

 

N



ηM = (2∑ bi / ai ) + qM  / ∑1/ ai 

(3)  i =1   i =1  where ai and bi are parameters depending on first ionization energy (IE) and first electron affinity (EA) value for an atom and these parameters are given as: ai = ( IE + EA) / 2 and

bi = ( IE − EA) / 2 , respectively. N and qM stand for the total number of atoms in the molecule and the charge of molecule, respectively. It should also be noted that many researchers have proposed useful empirical expressions relating the force constants of diatomic molecules. One of the well-known of mentioned empirical relations is Badger’s rule [22] given below.

k=

1.86 × 105 (re − dij )3

(4)

where, k is bond stretching force constant in dyn cm-1 units. re represents the internuclear distance in Angstrom (A0) units and dij is a function of the position of bonded atoms in the periodic table. After Badger’s rule, Gordy [23] put forward a useful equation that shows the relation between bond force constant, bond order, bond length, and the electronegativities of bonded atoms for a diatomic molecule. One of the most remarkable developments for the prediction of bond force constants of diatomic molecules has been provided by Pearson [24]. He used the following simple formula for bond force constants (kAB) of diatomic molecules AB, k AB = 2

Z A .Z B R0 3

(5)

where ZA, and ZB, are the effective charges on A and B atoms, respectively. R0 is the equilibrium nuclear separation. Nalewajski [25] who pointed out to some deficiencies of Pearson method, introduced a simple empirical force constant-bond length relation that is more accurate than Eq. (5) being also compatible with Badger’s rule, namely, (6) k AB = ( Z A + δ A ).( Z B + δ B ) / 2 nA + nA −2 Re 3 In the presentation of this equation, Nalewajski used the expression: z X = (Z X + δ X ) / 2nX −1/2 to calculate the effective charge (zX) in terms of the location of atom X in the periodic table. Here ZX represents the atomic number. For an atom X from groups IA-VA of the periodic table δX is equal to zero. δX for any X atom from groups VIA and VIIA is calculated as: δ X = 5 − υ X . Re is the bond length of diatomic molecule. nX and υX symbols stand for the period and the group of atom X, respectively. In 2000, Ohwada [26] proposed a simple method to calculate the heteronuclear diatomic force constant (kAB) and homonuclear diatomic force constants (kAA, kBB). Soon after, considering vibrational wave numbers of molecules, Kurita [27] modified the Badger’s rule and applied to CX (X= C, Si, Ge, N, P, As, O, S, Se, F, Cl and Br) type diatomic molecules. Investigating the correlation between bond stretching force constant and chemical hardness, recently, Kaya et. al. [28] derived the following useful equation to calculate the bond stretching force constants of diatomic molecules via their molecular hardness (ηM) and bond length (re) data. aη (7) k AB = 2M + b re

wherein kAB is bond force constant in N/m unit; ηM is molecular hardness in eV unit. re represents the bond length of diatomic molecules. a and b are parameters that have different numerical values for different molecule groups. The aim of this article is to present a useful method that exploits the relationship between lattice energy and bond force constant in diatomic ionic compounds. In the presentation of the mentioned relationship, Born-Haber Fajans thermochemical cycle and lattice energy equations derived by Jenkins and Kaya will be considered. 2. THEORETICAL APPROACH

