When an electric discharge is passed through hydrogen gas, the hydrogen molecules dissociate to produce excited to produce excited hydrogen atoms. These excited atoms emit electromagnetic radiation of discrete frequencies which can be given by the general formula
`vecv=109677[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]`
What points of Bohr's model of an atom can be used to arrive at this formula? Based on these points derive the above formula giving description of each step and each term.

Explanation

The two important points of Bohr's model that can be used to derive the given formula re as follows
(i) Electrons revovle around the nucleus in a circular path of fixed radius and energy. These paths are called orbits. Stationary states or allowed energy states.
(ii) Energy is emitted or abosrbed when an electron moves from higehr stationary state to lower stationary state or from lower stationary state to higher stationary state respectively
Derivation The energy of the electron in the `n^(th)` stationary state is given by the expression.
`E_(n)=-R_(H)((1)/(n^(2))) n=1,2,3`
Where `R_(H)` is called Rydberg cosntant and its vale is `2.18xx10^(-18)J`. The enregy of the lowest state also called the ground state, is
`E_(n)=-2.18xx10^(-18)((1)/(1^(2)))=-2.18xx10^(-18)J`
The energy gap between the two orbits is given by the equation,
`DeltaE=E_(f)-E_(i)`
On combining Eqs. (i) and (ii)
`DeltaE=(-(R_(H))/(n_(i)^(2)))-(-(R_(H))/(n_(i)^(2)))` ltbr. Where `n_(i)` and `n_(f)` stand for initial orbit and final orbit.
`DeltaE=R_(H)[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]=2.18xx10^(-18)J[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]`
Frequency v associated with the absorption and emission of the photon can be calculated as follows
`v=(DeltaE)/(h)=(R_(H))/(h)[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]`
`rArrv=(2.18xx10^(-18)J)/(6.626xx10^(-34)Js)[(1)/(n_(i^(2)))-(1)/(n_(f)^(2))]`
`v=3.29xx10^(15)[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]HZ`
`rArr vecv=(v)/(c)=(329xx10^(15))/(3xx10^(8)ms^(-1))[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]`
`vecv=1.09677xx10^(7)[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]m^(-1)`
`vecv=109677[(1)/(n_(i)^(2))-(1)/(n_(f)^(2))]cm^(-1)`