CBSE 2025, Set 3

Question

Write the mathematical forms of three postulates of Bohr's theory of the hydrogen atom. Using them prove that, for an electron revolving in the $n^{	ext{th}}$ orbit,
(a) the radius of the orbit is proportional to $n^2$, and
(b) the total energy of the atom is proportional to $\left(\frac{1}{n^2}ight)$.

Edvaya · Accepted Answer

The correct answer is Subjective. **Mathematical forms of Bohr's Postulates:** 
 1. Electrons revolve in circular orbits where the centripetal force is provided by the electrostatic force of attraction:
 $$\frac{mv^2}{r} = \frac{1}{4\pi\epsilon_0} \frac{Ze^2}{r^2} \quad 	ext{--- (1)}$$ 
2. Angular momentum is quantized ($L = mvr$): 
 $$mvr = \frac{nh}{2\pi} \quad 	ext{--- (2)}$$
3. Energy of emitted/absorbed radiation corresponds to the difference in energy levels:
 $$h
u = E_f - E_i \quad 	ext{--- (3)}$$

**(a) Proof that $r \propto n^2$:**
From equation (2), find velocity $v$:
$$v = \frac{nh}{2\pi mr}$$
Substitute $v$ into equation (1):
$$m \left( \frac{nh}{2\pi mr} ight)^2 \frac{1}{r} = \frac{1}{4\pi\epsilon_0} \frac{Ze^2}{r^2}$$
$$\frac{m n^2 h^2}{4\pi^2 m^2 r^3} = \frac{Ze^2}{4\pi\epsilon_0 r^2}$$
Solving for $r$:
$$r = \frac{n^2 h^2 \epsilon_0}{\pi m Z e^2}$$
For a given atom, all terms except $n$ are constant, so:
$$\boxed{r \propto n^2}$$

**(b) Proof that $E_n \propto \frac{1}{n^2}$:**
The total energy $E_n$ is the sum of Kinetic Energy ($K$) and Potential Energy ($U$).
$K = \frac{1}{2}mv^2 = \frac{1}{2} \left( \frac{1}{4\pi\epsilon_0} \frac{Ze^2}{r} ight)$ [Using eq 1]
$U = \frac{1}{4\pi\epsilon_0} \frac{(Ze)(-e)}{r} = - \frac{1}{4\pi\epsilon_0} \frac{Ze^2}{r}$

Total Energy $E_n = K + U = - \frac{1}{8\pi\epsilon_0} \frac{Ze^2}{r}$

Substitute the expression for $r$ obtained in part (a):
$$E_n = - \frac{Ze^2}{8\pi\epsilon_0 \left( \frac{n^2 h^2 \epsilon_0}{\pi m Z e^2} ight)}$$
$$E_n = - \frac{m Z^2 e^4}{8 n^2 \epsilon_0^2 h^2}$$
Since $m, Z, e, \epsilon_0, h$ are constants:
$$\boxed{E_n \propto \frac{1}{n^2}}$$

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