Atoms - Bohr’s Model of Hydrogen Atom

Bohr's First Postulate: Electrons revolve in certain stable, non-radiating circular orbits called 'stationary states'. The centripetal force is provided by the electrostatic attraction: $\frac{mv^2}{r} = \frac{1}{4\pi\epsilon_0} \frac{Ze^2}{r^2}$

Bohr's Second Postulate (Quantization Condition): An electron can revolve only in those orbits for which its orbital angular momentum ($L$) is an integral multiple of $\frac{h}{2\pi}$. Thus, $L = mvr = n\hbar = \frac{nh}{2\pi}$, where $n = 1, 2, 3, ...$ is the principal quantum number.

Bohr's Third Postulate (Frequency Condition): An electron might make a transition from a higher energy state ($E_i$) to a lower energy state ($E_f$) by emitting a photon of energy $h\nu = E_i - E_f$.

The radius of the $n^{th}$ orbit is proportional to $n^2$: $r_n \propto n^2$. For a Hydrogen atom ($Z=1$), the first Bohr radius is $a_0 \approx 0.529 \text{ \AA}$.

The velocity of an electron in the $n^{th}$ orbit is inversely proportional to $n$: $v_n \propto \frac{1}{n}$.

Energy of the electron: The total energy $E_n$ is negative, indicating that the electron is bound to the nucleus. Relationship between energies: $K.E. = -E_n$ and $P.E. = 2E_n$.

Hydrogen Spectrum: When an electron jumps from $n_2$ to $n_1$, the wavelength $\lambda$ of emitted radiation is given by the Rydberg formula: $\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$, where $R$ is the Rydberg constant ($1.097 \times 10^7 \text{ m}^{-1}$).