One of the main aims of theoretical chemistry is to present useful correlations between chemical concepts given that they provide important clues about the reactivity and stability of chemical species. In the present work we investigate for a simple and useful correlation between bond stretching force constant and lattice energy for diatomic ionic compounds. Bond dissociation energy (BDE), bond length and vibrational stretching frequency are experimentally measurable properties of a chemical bond. Bond force constant can be easily determined from IR and Raman spectroscopy. The bond stretching force constant of a chemical bond is closely associated with parameters such as bond dissociation energy, bond length, and bond order indices. For instance, Gordy [23] successfully pointed out that there is a linear correlation between bond length, bond order, and bond force constant. Also, in a study including diatomic hydrides, Pearson [29] showed that there is a linear correlation between bond dissociation energy and k1/2, where k is bond force constant. It is known that a stronger chemical bond has a shorter bond length and a higher stretching frequency. It can be said that the bond force constant is a measure of the strength of a chemical bond and, for a diatomic molecule, this quantity can be considered a criteria for stability. As we have above emphasized, lattice energy also is one of the important tools in terms of the stability analysis of chemical species. In the present work we will stress that lattice energy of ionic compounds may be closely associated with its bond stretching force constants because both quantities are used in the comparison of stability of chemical species [30,31]. In the literature, bond force constant value obtained from IR spectroscopy for alkali halides are easily accessible. Bond stretching force constant values of diatomic molecules considered in the present study have been computed with the simple harmonic formula in Eq. (8) [32], (8) ν = (1 / 2π )( k / µ )1/ 2 where k and ν is bond stretching force constant and vibrational frequency of the diatomic molecule AB, respectively. µ represents the reduced mass, µ = (mA × mB ) / (mA + mB ), with mA and mB representing the atomic masses of A and B atoms. Recently, Zavitsas [33] presented the relation between stretching frequencies and bond lengths for diatomic and polyatomic species, namely Eq. (9), (9) ν = a / ( r µ 1/ 2 ) + b where r stands for bond length of molecule, and the coefficients a and b that takes different numerical values for different sets of molecules. It is important to note that the results of the method introduced by Zavitsas are compatible with experimental vibrational frequency

available data. We also considered this method by using results of Eq. 9 into Eq. 8 in cases where no experimental values for the bond stretching force constant are available. To ascertain the relation between bond stretching force constant and lattice energy, we use results from the associated linear correlation analysis. Experimental lattice energy (BHF) and calculated lattice energy values via Kaya and Jenkins methods, as well as bond force constant values obtained from experiments molecular hardness, and molar volume values for 49 simple ionic compounds are reported in Table 1. Table 1, Lattice energy (U) obtained using various methods and bond stretching force constant (k), chemical hardness (ηM) and molar volume (Vm) values for selected diatomic molecules Alkali UBHF UKaya UJenkins ηM Vm k halide (kJ/mol) (kJ/mol) (kJ/mol) (eV) (nm3) (N/m) 1036 1046 6,85 0,01634 249,50 LiF 1029 853 839 5,99 0,03403 150,80 LiCl 827 807 800 5,85 0,04163 126,40 LiBr 780 757 746 5,56 0,05452 98,17 LiI 721 923 924 6,62 0,02725 175,40 NaF 905 787 787 5,81 0,04482 110,12 NaCl 764 747 753 5,63 0,05322 95,82 NaBr 727 704 713 5,43 0,06782 76,40 NaI 678 821 803 5,76 0,03889 137,84 KF 796 715 701 5,08 0,06239 86,32 KCl 695 682 679 4,95 0,07212 70,05 KBr 667 649 650 4,77 0,08826 52,58 KI 630 785 761 5,59 0,04875 130,84 RbF 723 689 679 4,94 0,07170 76,57 RbCl 668 660 659 4,82 0,08197 67,09 RbBr 644 630 623 4,65 0,11338 49,31 RbI 610 740 714 5,26 0,06129 124,85 CsF 723 659 664 4,66 0,07006 71,94 CsCl 672 631 646 4,54 0,07959 56,85 CsBr 648 604 623 4,39 0,09565 39,06 CsI 616 1088 1189 978 8,75 0,01930 332,00a CuF 996 928 791 7,49 0,03965 231,00a CuCl 978 867 738 7,21 0,05057 206,00a CuBr 966 829 717 6,85 0,05577 174,00a CuI 918 1072 1017 7,15 0,01692 101,69a LiH 807 945 871 6,93 0,02853 77,54a NaH 713 806 755 6,12 0,04656 55,28a KH 684 772 719 5,94 0,05522 51,10a RbH 653 731 687 5,61 0,06502 46,21a CsH 837 929 766 7,79 0,04437 232,85a TlF 711 827 713 6,86 0,05685 143,15a TlCl 692 801 695 6,66 0,06246 124,68a TlBr 672 760 658 6,38 0,07546 103,80a TlI 974 1027 814 8,59 0,03600 250,60a AgF 918 904 774 7,35 0,04279 183,80a AgCl 905 866 748 7,08 0,04817 167,30a AgBr 892 792 676 6,72 0,06869 146,70a AgI

