The revision on Bohr's theory of hydrogen atom provides an acceptable explanation in classical way for the non-existence of hydrogen negative ion and for the limiting reachability of the attracted electron towards the nucleus.
H- ion (negative hydrogen ion) is not existing- Why ?
The first (1s) orbit can accommodate two electrons , but in the hydrogen atom its first orbit cann't be filled with two electrons . Two electrons can stay in the 1s orbit only when one more proton is present in the nucleus . It is confirmed with the existence of Helium atom.
Fig. Negative Hydrogen ion
If one more electron is allowed in the first orbit of hydrogen , its total energy becomes positive and hence it moves away from the nucleus. and the system transforms into a less potential stable state
The Kinetic energy of one of the electrons = (1/2) mv^2 and for both the electrons which are identical in all respect K.E = mv^2.
Electrostatic force (attraction) due to the nucleus = e^2/Kr^2 where K = 4πεo . Since both the electrons are in the same orbit ,there must be mutual interaction between them, that is why they attain stability by staying exactly diametrically opposite. Electrostatic force (repulsion) due to the presence of second electron is e^2/4Kr^2. The resultant force experienced by the electron is e^2 /Kr^2 - e^2/4Kr^2 = (3/4) e^2/Kr^2 .It is counter-balanced by centrifugal force mv^2/r .K.E of both electrons is (3/4) e^2 /Kr .
The potential energy of the first electron due to the presence of the nucleus only is - e^2 /Kr. When the second electron moves towards the neutral hydrogen atom , its electrostatic potential energy is zero until it reaches the electronic orbit of the first electron. Due to additional electron -electron interaction the potential energy is increased by e^2/2Kr . The total energy associated with the system = (3/4) e^2/Kr - e^2/Kr + e^2 /2Kr = (e^2 /4Kr) . Since there is no binding energy, the system gets transformed into normal hydrogen atom by expelling out the additional electron.
Why electron in general cannot reach beyond the innermost orbit?
When an electron is attracted by a nucleus ,it gets accelerated towards the nucleus until it reaches a point where its velocity is exactly equal to the orbital velocity required to keep the electron stable in the orbit
The loss of potential energy = gain in kinetic energy + relativistic increase of mass
Since the gain in kinetic energy of electron is half of the loss of its potential energy, the energy equivalent of relativistic increase of mass of the electron comes from the remaining half of the loss of potential energy.
(1/2) e^2/Krn = (1/2) mn vn^2 or vn^2 = e2/Kmnrn, where rn = n^2ao (1-vn^2 /c^2)^1/2 and mn = mo (1-vn^2/c^2)^-1/2
or vn^2 =(1/n^2) [e^2/ Kmo ao]
The highest value of vn is v1 with n = 1; v1=[e^2/ Kmo ao]
For stability of the electron in the orbit electrostatic force = centrifugal force
e^2/Krn^2 = mn vn^2/rn
or vn^2 = e^2/ Krn mn = (1/n^2) [e^2/ Kmo ao]
When an accelerated electron moves towards the nucleus, usually it will not make a straight line motion, If so,it cannot be stopped abruptly in an allowed orbit The accelerted electron moves along a curved path and ultimately it attains stable orbital motion with uniform velocity.
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