Answer to Question #2477 in Atomic and Nuclear Physics for berkan

Question

Evaluate the Rydberg constant R using Bohr theory and show that its value is R = 1.0974x10-1.

Answer 1

The energy of electron in atom
E=(mV²)/2-(Ze²)/(4πε₀ r.) (1)

The balance of forces
(Ze²)/(4πε₀ r² )=(mV²)/r (2)

From here we get
E=-(Ze²)/(8πε₀ r.) (3)

Multiplying (2) by mr^3 and using L=nħ we get
(Ze^2 mr)/(4πε_0 )=(mVr)^2=n^2 ħ^2

And finally
E=-(Ze^4 m)/(32 ε_0^2 π^2 n^2 ħ^2.)

Then
hν=(Ze⁴ m)/(32 ε_0² π² ħ² )(1/(n₁² )-1/(n₂² ))

but as we know from atomic physics

hν=hcR(1/(n₁² )-1/(n₂² ))

R = (Ze⁴ m)/(32ε₀² π² ħ³ c) = 1.0974