3526 3354 2721 11,05 BeF2 3033 2647 2310 9,20 BeCl2 2950 2494 2217 8,71 BeBr2 2780 2299 2085 8,07 BeI2 2978 3298 2866 10,15 MgF2 2540 2495 2326 8,23 MgCl2 2451 2341 2207 7,80 MgBr2 2340 2166 2067 7,26 MgI2 2640 2919 2694 9,03 CaF2 2271 2271 2182 7,47 CaCl2 2134 2167 2096 7,14 CaBr2 2087 2029 1974 6,70 CaI2 2476 2728 2551 8,61 SrF2 2170 2220 2178 7,16 SrCl2 2040 2128 2106 6,85 SrBr2 1976 1991 1969 6,45 SrI2 2373 2529 2417 8,06 BaF2 2069 2140 2154 6,74 BaCl2 1995 2052 2072 6,47 BaBr2 1890 1938 1962 6,09 BaI2 a force constant values were calculated using Zavitsas method and Eq. 8

0,03930 0,06987 0,08090 0,10090 0,03286 0,06814 0,08218 0,10424 0,04070 0,08571 0,09899 0,12336 0,04919 0,08625 0,09745 0,12460 0,05950 0,08967 0,10322 0,12611

515,00 328,00 253,00 196,00 323,00 205,00 167,00 145,00 190,00 118,00 103,00 86,00 162,00 104,00 89,00 74,00 151,00 97,00 77,00 65,00

250

Force constant, N/m

200

150

100

50

0 600

700

800

900

1000

1100

Lattice energy, kJ/mol

Fig. 1. Plot of lattice energy (UBHF) vs. bond stretching force constant of alkali halides with a correlation coefficient, R2=0.9689, N=20

600

Force constant, N/m

500

400

300

200

100

0 1600

2000

2400

2800

3600

3200

Lattice energy, kJ/mol

Fig.2. Plot of lattice energy (UBHF) vs. bond stretching force constant of earth alkali metal halides with a correlation coefficient, R2=0.9799. N=16

120

Force constant, N/m

100

80

60

40 600

700

800

900

1000

Lattice energy, kJ/mol

Fig. 3. Plot of lattice energy (UBHF) vs. bond stretching force constant of alkali hydrides with a correlation coefficient, R2=0.9965. N=5

280

Force constant, N/m

240

200

160

120 880

900

920

940

960

980

Lattice energy, kJ/mol

Fig. 4. Plot of lattice energy (UBHF) vs. bond stretching force constant of silver monohalides with a correlation coefficient, R2=0.9975. N=4

360

Force constant, N/m

320

280

240

200

160 960

1000

1040

1080

1120

Lattice energy, kJ/mol

Fig. 5. Plot of lattice energy (UBHF) vs. bond stretching force constant of copper (I) halides with a correlation coefficient, R2=0.9820. N=4

240

Force constant, N/m

200

160

120

80 640

680

720

760

800

840

Lattice energy, kJ/mol

Fig. 6. Plot of lattice energy (UBHF) vs. bond stretching force constant of thallium monohalides with a correlation coefficient, R2=0.9949. N=4

Figs 1-6 are plots of bond stretching force constants vs. lattice energy of alkali halides, earth alkali metal halides, alkali hydrides, silver monohalides, copper (1) halides and thallium halides, respectively. As it can be seen from the plotted graphs above, there is a linear correlation between lattice energy (U) and bond stretching force constant (k) of simple ionic compounds. The correlation coefficients of plotted graphs can be considered as a proof of the reliability for the relationship between lattice energy and bond stretching force constants in simple ionic compounds. The following simple formula for the calculation of bond stretching force constants of simple ionic compounds considered in this study. k = mU + n (10) wherein k represents the bond stretching force constant in N/m unit of simple ionic compound. U is the experimental lattice energy (determined by Born-Fajans-Haber thermochemical cycle) in kJ/mol unit for simple ionic compound. m and n are constants that take different numerical values for different sets of ionic compounds. The numerical values and the units of the mentioned coefficients are reported in Table 2. Table 2. Numerical values and units of the constants required to use in Eq.10(a) Molecule m (N m-1 mol kJ-1) n (N m-1) 0.4627 -241.93 Alkali halides 0.2691 -471.65 Earth alkali metal halides 0.2139 -95.16 Alkali hydrides 1.2443 -960.49 Silver monohalides 1.2215 -994.33 Copper (1) halides 0.7617 -403.4 Thallium monohalides a)

determined from linear correlations using the test set of diatomic molecules in Table 1.

