Question

Derive an expression for radius of n^th Bohr orbit.

Answer 1

Let, m_e = Mass of electron

e = Charge on electron

r_n = Radius of n^th Bohr’s orbit

v_n = Linear velocity of electron in n^th orbit

Z = Number of electrons in an atom

n = Principal quantum number

From Bohr’s first postulate,

Coulomb’s force (F_e) = Centripetal force (F_cp)

∴ `(Z e^2)/(4 pi epsilon_0 r_n^2) = (m_e V_n^2)/r_n`

∴ `V_n^2 = (Z e^2)/(4 pi epsilon_0 r_nm_e)`    ...(i)

From Bohr’s second postulate,

m_er_nv_n = `(n h)/(2 pi)`

∴ V_n = `(n h)/(2 pi m_e r_n)`

∴ `V_n^2 = (n^2 h^2)/(4 pi^2 m_e^2 r_n^2`    ...[Squaring both sides]    ...(ii)

From (i) and (ii) we get,

`(Z e^2)/(4 pi epsilon_0 r_n m_e) = (n^2 h^2)/(4 pi^2 m_e^2 r_n^2)`

∴ r_n = `(n^2 h^2 epsilon_0)/(pi m_e Z e^2)`

= `(epsilon_0 h^2)/(pi m_e Z e^2) n^2`

This is the required expression for the radius of the n^th orbital.