Hydrogen’s ionization energies and shell transitions were possible to calculate with the wave equations after knowing the distance between the electron and nucleus (n), and the amplitude difference for destructive wave interference between one proton and one electron (δ). As atoms become more complex with additional protons and electrons, the orbitals (distance) and amplitude factor begin to change and the equations become more complex. If the amplitude factor and orbital distances are known, the wave equations can be applied to any atomic element.
Elements from helium to calcium can be calculated for the first orbital (1s) because the amplitude factor can be calculated and the orbital distance can be approximated. Even though various protons in a nucleus have an affect on orbitals, elements up to calcium have a 1s orbital distance similar to hydrogen. After calcium, the electron filling sequence goes from 4s to 3d, complicating the estimated orbital distances, diverging even further from hydrogen’s distance. Thus, the calculations in this section are applied as a general approximation for elements from helium (2 protons) to calcium (20 protons), but they are using the orbital distances calculated for hydrogen.
As illustrated above, the distance in wavelengths (n) to the wave cancellation gaps that allow for electron orbitals will change based on the structure of the nucleus (proton and neutron arrangement). Hydrogen’s distances for s orbital shells can be used as an approximation for atoms through calcium.
The amplitude factor is the other variable for consideration in the wave equations. For ionized atoms that only have one electron in the 1s orbital, it is easy to calculate the amplitude factor. The one electron is only affected by protons in the nucleus. However, it is more common for atoms greater than hydrogen to have multiple electrons in the atom. These electrons are also producing waves that will affect the amplitude factor of other electrons and must be considered (pictured in red above). This changes the amplitude factor of the transitioning electron (pictured in black above). The amplitude factors have been calculated for a single electron being ionized from the 1s shell, for an ionized atom with two electrons, and for an electron in the 1s shell that is affected by all other electrons in a neutral atom (e.g. 20 electrons for calcium).
Evidence
This section details the use of the amplitude factor for the 1s orbital. In this section, it is assumed that the 1s orbital distance in wavelengths (n) is the same as hydrogen, which appears to be a good approximation, but it is not exact. Specifically, n=187,789 is used as the orbital distance in the equations.
Amplitude factor is dependent on a number of variables including the number of protons in the nucleus and other electrons within the atom. The easiest calculation is an ionized atom that has only one electron in the 1s1 orbital. When this only electron is removed, the atom is fully ionized. It has no electrons remaining in orbit. This is referred to here as 1s(0). An asterik (*) denotes the atom is ionized.
1s(0)* Orbital (remove last electron from 1s1 orbital)
The calculations are straightforward for an ionized atom that has only one electron, and this one electron will be removed from orbital 1s. The Amplitude Factor – 1s(0)* equation is used to determine the amplitude factor of these atoms, where:
δ – Amplitude factor
Z – Number of protons in nucleus
* Ionized Atom
Amplitude Factor – 1s(0)*
Helium (He) with two protons (Z) in the nucleus is shown in the example below.
Helium has an amplitude factor of 4. This can be replicated for any atom, including calcium with 20 protons and thus has an amplitude factor of 202 = 400. With the amplitude factor known, and the orbital distance for 1s (n=187,789), the Transverse Energy Equation and wave constants can be used. K=10 for the electron. ni is the initial position for the electron in the 1s orbital and nf is set to infinity. The equation for calcium with an amplitude factor of 400 is therefore:
The energy required to remove the electron from 1s1 is -8.71 x 10-16 joules. A negative sign in this equation means a photon is absorbed (positive sign means a photon is created). Data from experimental evidence is listed in MJ per mole, so joules can be converted using the following:
The transverse wave energy for an electron (K=10) in a calcium atom (Z=20) at the 1s orbital distance (n=187,789) is –524.5 MJ per Mole, matching experimental data. This process was repeated for the first 20 elements and placed into the table below in the –MJ/Mol – Calculated column in red. It is compared against energy data from experiments in the far right column (MJ/Mol – Measured).