Kaya and Jenkins have previously calculated the lattice energies of ionic compounds from the light of their ηM/Vm1/3 and 1/Vm1/3 ratios. We can also present new bond stretching

force constant equations based on ηM/Vm1/3 and 1/Vm1/3 ratios for the selected set of simple ionic compounds because there is a remarkable correlation as given in Eq. 10. In addition, we propose the following equations to predict the bond stretching force constants of simple ionic compounds, namely, k=p k =t

ηM Vm1/ 3 1

Vm1/3

+q

+z

(11) (12)

The numerical values and units regarding p , q, t and z constants appearing in Eq. 11 and Eq.12 are presented in Table 3. Table 3. Numerical values and units of the constants required to apply Eq. 11 and Eq. 12 To use in Eq.11 To use in Eq.12 Molecule p q t z 11.278 -60.541 112.52 -195.36 Alkali halides 18.829 -177.63 323.61 -586.21 Earth alkali metal halides 4.0109 -11.522 40.071 -54.334 Alkali hydrides 11.064 -42.187 162.93 264.12 Silver monohalides 10.244 0.102 133.81 -165.08 Copper (1) halides 19.198 -193.15 287.95 -591.22 Thallium monohalides

3. RESULTS and DISCUSSION

This work presents new and useful correlations between chemical concepts which are used to characterize the stability or reactivity of diatomic ionic molecules. We have found a linear correlation between their lattice energies and bond stretching force constants. On the light of mentioned linear correlation, we have introduced some new and useful equations that can be used to calculate the bond stretching force constants of such a ionic compounds. Table 4 contains bond force stretching force constant values calculated using Eq.10, Eq.11 and Eq. 13 for the molecules being considered in this study. As one can see from Tables 1 and 4, the equations presented in this study provides quite reliable and accurate results in terms of the prediction of bond stretching force constants of simple ionic compounds. It is apparent that the most useful among our bond force constant equations is Eq. 10. This equation requires the knowledge of the experimental lattice energy to calculate the bond force constant. On the other hand, to calculate the bond stretching force constant of a simple ionic compound via Eq. 11 and Eq. 12, we need additional calculations.

Table 4. Bond stretching force constants (in N/m units) obtained via the derived equations in the study Molecule Eq.10 Eq.11 Eq.12 237 244 248 LiF 153 148 152 LiCl 131 130 129 LiBr 108 105 101 LiI 185 187 179 NaF 122 124 121 NaCl 104 108 104 NaBr 84 89 80 NaI 138 131 137 KF 89 84 88 KCl 74 74 75 KBr 58 60 57 KI 121 112 113 RbF 77 74 75 RbCl 63 64 64 RbBr 49 48 37 RbI 100 90 90 CsF 63 67 77 CsCl 50 58 66 CsBr 37 47 50 CsI 477 434 360 BeF2 344 243 197 BeCl2 329 419 418 MgF2 211 202 201 MgCl2 187 160 154 MgBr2 158 113 97 MgI2 139 141 143 CaCl2 102 112 109 CaBr2 90 76 59 CaI2 112 127 141 SrCl2 77 102 112 SrBr2 60 65 57 SrI2 167 211 237 BaF2 85 105 132 BaCl2 65 82 99 BaBr2 37 51 55 BaI2 101 100 101 LiH 77 79 77 NaH 57 57 57 KH 51 51 51 RbH 44 44 45 CsH 251 245 229 AgF 181 190 201 AgCl 165 173 183 AgBr 149 139 133 AgI 334 334 334 CuF 222 225 227 CuCl 200 199 197 CuBr 186 183 185 CuI TlF 229 234 222 149 TlCl 138 157

