Question

State Bohr’s postulates and derive the expression for the radius of the \( n^{\text{th}} \) orbit.

Hint:

Important formulas: - Angular momentum: \( mvr = \frac{nh}{2\pi} \) - Orbit radius: \( r_n = n^2 a_0 \) - Bohr radius: \( a_0 = 0.529 \, \text{Å} \) Radius increases rapidly with quantum number \( n \).

Answer 1

Concept: Bohr proposed a model for hydrogen atom to explain atomic stability and discrete spectral lines. He introduced quantized orbits where electrons revolve around the nucleus without radiating energy.
Bohr’s Postulates:

• 
  Stationary Orbits: Electrons revolve in certain stable circular orbits without emitting radiation.
• 
  Quantization of Angular Momentum: Angular momentum of electron is quantized: \[ mvr = \frac{nh}{2\pi}, \quad n = 1,2,3,\dots \]
• 
  Emissiobsorption of Radiation: Radiation is emitted or absorbed only when an electron jumps between orbits: \[ h\nu = E_2 - E_1 \]

Derivation of Radius of \( n^{\text{th}} \) Orbit Consider hydrogen atom:

• Electron mass = \( m \)
• Electron charge = \( e \)
• Orbit radius = \( r \)

Step 1: Electrostatic Force = Centripetal Force Attraction between nucleus and electron provides centripetal force. Coulomb force: \[ F = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{r^2} \] Centripetal force: \[ F = \frac{mv^2}{r} \] Equating: \[ \frac{mv^2}{r} = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{r^2} \] \[ mv^2 = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{r} \quad \cdots (1) \]
Step 2: Quantization of Angular Momentum From Bohr’s second postulate: \[ mvr = \frac{nh}{2\pi} \] \[ v = \frac{nh}{2\pi mr} \quad \cdots (2) \]
Step 3: Substitute velocity in equation (1) Substitute \( v \) from (2) into (1): \[ m \left(\frac{nh}{2\pi mr}\right)^2 = \frac{1}{4\pi \varepsilon_0} \frac{e^2}{r} \] \[ \frac{n^2 h^2}{4\pi^2 mr^2} = \frac{e^2}{4\pi \varepsilon_0 r} \]
Step 4: Solve for \( r \) Multiply both sides by \( 4\pi^2 mr^2 \): \[ n^2 h^2 = \frac{e^2}{4\pi \varepsilon_0 r} \cdot 4\pi^2 mr^2 \] \[ n^2 h^2 = \frac{\pi m e^2 r}{\varepsilon_0} \] \[ r = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2} \] \[ \boxed{r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2}} \]
Bohr Radius: For \( n = 1 \): \[ a_0 = \frac{h^2 \varepsilon_0}{\pi m e^2} \] Numerically: \[ a_0 = 0.529 \, \text{Å} \] Thus: \[ \boxed{r_n = n^2 a_0} \]
Key Results:

• Radius increases as \( n^2 \)
• Higher orbits are farther from nucleus
• Explains hydrogen spectrum

Limitations of Bohr Model:

• Works only for hydrogen-like atoms
• Cannot explain fine spectral lines
• Ignores wave nature of electron