Key Points

Proposed in 1898, this model describes the atom as a uniformly distributed sphere of positive charge with negatively charged electrons embedded in it, much like seeds in a watermelon. It is also known as the plum pudding model.

This experiment involved bombarding a thin gold foil with alpha-particles. The observation that a few particles were deflected at large angles led to the discovery of a small, dense, positively charged nucleus.

Based on the scattering experiment, this model suggests that an atom consists of a tiny, massive, positively charged nucleus at the center, with electrons revolving around it like planets around the sun.

The impact parameter (b) is the perpendicular distance of the initial velocity vector of an alpha-particle from the nucleus. A small impact parameter results in a large scattering angle, with a head-on collision (b=0) causing a deflection of nearly $180^{\circ}$.

For a head-on collision, the initial kinetic energy of the alpha-particle is converted into electrostatic potential energy. The distance of closest approach (d) is given by $K = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{d}$.

According to classical physics, an accelerating electron must radiate energy and spiral into the nucleus, making the atom unstable. It also failed to explain the discrete line spectra observed for elements like hydrogen.

When excited, rarefied gases emit radiation at specific, discrete wavelengths, forming an emission line spectrum. When white light passes through a gas, it absorbs these same wavelengths, creating an absorption line spectrum.

An electron in an atom can revolve in certain stable orbits, called stationary states, without emitting any radiant energy, contrary to classical electromagnetic theory.

An electron revolves only in those orbits for which its angular momentum (L) is an integral multiple of $h/2\pi$. The condition is given by $L = mvr = n \frac{h}{2\pi}$, where n is the principal quantum number.

An atom emits a photon when an electron transitions from a higher energy state $E_i$ to a lower energy state $E_f$. The frequency of the emitted photon is given by $h\nu = E_i - E_f$.

The radius of the nth stationary orbit is directly proportional to the square of the principal quantum number, $r_n \propto n^2$. The radius of the innermost orbit (n=1) is $5.3 \times 10^{-11} \text{ m}$.

The energy of an electron in the nth orbit is quantized and given by the formula $E_n = -\frac{13.6}{n^2} \text{ eV}$. The negative sign indicates that the electron is bound to the nucleus.

The lowest energy state of an atom (n=1) is called the ground state. States with higher energy (n > 1) are called excited states. For hydrogen, the ground state energy is -13.6 eV.

Ionization energy is the minimum energy required to remove an electron from the ground state to infinity (n = $\infty$). For a hydrogen atom, this energy is $0 - (-13.6 \text{ eV}) = 13.6 \text{ eV}$.

De Broglie explained the quantization of angular momentum by considering the electron as a particle wave. A stable orbit is one where the circumference is an integral multiple of the electron's de Broglie wavelength: $2\pi r_n = n\lambda$.

Bohr's model is only applicable to hydrogenic (single-electron) atoms and cannot be extended to multi-electron atoms. It also fails to explain the relative intensities of spectral lines.