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A Generalization of the
Bohr Equation for the
Ionization Energy of
the Outer Electron of
Hydrogen-like Atoms

This is a generalization of the equation developed by Niels Bohr for the ionization energy of an electron in an atom in which there is only a single electron in the outer shell. This generalization applies as well to a nucleus consisting of a core with equal numbers of protons and neutrons surrounded by halo neutrons in orbits around it.

The Bohr Equation

The Bohr model of a hydrogen-like atom or ion predicts that the potential energy V of an outer shell electron is given by

V = −RZ²/n²

where Z is the number of protons in the nucleus less the number of electrons in the inner shells. The parameter n is the principal quantum number and R is a constant equal to approximately 13.6 electron volts (eV). This formula is the result of the potential energy V being given by equal to

V = −α Ze²/s

where e is the charge of an electron, Ze is the net charge experienced by the electron, s is the separation distance of the centers of the nucleus and the electron and α is a constant. Thus at infinite separation the potential energy of the pair is zero.

It is assumed that the nucleus is so massive compared to the electron that separation distance is the same as the orbit radius of the electron. Let rn be the orbit radius when the principal quantum number is n. The orbit radius is given by

rn = n²h/(Zmee²)

where h is Planck's constant divided by 2π and me is the mass of the electron.

Shell Structure

Electrons in atoms are organized in shells whose capacities are equal to 2m², where m is an integer. Thus there can be at most 2 electrons in the first shell, 8 in the second shell and 8 in the third shell and 18 in each of the fourth and fifth shells.

Shielding by Electrons

The term hydrogen-like ion means that there is but one electron in a shell, the inner shells being completely filled. The value of Z in the above formulas was considered to be the number of protons p in the nucleus less the number of electrons in inner shells. ε0; i.e., Z=p−ε0. Thus ε0 positive charges of the nucleus are said to be shielded by the ε0 electrons which are closer to the center of the atom than the electron being considered. On the other hand, there is no shielding by any electrons which are farther from the center of the atom. Shielding is an inapproppriate term; cancelling is more accurate.

Shielding by Electrons
in the Same Shell

The Bohr model may be generalized a bit by taking into account shielding by electrons in the same shell. When a charge is distributed uniformly on a spherical surface it has the effect on another charge outside of the spherical equal to what that same charge would have concentrated at the center of the sphere. The effect on a charge within the sphere is zero. The effect on a charge located on the same sphere is equal to what half the charge would have located at the center of the sphere. Thus if the number of electrons in the same shell is denoted as ε1 then

Z = p − ε0 − ½(ε1-1)

Thus the energy required to remove an electron from a shell should decrease with the number of electrons in that shell. This is due to the shielding of some of the positive charge of the nucleus by electrons in the same shell.

A shell may be composed of subshells so the above terms need to be modified; i.e.,

ε1 = number of electrons in the same subshell
ε0 = number of electrons in the inner shells or subshells

A Generalization

Let one particle of generic charge q be in a circular orbit around the center of a particle of generic charge Q. The case of just the two particles will be considered first. The force F on each of the particles is assumed to be of the form

F = HqQ/r²

where H is a constant and r is the orbit radius.

The potential energy V of the system is then

V(r) = −HqQ/r

The orbit radius is given by a balance of the force of attraction and the centripetal acceleration of particle of charge q; i.e.,

HqQ/r² = mv²/r

where m is the mass of the charge q particle and v is its tangential velocity.

Thus

r = HqQ/(mv²)

Hence

V(r) = −HqQ/r = −HqQ/(HqQ/(mv²) = − mv²

The kinetic energy K is given by ½mv². Thus the changes in energy when the pair goes from infinite separation with V=0 and K=0 to a separation of r with V=−mv² and K=½mv² are ΔK=−½ΔV. Thus one half of the loss of potential energy in the formation of the atom goes into the increase in kinetic energy of the electron. The other half goes into the formation of a vibration in the field of the charges.

The angular momentum pθ of the charge q particle is given by

pθ = mvr = mv(HqQ/(mv²)) = HqQ/v

Angular momentum is quantized; i.e.,

pθ = lg

where l is a positive integer, called the principal quantum number and g is the field constant for the force. For the electromagnetic field it is Planck's constant divided by 2π. (For more on this matter see Field constants.)

Therefore

HqQ/v = lg
and hence
v = HqQ/(lg)

Thus potential energy V is given by

V = −mv² = −m(HqQ/(lg))²
and therefore
V = −m(H/g)²q²(Q/l

The net charge experienced by a q particle, X, may be a core charge Q less the shielding by particles in inner shells and in the same subshell. Let ε0 the number of partticles in inner shells and subshells relative to the q particle and ε1 in the same subshell. Then

X = Q − qε0 −½q(ε1−1)

Thus the energy E required to remove a particle of generic charge q from structure in which it is experiencing a generic charge of X is

E = m(H/g)²q²(X/l

This may be expressed as

E = R(X/l

where
R = mq²(H/g

Conclusions

The Bohr equation exactly generalizes for any generic charge whose force is dependent upon inverse distance squared.


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