Derive the energy expression for hydrogen atom using Bohr atom model.

Explanation

The energy of an electron in the `n^(th)` orbit
Since the electrostatic force is a conservative force, the potential energy for the `n^(th)` orbit is
`U_(n) = (1)/(4piepsilon_(0))((+Ze)(-e))/(r_(n)) = - 1/(4 piepsilon_(0))(Ze^(2))/(r_(n))`
`= -(1)/(4 epsilon_(0)^(2))(Z^(2)me^(4))/(h^(2)n^(2))(because r_(n) = (epsilon_(0)h^(2))/(pime^(2))(n^(2))/(Z))`
The kinetic energy for the `n^(th)` orbit is `KE_(n) = 1/2 mv_(n)^(2) = (me^(2))/(8epsilon_(0)^(2)h^(2))(Z^(2))/(n^(2))`
This implies that `U_(n) = -2KE_(n)`. Total energy in the `n^(th)` orbit is
`E_(n) = KE_(n) + U_(n) = KE_(n) - 2KE_(n) = - KE_(n)`
`E_(n) = -(me^(4))/(8epsilon_(0)^(2)h^(2))(Z^(2))/(n^(2))`
For hydrogen atom (Z = 1),
`E_(n) = -(me^(4))/(8 epsilon_(0)^(2)h^(2)) (1)/(n^(2))`joule ....(1)
where n stands for principal quantum number. The negative sign in equation (1) indicates that the electron is bound to the nucleus.
Substituting the values of mass and charge of an electron (m and e), permittivity of free space`epsilon_(0)` and Planck.s constant h and expressing in terms of eV, we get
`E_(n) = -13.6(1)/(n^(2))eV`
For the first orbit (ground state ), the total energy of electron is `E_(1) = -13.6 eV`. For the second orbit (first excited state), the total energy of elctron is `E_(2) = -3.4 eV`. For the third orbit(second excited state), the total energy of electron is `E_(3) = -1.51 eV` and so on.
Notice that the energy of the first excited state is greater than the ground state, second excited state is greater than the first excited state and so on. Thus, the orbit which is closet to the nucleus `(r_(1))` has lowest energy (minimum energy compared with other orbits). So, it is often called ground state energy (lowest energy state) . The ground state energy of hydrogen`(-13.6 eV)` is used as a unit of energy called Rydberg(1 Rydberg = -13.6 eV).The negative value of this energy is because of the way the zero of the potential energy is defined. When the electron is taken away to an infinite distance (very far distance) from nucleus both the potential energy and kinetic energy terms vanish and hence the total energy also vanishes.