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The Ionization Energies of
Electrons in Subshells of Atoms

For the Bohr model extended to hydrogen-like atoms the energy E of the outermost electron is given by the formula

E = −mZ²K²/(2h²n²)

where m is the mass of an electron, Z is the net positive charge experienced by the electron, K is the constant for the electrostatic force between two charges, h is Planck's constant divided by 2π, and n is an integer called the principal quantum number.

The net charge experienced by the electron Z is the positive charge of the nucleus less the shielding by the inner electrons. The charge shielded by the inner electrons may be equal to the number of inner electrons, but it could be different as a result of the deviations from sphericity of the charge distributions of the inner electrons.

The model can be extended beyond the hydrogen-like atoms by allowing some portion of the charge of the electrons in the same shell shield an electron. That proportion or shielding factor should be in the vicinity of one half. For more on this see Ionization.

The potential energy of an electron separated from a net charge of Z by a distance s is

E = −KZ/s

In an atom with the nucleus being so much more massive than an electron the separation distance s and the orbit radius r are essentially the same. Thus

−mZ²K²/(2h²n²) = −KZ/r
which reduces to
mZK/(2h²n²) = 1/r
and hence
r = 2h²n²/(mZK)

If the orbit radius for electrons depends only upon the principal quantum number n then any electrons with the same n would be in the same shell.

There are however three other quantum numbers for an electron state. There is its magnetic quantum number k which can range from 0 to (n-1). The orbital quantum number l can range from −k to + k. Finally there is the spin quantum number s which can have the values ±½. This means there are 2(2k+1) electron states with the same magnetic quantum number k. The electron states having the same magnetic quantum number k can be characterized as belonging to the k subshell for the shell corresponding to the principal quantum number n.

The Subshells and Their Structure

For each principal quantum number n there is a set of subshells containing {2, 6, 10, …} states. The value of the magnetic quantum number k, in principal could go up to (n-1) but in practice it often does not do so. Based upon the atomic numbers for the noble gases the capacities for the electron shells are {2, 8, 8, 18, 18, 32}. This means that the value of k for the first shell is 0; i.e., n=1 minus 1. For the second shell k equals 1; i.e., n=2 minus 1. But for the third shell k is equal 1 also. This is because the energy for n=3 and k=2 is greater than the energy for n=4 and k=0. Thus the fourth shell starts filling after the third shell has eight electrons. Likewise for the fourth shell the maximum k is 2 rather than 3. And in the fifth shell the maximum k is 3 rather than 4.

For each shell, the subshell corresponding to a value of k has 2(2k+1) states and the same spatial structure in each shell.

In the first shell there is only the subshell corresponding to the magnetic quantum number being zero. These are depicted in the above diagram as the circles colored violet. In the second shell there are two subshells, depicted in the above diagram as violet and green, corresponding to the magnetic quantum numbers of 0 and 1. In the third shell there are three subshells corresponding to the magnetic quantum numbers of 0, 1 and 2 and depicted as violet, green and yellow circles. In the fourth shell there are the subshells for the magnetic quantum number ranging from 0 to 3, with the subshell for 3 depicted as a red circle.

The Sommerfeld Model for the
Electron States in an Atom

Arnold Sommerfeld developed a model of the atom in which electrons may have elliptical orbits. In his model there are two quantum numbers, nθ and nr. The quantum number nθ is called the azimuthal quantum number and represents the number of angular momentum units of h for the state. The other quantum nr quantizes the degree of eccentricity of the electron orbit. It can take on any nonnegative integral value; i.e.,

nr = 0, 1, 2, …

The sum of the two quantum numbers, nθ+nr, is defined as the principal quantum number n. The azimthal quantum number nθ can then take on the integral values 1, 2, …, n.

The energy E of an electron is then given by

E = −μZ²K²/(2h²n²)[1 + (α²Z²/n)(1/nθ − (3/4)(1/n)]

where μ is the reduced mass of the system and α is the fine structure constant, which is approximately 1/137.

The maximum deviation from the Bohr model occurs for n=1, nθ=1. For Z=1 the proportional deviation is then

[1 + (1/137)²(1/4) = 1 + 1.33×10-5

For Z=10 this becomes

[1 + (1/137)²(10²)(1/4) = 1 + 1.33×10-3

The Dependence of the Energy of an Electron State
Upon the Magnetic Quantum Number

Rather than approach this question through the use of theory, an empirical approach will be used. Theories are only worthwhile if they are empirically validated. Later theory will be used to refine the analysis.

