MP Board · Class 12 · Physics · AtomsState the postulates of Bohr's model of the hydrogen atom and derive the expression for the radius of the $n$-th orbit of a hydrogen atom using these postulates.

Postulates of Bohr's Model of the Hydrogen Atom

\nNiels Bohr proposed a model for the hydrogen atom in 1913, combining classical physics with early quantum concepts. The model is based on the following three fundamental postulates:

• Nuclear Concept: An atom consists of a central, positively charged nucleus where almost the entire mass of the atom is concentrated. Electrons revolve around this nucleus in circular orbits under the influence of electrostatic force of attraction, similar to planets revolving around the sun.
• Quantization Condition: Electrons can revolve only in those stable, non-radiating circular orbits for which the angular momentum ($L$) of the revolving electron is an integral multiple of $\frac{h}{2\pi}$. Mathematically, this is expressed as $L = mvr = \frac{nh}{2\pi}$, where $n = 1, 2, 3, \dots$ (known as the principal quantum number), $m$ is the mass of the electron, $v$ is the velocity, $r$ is the radius of the orbit, and $h$ is Planck's constant.
• Frequency Condition: An electron does not radiate energy while revolving in its permitted stationary orbits. Energy is emitted or absorbed only when an electron jumps from one allowed higher energy orbit to another lower energy orbit. The frequency ($\nu$) of the emitted or absorbed radiation is given by the relation $h\nu = E_2 - E_1$, where $E_2$ and $E_1$ are the energies of the higher and lower orbits, respectively.

Derivation of the Radius of the $n$-th Orbit

\nLet us consider an electron of mass $m$ and charge $-e$ revolving around a nucleus of atomic number $Z$ (charge $+Ze$) with a velocity $v$ in a circular orbit of radius $r$.

1. Centripetal Force Requirement: The necessary centripetal force for the circular motion of the electron is provided by the electrostatic force of attraction between the positively charged nucleus and the negatively charged electron according to Coulomb's law: $$\frac{mv^2}{r} = \frac{1}{4\pi \varepsilon_0} \frac{(Ze)(e)}{r^2}$$ $$\frac{mv^2}{r} = \frac{1}{4\pi \varepsilon_0} \frac{Ze^2}{r^2} \quad \text{--- (Equation 1)}$$

2. Bohr's Quantization Condition: According to Bohr's second postulate, the angular momentum is quantized: $$mvr = \frac{nh}{2\pi}$$ Solving for velocity $v$: $$v = \frac{nh}{2\pi mr} \quad \text{--- (Equation 2)}$$

3. Substituting Velocity: Substitute the value of $v$ from Equation 2 into Equation 1: $$\frac{m}{r} \left( \frac{nh}{2\pi mr} \right)^2 = \frac{1}{4\pi \varepsilon_0} \frac{Ze^2}{r^2}$$ $$\frac{m}{r} \cdot \frac{n^2 h^2}{4\pi^2 m^2 r^2} = \frac{1}{4\pi \varepsilon_0} \frac{Ze^2}{r^2}$$ $$\frac{n^2 h^2}{4\pi^2 m r^3} = \frac{1}{4\pi \varepsilon_0} \frac{Ze^2}{r^2}$$

4. Solving for Radius ($r$): Rearranging the terms to isolate $r$: $$r = \frac{n^2 h^2 \cdot 4\pi \varepsilon_0}{4\pi^2 m Z e^2}$$ $$r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m Z e^2} \quad \text{--- (General expression for radius)}$$ \nFor a hydrogen atom, $Z = 1$. Substituting the standard values of fundamental constants ($h = 6.63 \times 10^{-34} \text{ J s}$, $\varepsilon_0 = 8.854 \times 10^{-12} \text{ C}^2\text{N}^{-1}\text{m}^{-2}$, $m = 9.1 \times 10^{-31} \text{ kg}$, $e = 1.6 \times 10^{-19} \text{ C}$), the radius of the first orbit ($n = 1$) is obtained as $r_1 \approx 0.53 \text{ Å}$ or $5.3 \times 10^{-11} \text{ m}$. Thus, the radius of the $n$-th orbit is directly proportional to the square of the principal quantum number ($r_n \propto n^2$).