Regularities and Correlations of Atomic Ionization

Iraqi J. Chem., 11, 3, 1993

Regularities and Correlations of Atomic Ionization Energies 1. The Noble Gases Bahjat R. J. Muhyedeen

Anwar T. Al- Thib

and Ghazi A. W. Derwish* Department of Chemistry, College of Science University of Baghdad, Jadriyah, Baghdad, Iraq

Abstract: Examination of the first ionization energies of the elements of Group VIII of the Periodic Table, the so-called Nobel Gases, showed their dependence on the period number N, the atomic number Z and the size of the atom as given by their covalent radii. Empirical relationships which were developed in terms of N and Z, revealed that the elements of Group VIII fall into two subsets, one with oddN values, viz., He, Ar, Xe, and the other with even-N values, viz., Ne, Kr, and Rn. This behavior was explained in terms of the characteristics of the orbitals in the outer shell of the Atoms. Values were obtained for the effective nuclear charge and effective period number in terms of the screening constant and quantum defects, which were compared with the corresponding terms of others, obtained from quantum mechanical considerations.

*

To whom all correspondence should be addressed

1

Anwar T. M. Al-Thib, Bahjat R. J. Muhyedeen and

Ghazi A. W. Derwish

1. Introduction: Ionization energies have always received great attention whether by theoreticians or experimentalists. For the former, calculations of ionization energies have been the proving grounds for their theories; but for the latter, values of ionization energies as determined by their various methods have proved a challenge for their experimental skills and the precision of their instruments. The empiricists, on the other hand, through the establishment of patterns of behavior and correlations of ionization energy values, have always played a critical role for both by anticipating the results of theories and in testing for irregularities and deviations of the experimentally determined values. Ahrens (1) had presented a lucid and comprehensive account of atomic ionization energies of the elements and their regularities and correlations with particular emphasis on empirical relationships. Castro and Fernandez (2) undertook an interesting comparison of a number of semi-empirical relations and had come to the conclusion that there was very little to choose from them. Correlations of ionization energies have proved to be of interest in other respects, thus for example in obtaining good estimates of atomic polarizabilities (3) and in developing a scale of electronegativities of Main-Group atoms (4,5). In another vein, Homer and Muhammadi (6) were able to optain good estimates of molecular ionization energies from values of the polarizibilites and the first ionization energies of the Noble-Gas atoms from which the molecules were hypothetically constructed according to an interesting empirical model. The periodic nature of atomic ionization energies of the elements had been observed almost since the inception of the modern periodic classification of the elements. In the present work, it is proposed to obtain empirical relationships of the first ionization energies of the Main-Group

Iraqi J. Chem., 11, 3, 1993

elements of the Periodic Table, based upon the rather elmple observation that, when the ionization energies of the elements of each group were normalized to that of the first member of the group, they all exhibited very similar patterns reminiscent of atomic-shell energy levels (7) as can be seen in Figure 1.

2. Method: In order to treat the subject methodically, it was decided to deal first with the elements of Group VIII, i.e., the Noble Gases, and to try to correlate their first ionization energies with some fundamental parameters such as the period number (N) of the elements, i.e., the principal quantum number of the outermost shell of the atom, the atomic number (Z), and the atomic radius (R). Data of atomic ionization energies were obtained from standard sources (8) and to avoid the problem of different units used, the results were expressed in terms of relative ionization energies ( Er ) normalized to the value for helium assumed to be 100. All relationships were obtained using either linear or polynomial regression analysis programmes, and the best functions were judged by the values of the coefficients of correlation obtained. The relative ionization energies were first correlated to either N, Z or R, and then to some combinations thereof. The chosen relationships were also examined graphically in order to detect any marked deviations or possible curvature that might necessitate any special consideration. Although the approach is empirical but many of the correlations were guided by theoretical considerations and some comparisons were made between the present results and those of others (2,9).

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Ghazi A. W. Derwish

3. Results and Discussions: 3.1 Correlation with the Period Number N Based on the similarity in the patterns exhibited by the elements of the different groups of the periodic Table as shown in Figure 1, it was decided to examine first the correlation of Er's of Group VIII elements with period number N. Figure 2, shows the simple plot of Er vs. N, from which it can be seen that there are two subsets of Group VIII elements, one contains the elements with odd N (viz., He, Ar,Rn ) and the other contains the elements with even N (viz., Ne, Kr, Xe ). Regression analysis showed that the two subsets could be represented by the functions given in Table 1, and that these functions have a much higher coefficients of correlation than any of the functions that contain all the elements of the Group. Basing the selection of the "best" functions on the values of the coefficients of correlation, the following cosine functions were selected for the two subsets: Er( odd-N)= 71.958 + 39.599 cosΠ( N/N+1)2

(1)

Er(even-N)= 78.771 + 51.874 cosΠ( N/N+1)2

(2)

and Figure 3 clearly shows how distinct were these two subsets.

