Correct Answer - Option 4 :
\(\frac h {2\pi}\)
Concept:
Bohr Model of Hydrogen Atom
-
Bohr proposed a model for the hydrogen atom in which a single electron revolves around a stationary nucleus of positive charge Ze (called hydrogen-like atom).
- A moving electron in its circular orbit behaves like a particle-wave.
- As a result, standing waves are produced and the total distance traveled by a wave is an integral number of wavelengths.
This gives the relation: \(2πr_k = \frac{kh}{mv_k}\) = kλ ----(1)
Where rk is the radius of the kth orbit, and λ is the wavelength.
Also The de Broglie wavelength is given by:
λ = \(\frac{h}{mv_k}\) ----(2)
Where vk is the velocity of the electron in kth orbit.
From (1) and (2) we get:
\(2πr_k = \frac{kh}{mv_k}\)
⇒\(mv_kr_k =\frac{kh}{2π}\) ⇒\(m\omega_k =\frac{kh}{2π}\)
∴ Angular momentum = \(m\omega_k =\frac{kh}{2π}\)
Explanation:
So, the angular momentum is given by
\(L =m\omega_k = mvr =\frac{kh}{2π}\)
k and h are constant
So, we can say that
L ∝ h / 2 π
So, \(\frac h {2\pi}\) is the correct option.