Note that experimental data from the reference on Wikipedia is listed in KJ per mole, which has been converted to MJ per mole in the table. The experimental values used are the ionization energies for a fully ionized atom (values at the far right column of each element in the ionization table on Wikipedia).
Note that the ionization energies exactly match the first 13 elements, then start to deviate slightly as elements become larger. This is an indication that their 1s orbital distance is not the same as hydrogen. Hydrogen appears to be a good approximation, but the exact orbital distances will eventually need to be worked out for each nucleus configuration to get an exact match of ionization energies for larger elements.
1s(1)* Orbital (remove first electron in 1s2 from an ionized atom)
The calculations for an ionized atom that has only two electrons, both in shell 1s, need to consider the amplitude effect of other electrons, not just protons in the nucleus. The Amplitude Factor – 1s(1)* equation is used to determine the amplitude factor. Note that in addition to the number of protons (Z), amplitude factor is also dependent on the number of electrons remaining in 1s (N1e = number of electrons in 1s). In this case, N1e is 2 electrons.
N1e – Number of electrons in 1s shell
* Ionized Atom
Amplitude Factor – 1s(1)*
An example calculation is heavily ionized neon with 10 protons and 2 electrons in the 1s shell (N1e):
Similar to 1s(0)*, the ionization energy can be calculated using the Transverse Energy Interaction equation. The calculations are the same with the exception of amplitude factor. The same process has been used to calculate the required energy, and conversion to MJ per mole and placed into the following table:
The ionization energy calculations (red) in MJ/mol are nearly an identical match with experimental data. Note that experimental data on Wikipedia is listed in KJ/mol and converted to MJ/mol in these calculations. In the Wikipedia table, the energies to remove the first electron in the 1s shell is the second value from the right column for each of the elements (atoms with two electrons).
Once again, deviations between the calculated and measured values becomes larger as the nucleus grows in size (protons). It is very possible that the amplitude factor is correct using these equations, but orbital distance (n) needs to be revised in the equations once the cancellation points have been determined, like they have been for hydrogen.
1s(1) Orbital (removing the first electron in 1s2 from a neutral atom)
A similar process was used to compare the results of ionization of the first electron in the 1s orbital, including electrons that are in outer shells. Although the equation is the same as above, a different amplitude factor is used. This is because the electron being removed is affected by other electrons in the atom. The ionization energy for Orbital 1s(1) requires an amplitude factor that takes into consideration not only the protons (Z) in the nucleus, but also the surrounding electrons in outer orbitals.
The equation for Amplitude Factor 1s(1) requires the number of protons (Z) and electrons for each shell where N1e = 1s shells, N2e = 2s, 2p shells, N3e = 3s, 3p shells, N4e = 4s, 4p shells. Note the denominators are 2, 8, 8, 8 representing the number of electrons for these shells.
Amplitude Factor – 1s (1)
An example calculation is calcium with 20 protons, 2 electrons in N1, 8 electrons in N2 and N3 each, and 2 electrons in N4:
Calcium, when it was heavily ionized and only had one electron, only had destructive wave interference with 20 protons in the nucleus and its amplitude factor was 400. Here, in this calculation, it also has the consideration of other electrons in the atom that add to this wave interference. In this case, the amplitude factor for calcium is 296.6.
A comparison of the amplitude factors for 1s(0)*, 1s(1)* and 1s(1) is provided below. Elements from hydrogen to calcium have been calculated using the same equations as above.
Again, knowing the amplitude factor and orbital distance, the ionization energy can be calculated using the Transverse Energy Interaction equation. The same process has been used to calculate the required energy, and conversion to MJ per mole and placed into the following table:
The ionization energy calculations (red) in MJ/mol are nearly an identical match with experimental data (last column in italics) in the table above. As seen in the other calculations, as elements become larger, there is a slight deviation, likely due to the approximation of the 1s orbital distance using the value for hydrogen.
The wave equations are thus shown to not only exactly match hydrogen energies and wavelengths, but can be further used to calculate larger elements through calcium, at least the first orbital shell (1s). With further modeling to determine exact orbital distances based on wave cancellations, these ionization energies will be refined and will likely be exact matches.