TlBr TlI

123 108

129 96

134 90

In general, the calculated bond stretching force constant values using Eq. 10 are in close agreement with the data obtained from experiments (last column in Table 1). In order to test the reliability and limitations of the use of the linear correlation between lattice energy and bond stretching force constant, we applied the proposed equations to earth alkali metal halides including three atoms in addition to diatomic molecules like alkali halides, alkali hydrides, silver monohalides, thallium halides and copper (I) halides. It can be seen from Table 4 that results obtained using the three different proposed equations are very useful for diatomic molecules. However, the calculated bond stretching force constant values obtained via Eq. 11 and Eq. 12 exhibit notable deviations from experimental data. Note however that Eq. 10 gives valid and reliable results even for earth alkali metal halides. In this case, it should be noted that the use of Eq. 10 compared to Eq. 11 and Eq. 12 is more simple and more logical for earth alkali metal halides. In terms of the prediction of lattice energies of inorganic ionic compounds, it is apparent that Jenkins equation given by Eq. 1 will work for all materials for which their "formula unit volume" V is available which as well as being obtainable from density, as is stated, their values also are taken from crystal structure data. On the other hand, Kaya lattice energy equation given via Eq.2 in contrast needs value for the hardness parameter which is surely limiting the equation to apply to molecules with only one type of bond. A similar comparison is valid for Eq. 11 and Eq. 12. The use of the Eq. 11 including chemical hardness concept is more complex and limited to the use of Eq. 12 including only molar volume concept. On the other hand, it should be noted that Eq. 11 in general provides more close results to experimental bond stretching force constant values compared to Eq. 12.

Table 5, Comparison of bond force constant values (in N/m units) using various theoretical methods for alkali halides Alkali Nalewajski Pearson Kaya Experimental halide 227 207 237 249,50 LiF 138 135 153 150,80 LiCl 131 110 131 126,40 LiBr 78 91 108 98,17 LiI 222 109 185 175,40 NaF 157 86 122 110,12 NaCl 158 75 104 95,82 NaBr 99 63 84 76,40 NaI 135 86 138 137,84 KF 94 66 89 86,32 KCl 94 58 74 70,05 KBr 59 49 58 52,58 KI 116 75 121 130,84 RbF 81 58 77 76,57 RbCl 81 51 63 67,09 RbBr 51 43 49 49,31 RbI 78 72 100 124,85 CsF 53 53 63 71,94 CsCl 54 49 50 56,85 CsBr 34 41 37 39,06 CsI

obtained % SD 5,01 1,45 3,63 10,01 5,47 10,78 8,53 9,94 0,11 3,10 5,63 10,30 7,52 0,56 6,09 0,62 19,90 12,42 12,04 5,27

In the calculation of bond vibrational frequencies of molecules, Eq. (9) proposed by Zavitsas become quite useful and trustworthy. This equation provides almost the same results that bond vibrational frequencies determined by IR spectroscopy. The linear correlation coefficients of ν vs. 1/reµ1/2 plotted for several sets of molecules in Ref 33 proofs the reliability of Zavitsas method. On this basis, by inserting into Eq. (8) the bond stretching frequencies calculated using Eq. (9), bond stretching force constants of diatomic molecules can be accurately obtained. In the Table 1, bond stretching force constant values labeled with “a” superscript have been calculated via Zavitsas method. In the same Table, bond stretching force constant values given for earth alkali halide molecules have been taken from Ref. [34]. As is known, bond vibrational frequencies of diatomic molecules are determined in the light of Eq. 8. Experimental bond force constant value for a diatomic molecule is calculated using Eq. 8 and considering bond vibrational frequency value that was determined by IR spectroscopy for mentioned diatomic molecules. Zavitsas method calculates the bond vibrational frequencies of diatomic and some polyatomic molecules in a great agreement with experimental results. Already, the data given and the plotted graphs in Ref. 33 support our this idea. For that reason, in situations that we could not reach to experimentally determined bond stretching force constant values, we determined the bond stretching frequencies of diatomic molecules using Zavitsas method and calculated the bond stretching force constants of these molecules with the help of Zavitsas data.