3.2 Correlation with the Atomic Number Z The dependence of ionization energies upon atomic number or effective atomic number is well-known, and a number of empirical and semi-empirical relationships have been developed (1,2,9). However, the simplest plot of Er vs Z as shown in Figure 4 indicated strongly that in this instance the elements of Group VIII fall also into two distinct subsets, one for those elements with odd-N values, and the other for the even-N elements. Table 2 gives the functions for both subsets as well, as for the whole group. Here again the coefficients of correlation for the functions of the odd-N and even-N subsets were much better than that for the whole group.

Iraqi J. Chem., 11, 3, 1993

The "best " functions for the two subsets are: lnEr( odd-N)= 4.7582 - 0.21267 lnZ

(3)

lnEr(even-N)= 5.2167 - 0.32460 lnZ

(4)

and are illustrated in Figure 5. However, it is interesting to note that the functions in terms of Z do not correlate as well as those in terms of N.

3.3 Correlation with Atomic Radius R It is rather difficult to define meaningfully the term atomic radius, but for the purpose of the present work the following parameters were considered: covalent radii (Rcov), van der Waals radii (R v) and van der Waals coefficient (Bv). The latter of course gives a measure of atomic volume. It is well-known that ionization energies were found to decrease with the size of the atom (10), however it would be interesting to examine which of the above parameters gives the best correlation. The results are summarized in Table 3, and it can easily be seen that the relationships in terms of the covalent radius (Rcov) have the highest coefficient of correlation.

Having settled this point, it is interesting to note that here again the simple plot of Er vs Rcov, as seen in figure 6, indicates that the odd-N and the even-N elements follow two distinct functions. Regression analysis gave the functions outlined in Table 4 for both subsets as well as for the whole group, from which the same observation could be made as with the other two parameters N and Z. The best functions for the two subsets are: lnEr( odd-N)= 4.1037 - 0.7326 lnRcov lnEr(even-N)= 4.1082 - 0.8616 lnRcov

(5) (6)

and are illustrated in Figure 7. Comparison of the dependence of Er upon N, Z or R reveals that the "best" relationships were obtained between Er and N. 5


Ghazi A. W. Derwish

3.4 Correlation with N , Z and R In investigating the possible correlation of Er with N, Z and R use is made of the known quantum theoretic expressions (7,9) of electron energy and its distance from the nucleus in terms of the principal quantum number (n), and the atomic number (Z). Taking these into consideration, attempts were made to find the function with "best" coefficient of correlation. It is interesting to note that when in accordance with the old quantum theory, R was taken in terms of (N2/Z) instead of the covalent radius van der Waals radius, better correlations were obtained. The results, which are given in Table 5 and plotted in Figure 8, show that the function with "best" coefficient of correlation is : Er= a - b (Z/N2)0.2 Exp[-(N/N+1)4]

(7)

Opting for the simple functions of the general form Er= a + b Zp/Nq

(8)

with p,q = 1,2 ...., the results were of rather poor coefficients of correlation when compared with the logarithmic form : Er= a - b ln(Zp/Nq)

(9)

It may be of interest to note that equation 8 with p=q=2 and R is taken in terms of N2/Z resembles the relationship developed by Luo and Benson (4) if Z is replaced by the number of electrons of the outermost shell of the atom. Table 6 shows the same trend which has been noted in the previous sections ,i.e., the elements of group VIII fall into two subsets whose function have better coefficients of correlation than that for the whole elements of the group as illustrated in Figure 9 which clearly supports this observation.

Iraqi J. Chem., 11, 3, 1993

3.5 Quantum Defects and Screening Constants Preserving the simplicity of the present approach, it is interesting to examine the expected regularities in atomic ionization energies in, the light of quantum theoretic considerations. In its simplest form Rydberg formula gives the ionization energy in terms of Z2/N2, however a more sophisticated form expresses the energy with due consideration given to the so-called effective nuclear charge Z* and a modified quantum number N* (7). As stipulated by quantum mechanics, it is expected that quantum numbers and central potential fields can only serve as simplified models for atoms heavier than Helium (9), and to preserve these models terms such as screening constant (5) and quantum defects (6) were introduced (9,11). Following on this approach and recognizing that the "best" simple functions in terms of Z and N and of a form similar to that of the old quantum theory, were in the logarithmic form, i.e., equation 9, it was decided to find if by the use of effective "values" for Z and N, formula could be obtained with good coefficients of correlation that would treat the elements of group VIII as a whole rather than didividing them into two subsets with odd and even-N. The modified equation has the general form: Er= a + b ln[(Z-S)p/(N-D)q]

(10)

where S is the screening constant and D is the quantum defect. These two terms may be taken to correspond to Slater's terms, with the necessary observation that the present terms were obtained solely on the basis of the periodicity of ionization energies of the atoms whereas Slater's were based on the stipulation that the energy levels of any one atom must follow the quantum mechanical expression obtained from central field theory. The values* of S and D that gave the best fit for the ionization energies were found to be those given by equation 10 with p=3 and q=4 and are presented in Table 7. It could be seen from inspection of Figures 10 and 11 that these values *

The evaluation of D and S was carried out by a minimization procedure for the uncertainty in the values of N and Z respectively to obtain the “exact” experimental values of Er.

