The semiclassical Bohr Model of the hydrogen atom assumes that an electron is in a classical circular orbit moving at a speed such that its orbital angular momentum is nh = nh / 2π , where n is identified as the principal quantum number, a positive integer. In the following parts, give your answers in terms of fundamental constants (e, m, ε0, and h) and n.
a. Using Newton's Second Law and the definition of angular momentum, derive the energy of the hydrogen atom in terms of fundamental constants and n.
b. Derive the radius of the electron's orbit in state n in terms of fundamental constants and n. c. When the electron is in state n, with what frequency f does it orbit the nucleus? That is, if it takes a time T to complete one orbit, the orbital frequency is f = 1/T.

a)   E = - (2π² k m e²/h²)   1 / n²

b)  r = n² h² / (4π² m k e²)

c)   f = - (2π² k m e² / h³)     1 / n²

Explanation:

a) It is asked to find the energy of the Hydrogen atom, in this case the electron is subjected to an electrical force by the nucleus that has a positively charged proton, let's write Newton's second law

        F = m a

the electric force is

        F = k qq / r²

the acceleration is centripetal

        a = v² / r

we substitute

        k q² / r² = m v² / r

        v² = k q² / r m         (1)

now we can use the energy of the electron

        E = K + U = ½ m v² + k q² / r

        E = ½ m (k q² / r m) - k q² / r

        E = - ½ q² / r

the charge q is the charge of the electron q = e

as the energy is negative it indicates that the system is stably linked

now let's use the angular momentum

      L = n h / 2π

the angular momentum of an electron is

       L = p r = m v r

we substitute

       m v r = n h / 2π

        v = n h / 2π      1/mr        (2)

we substitute the equation 2 in 1

        n² h² / (4π² m² r²) = k e² / (m r)

        n² h² / (4π² m r) = k e²

         r = n² h² / (4π² m k e²)

we substitute in the energy equation

         E = - ½ e² (4π² m k e²) / h² n²

         E = - (2π² k m e²/h²)   1 / n²

b) r = n² h² / (4π² m k e²)

c) to find the frequency we use the Planck equation

      E = h f

      f = E / h

       f = - (2π² k m e² / h³)     1 / n²