The Data Set

The regression of the ionization energies on these variables gives

E = 207.02 + 29.3129P − 92.36185n − 12.84919k − 23.89817IE − 26.95341SS
which can be rearranged to
E = 207.02 + 29.3129(P − 0.815IE − 0.920SS) − 92.36185n − 12.84919k

The quantity within the parentheses is in the nature of a net charge, the positive charge of the nucleus less the shielding by electrons in the inner shells and in the same shell.

The coefficient of determination (R²) for this equation is 0.629. The signs of all the coefficients are what would be expected. What is a surprise is the plot of the actual ionization energies versus the estimates of the ionization energies derived from the above regression equation.

What the above graph indicates is that there are quadratic relationships between the ionization energies and the variables used in the regression. This is not a surprise. As was previously stated the Bohr model for hydrogen-like atoms has the following relationship with the explanatory variables.

E = −mZ²K²/(2h²n²)

where m is the mass of an electron, Z is the net positive charge experienced by the electron, K is the constant for the electrostatic force between two charges, h is Planck's constant divided by 2π, and n is the principal quantum number.

In an attempt to capture the quadratic relationships a new variable Z was created to represent the net positive charge experienced by an electron. The preliminary definition of Z is

Z = P − IE − ½SS

The inclusion of Z and Z² in the regression produces the following result.

E = 172.49902 + 45.40487Z − -1.15657Z² − 99.48007n
− 23.81374k + 6.122622591IE − -9.74682SS

Now the coefficients for the number of inner shell electrons IE and the number of same shell electrons SS reflect the deviation of the true shielding rations from the ones used in defining Z.

Again the plot of the actual values versus the regression estimates of the values indicates a quadratic relationship.

The coefficient of determination for this equation is 0.64.

Varying the shielding ratios can improve the statistical fit (R²) but unless the perfect values of the shielding ratios are used in defining Z there is a quadratic relationship between the data and the regression estimates. For example for a shielding ratio for inner shell electrons of 0.9 and for same shell electrons of 0.6 the value of R² increases to 0.658, but the quadratic dependence is still there.

When both shielding ratios are 0.7 the value of R² is the higher value of 0.7065, but the quadratic dependence is still there

The Dependence of Ionization Energy of
Electron States on the Shell Number

The negative signs for the coefficients for n in the regression indicates an inverse relationship between shell number and ionization energy. The Bohr model makes the relationship for explicit; i.e., the ionization is inversely proportional to the square of the shell number. The Bohr relationship can be put into the form

E = −m(Z/n)²K²/(2h²)

This suggests a regression equation involving (Z/n) and (Z/n)² in the place of Z and Z². Such a regression produces the following result:

E = 9.67930 + 1.99041(Z/n) + 12.87402(Z/n)² − 0.047683IE − 3.98931SS − 0.87273k

Not only is the coefficient of determination greatly improved, to 0.972, but the quadratic dependence in the graph of actual ionization energies versus the regression estimates of those energies has disappeared.

However the regression coefficient for the magnetic quantum number k is not significantly different from zero at the 95 percent level of confidence. (Its t-ratio is only 0.56.) However, from the Sommerfeld model it is known that the effect of another quantum numbers beside the principal one is very small. The effect could be there but of insignificant magnitude compared with the intrinsic uncertainties of the ionization energies. The coefficient for the number of Inner Shell Electrons is also not significantly different from zero but that only indicates that the effect of the inner shell electrons is adequately captured in the definition of Z.

The above equation can be rearranged to the form

E = 9.67332 + 12.87402[(Z/n)² + 0.07730]² − 0.047683IE − 3.98931SS − 0.87273k

Conclusions

The most significant conclusion of the analysis is that the relevant variable determining the ionization energy of an electron state is the ratio of the net charge experienced by the state, Z, to the shell number n. The ionization energy is a quadratic function of (Z/n).

The shielding ratio for the inner shell electrons is less than unity and the shielding ratio for the electrons in the same shell is greater than one half.

When the dependence of ionization energy on (Z/n) is taken into account there appears to be no significant effect of the magnetic quantum number k on ionization energy. The magnetic quantum number is just the subshell number less one.

(To be continued.)


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