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Ghazi A. W. Derwish

are internally consistent. Table 7 also shows that the values for S compare fairly with those for the screening constants of Slater (9) and Clementi (10). However, the values for D bear little resemblance to those of Slater's quantum defects. This is not surprising in view of the basically different approaches. The strong interdependence of S and D as illustrated in Figure 12 suggests that their ratio ( i.e., D/S ) which falls progressively as the group is descended is related to ease of ionization of the atom. Furthermore the ratio Z*3/N*4 may be taken to represent the effective force on the electronic outer-shell with effective quantum number N* exerted by the "core" potential (Z*/R). In other words, with the approximation of central field in mind one may consider for each of the elements as if the ionizing electron is moving in a field of point charge Z* at a distance represented by N*. Examination of the values of S and D as calculated in the present work, further reveals the division of the ionization energy values of group VIII elements into two subsets. Thus starting with equation 10 using the values of a and b obtained for the odd-N elements and the even- N elements, the as given in Table 7 and plotted in Figure 12 in terms of effective values for Z and N show quite clearly the different behavior of the two subsets.

4. Conclusions: The results of the present work show interestingly the periodicity of the first ionization energies of group VIII elements in their dependence on the period number and atomic number and may thus serve as prototype for the other groups of the main elements of the Periodic Table. Preliminary results seem to support this notion.

Furthermore, close examination of the various relationships

developed in the present work revealed the division of the group into two subsets, one containing the elements He, Ar and Xe with odd-N values, and the other containing the elements Ne, Kr and Rn with even-values; the division may be explained in terms of the characteristics of the electron shells of the atoms.

Iraqi J. Chem., 11, 3, 1993

Thus, the even-N elements uniformly show higher ionization energies than otherwise would be expected if they were to follow the relationships of the oddN elements. This behavior is attributed starting with neon to the penetration of its L-electrons into the inner K-shell, for krypton to the diffused nature of the dorbital of the penultimate shell allowing the penetration of the N-shell electrons closer to the nucleus and thus enhancing, together with the doubling of the nuclear charge, the ionization energy for this element. Returning to the element argon, the empty d-orbitals of its M-shell act to diffuse its outermost electrons outwards thus counter the neon penetration effect. As for the remaining elements xenon and radon, the diffused nature of the dand f-orbitals and the strong overlapping of the energy levels as well as the break-up of Russel-Saunder coupling complicate the picture, but nevertheless careful examination of their behavior shows that while xenon may follow the helium-argon line thus completing the odd-N subset, radon falls nicely with neon and krypton with the even-N subset. However for these two heavy elements the two subsets may seem to converge.

5. Acknowledgment: This work was started while the senior author (G. A. W. D) was at University of Surrrey, England, on sabbatical leave of absence from University of Baghdad. Both institutes are cordially thanked. Special thanks are due to professor J.

R. Jo. Jones, Dr. G. A. Webb and Professor J. Pratt of the

Departments of Chemistry, University of Surrey, for their hospitality and kindness.

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Ghazi A. W. Derwish

6. References: 1- L. H. Ahrens " Ionization Potentials-Some Variations, Applications ", Pergamon Press, Oxford, 1983.

Implications and

2- E. A. Castro and F. M. Fernandez, J. Chem. Phys., 1982,76,1177. 3- B. Fricke, J. Chem. Phys., 1986,84,862. 4- Y. R. Luo and S. W. Benson, J. Phys. Chem., 1989,93,7333. 5- P. L. Dekock, J. Phys. Chem., 1990,94,1713. 6- J. Homer and M. S. Mohammadi, J. Chem. Soc. Faraday Trans.2, 1987,83,1975. 7- a. A. E. Ruark and H. C. Urey " Atoms, Molecules and Quanta ". McGrawHill, N.Y. 1930. b. G. Herzberg " Atomic Spectra and Atomic Structure " Dover Publication, N.Y. 1944. 8- a. J. L. Franklin, J. G. Dillard, H. M. Rosenstock, J. T. Herron, K. Draxl and F. H Field, "Ionization Potentials, Appearance Potentials, and Heats of Formation of Gaseous Positive Ions " National Standard Reference Data System, Washington, 1966. b. Handbook of Chemistry and Physics, CRC Press, Boca Raton, Florida, 70th Edition, 1989-90. 9- J. C. Slater " Quantum Theory of Atomic Structure " Volumes I & II McGraw-Hill, N.Y. 1960. 10- J. E. Huheey " Inorganic Chemistry " Harper & Row, Singapore, 1972. 11- W. Kauzmann " Quantum Chemistry " Academic Press Inc.,

N.Y., 1957.