600

Experimental force constant, N/m

500

400

300

200

100

0 0

100

200

300

400

500

Calculated force constant, N/m

Fig. 7. Graphical comparison of calculated and experimental values of bond stretching force constants for all of studied molecules, R2: 0.9848 (Note: Theoretical results were obtained using Eq.10)

Table 6. Bond stretching force constant values (in N/m units) predicted via Badger’s rule and Kaya equation (Eq. 10) for some selected diatomic ionic molecules Molecule Badger Kaya 94.15 101 LiH 84.73 77 NaH 45.50 57 KH 133.38 153 LiCl 172.61 185 NaF

Within the framework of perturbation theory [35], the simple bond-charge model proposed by Pearson for diatomic molecules demonstrated that bond stretching force constant of diatomic molecules, k, is proportional to R0-3 and the effective charges of the constituent atoms. Using a different mathematical technique, Nalewajski introduced Eq. (6) for the prediction of bond stretching force constants of diatomic molecules. Eq. (6) proposed by Nalewajski provides results in better agreement to experimental data as compared to Pearson method, although in some cases, Eq. (6) erroneously predicts the trends of bond stretching force constants along related series of compounds. For instance, bond stretching force constant values determined from IR spectroscopy for sodium halides follow the order: kNaF>kNaCl>kNaBr>kNaBr, whereas the Nalewajski method transposes the ordering for NaBr and NaCl. It is known that bond stretching force constant value of NaCl, having a higher bond ionicity than NaBr, should be greater than the bond stretching force constant of NaBr. Pearson method is more successful in the explanation of experimental trends of bond stretching force constants of diatomic molecules while Nalewajski method gives more close results to experimental data. As a further comparison, we have tabulated in Table 5 the bond stretching force constant values obtained using Eq. (5), Eq. (6) and Eq. (10) for alkali halides, and

computed the percentage standard deviations (% SD) regarding our results with the help of the following equation:

SD% =

kknown − kcalculated × 100 kknown

(13)

It is apparent from Table 5 that the calculated bond stretching force constant values via lattice energy based method is more accurate compared to the results previously reported on the basis of theoretical methods reported by Nalewajski and Pearson. The method proposed by us succeed in both the calculation of bond stretching force constants in accordance with experimental data, and in predicting the experimental trends of bond stretching force constants for diatomic molecules. Table 6 provides the bond stretching force constant values calculated via Eq. 10 and Badger’s rule for some selected diatomic molecules. Results are also rather remarkable. Here the lattice energy based method appears with comparable accuracy to that of the Badger. Compared to all examined theoretical models, our method provides the most striking advantage based only in the knowledge of the experimental lattice energy.

4. CONCLUSION

Linear correlations between two quantities, namely experimental lattice energy and bond stretching force constant for some groups of simple inorganic ionic compounds are here presented. We also suggested two bond stretching force constant equations based on ηM/Vm1/3 and 1/ Vm1/3 ratios in the framework of lattice energy equations derived by Kaya and Jenkins. It has been shown that bond stretching force constant equations are very successful for diatomic molecules, although these equations exhibit notable deviations from experimental data in earth alkali halide molecules. The results obtained using Eq.10, which is the most reliable among the equations presented in this study, were compared with both experimental data and results from popular equations for bond stretching force constants of diatomic molecules reported by Pearson, Nalewajski and Badger. In Fig. 7, a graphical comparison between experimental bond force constant values and theoretical bond force constant values calculated via Eq. 10 of diatomic and triatomic molecules considered in this study is given. Comparisons and results obtained in the present study stresses the validity and reliability of both the correlation between lattice energy and bond stretching force constant, and the usefulness of Eq. 10 proposed to calculate the bond stretching force constants of simple ionic compounds. We expect these results will contribute to stimulate the efforts in the building of simple theoretical models devoted to calculate the bond stretching force constants of molecules.

AUTHOR INFORMATION Corresponding author

E-mail: [email protected] Phone: +9005455936389

ACKNOWLEDGEMENTS

“EC thank the Millennium Science Initiative (ICM, Chile), FONDECYT (Chile) and UNAB through Grant Nos. NC120082, 1140343 and UNAB DI-806-15/R, respectively.” REFERENCES

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Experimental force constant, N/m

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Calculated force constant, N/m

Graphical comparison of calculated and experimental values of bond stretching force constants for all of studied molecules,