We know that the electron is revolving in a specified orbit.
We also know that these orbits are called stationery orbits. When the electron
is in the stationary orbit, it neither loose the energy nor gain the energy.
Hence it has a fixed energy. This energy is both in the forms of potential
energy and kinetic energy. We can derive the equations for both potential and
kinetic energies as shown below. By adding potential and kinetic energies we
can get the total energy of the electron. It is the observed by the results
that the total energy and the potential energy are negative values. It is a
symbolic way of expressing that the electron is associated with the nucleus.
It is also noticed that potential energy is twice the
magnitude of total energy. It is observed that the magnitude of total energy
and the kinetic energy are equal.
The expression for the kinetic energy is done taking the
basic equation of kinetic energy and substituting the value of the velocity of
the electron that is derived. Again potential energy is defined basing on the
very definition of electrostatic potential energy. By adding both these
energies we got total energy.
It is observed that all these energies are directly
proportional to Squire of the atomic number and inversely proportional to
Squire of principal quantum number.
Energy levels of electron in different orbits
We know that electrons revolve is in different orbits around
the nucleus. Basing on the orbit each electron will have different energy. As
the number of the orbit is increasing the total energy of the electron also
increases and at the infinite orbit it’ll becomes zero. We can express the
total energy of any of the electron in terms of the basic energy of the
electron in the ground state which is equal to -13.6 electron volt.
Energy of the electron in any other orbit is this numerical
value divided by Squire of the principal quantum number. Basing on the above
result, we can write energies of different orbits not only for the hydrogen
atom, even for the hydrogen like attempts as shown below.
Problem and solution
An electron in a hydrogen atom makes the transition from one
orbit to another orbit. Assume that the Bohr atomic model is valid. It is given
that the time period of the electron in the initial state is eight times of the
final state. What are the possible values of principal quantum numbers?
We have proved that the time period of an electron in a given
orbit is directly proportional to cube of principal quantum number. Taking this
into consideration we can derive a direct relation between principal quantum
numbers. As we know that the principal quantum number is not a fractional
value, basing on that and the relation we can write different sets of values of
principal quantum numbers for the above problem as shown below.
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