Iraqi J. Chem., 11, 3, 1993

Table 1 Correlation of Relative Ionization Energies (Er) of Group VIII Elements with the Period Number (N) The General Form Er = a + b *cos(N/ N + 1)2 Groups VIII Elements

a

b

C.C.

S.D.S.

S.D.I.

Whole- set

42.082

73.870

0.9873

3.3871

0.0806

Odd- set

39.604

71.956

0.9999

0.1821

0.0046

Even- set

51.874

78.771

0.9999

0.5164

0.0099

C.C = Coefficient of correlation S.D.S = Standard deviation of slope S.D. I= Standard deviation of intercept

Table 2 Correlation of Relative Ionization Energies (Er) of Group VIII Elements with the Atomic Number (Z) The General Form Er = a + b *lnZ Groups VIII Elements

a

b

C.C.

S.D.S.

S.D.I.

Whole- set

4.84671

0.23014

-0.9660

0.0031

0.1338

Odd- set

4.75821

0.21267

-0.9991

0.0089

0.0417

Even- set

5.21670

0.32460

-0.9996

0.0093

0.0287

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Ghazi A. W. Derwish

Table 3 Comparison of Relationships With Paramerters (R x) Represnting Atomio Radius of Group VIII Elements The General Form Er = a - b ln Rx Rx *

Group VIII Elements

Rcov

Whole- set

Rv Rv

a

He, Ne, Ar, Kr, Xe He, Ne, Ar, Kr, Xe

b

C.C.

S.D.S.

S.D.I.

4.1015

0.78202

-0.9958

0.0360

0.0461

5.5923

2.13167

-0.8557

0.7442

0.3491

2.0590

0.6239

-0.9057

0.1686

0.2702

 Rcov = Convalent radius Rv = Van der Waals radius Rv = Van der Waals coefficient Table 4 Correlation of Relative Ionization Energies (Er) of Group VIII Elements with the van der Waal Radius (R cov) The General Form Er = a + b * ln Rcov Group VIII Elements

a

B

C.C.

S.D.S.

S.D.I.

Whole- set

4.1015

0.78202

0.9958

0.0360

0.0461

Odd- set

4.1037

0.73258

0.9989

0.0347

0.0473

Even- set

4.1082

0.86162

0.9992

0.0341

0.0396

Table 5 Correlation of Relative Ionization Energies (Er) of Group VIII Elements with N and Z The General Form Er = a + b (Z/ N2)0.2 *EXP-[(N/ N+1)4 Groups VIII Elements

a

B

C.C.

S.D.S.

S.D.I.

Whole- set

145.44

56.58

0.9992

2.8667

0.0197

Odd- set

142.25

53.87

0.9992

0.4015

0.0380

Even- set

150.52

60.66

0.9999

0.4082

0.0027

Iraqi J. Chem., 11, 3, 1993

Table 6 Correlation of Relative Ionization Energies (Er) of Group VIII Elements with N and Z The General Form Er = a - b ln(Z2/ N2) Groups VIII Elements Whole- set Odd- set Even- set

p = 1, q = 1

p = 3, q = 4

a

b

C.C

S.D.S

S.D.I

a

b

C.C

S.D.S

S.D.I

124.554 120.404 153.838

30.488 30.380 42.224

-0.9583 -0.9980 -0.9873

4.546 1.926 6.787

0.149 0.063 0.161

133.27 130.07 173.84

14.51 14.87 21.44

-0.9412 -0.9968 -0.9792

2.604 1.187 4.439

0.180 0.080 0.207

Table 7 Values of screening constant (S) and quantum defect (D) Groups VIII Elements He Ne Ar Kr Xe Rn

N 1 2 3 4 5 6

Z 2 10 18 36 54 86

Present work Odd and even Whole set sets separately D S D S 0.4363 1.0000 0.3968 1.0000 1.0879 7.4802 1.0188 7.4802 2.0892 13.673 2.0102 13.673 2.9265 29.650 2.8299 29.650 3.8229 45.450 3.7128 45.450 4.7179 75.097 4.5946 75.097

13

Slater values(9) D 0.0 0.0 0.0 0.3 1.0 1.8

S 0.30 4.15 11.25 27.75 45.75 77.75

Clementi values(10) S 0.312 4.242 11.236 26.231 -------------

Present work Whole group ZEFF 1.0000 2.5198 4.3270 6.3500 8.5500 10.903

NEFF 0.5637 0.9121 0.9108 1.0735 1.1771 1.2821