Atoms
📐 Formula & Cheat Sheet (English)
Quick Revision Notes: Class 12 Physics
Chapter: Atoms
### 1. Introduction & Thomson's Model
- Atom: The basic building block of all matter, electrically neutral consisting of positively charged protons and negatively charged electrons.
- Thomson's Model (Plum Pudding / Watermelon Model): An atom is a sphere of positive charge in which electrons are embedded like seeds in a watermelon. It successfully explained the electrical neutrality of an atom, but failed to explain the results of alpha-particle scattering experiments.
### 2. Rutherford’s Alpha-Particle Scattering Experiment
- Setup: Fast-moving $\alpha$-particles ($\text{He}^{2+}$ nuclei) from a radioactive source were directed at a very thin gold foil.
- Observations:
- Most $\alpha$-particles passed straight through the foil without deflection.
- A small fraction of $\alpha$-particles was deflected at small angles.
- A very few ($\approx 1$ in $8000$) were deflected back by $180^\circ$.
- Conclusions:
- Most of the space inside an atom is empty.
- The entire positive charge and almost all the mass of the atom are concentrated in a tiny central core called the nucleus.
Distance of Closest Approach ($r_0$)
When an $\alpha$-particle is projected towards the nucleus, it momentarily stops at distance $r_0$ before reversing its direction. Its initial kinetic energy is converted entirely into electrostatic potential energy. $$\frac{1}{2} m v^2 = \frac{1}{4\pi\varepsilon_0} \frac{(2e)(Ze)}{r_0}$$ $$r_0 = \frac{1}{4\pi\varepsilon_0} \frac{2Ze^2}{K}$$ (where $K = \frac{1}{2}mv^2$ is the initial kinetic energy of the $\alpha$-particle)
Impact Parameter ($b$)
It is the perpendicular distance of the velocity vector of the $\alpha$-particle from the central line of the nucleus during its collision. $$b = \frac{1}{4\pi\varepsilon_0} \frac{Z e^2 \cot(\theta/2)}{K}$$ (where $\theta$ is the scattering angle)
### 3. Bohr’s Atomic Model (Postulates)
Bohr modified Rutherford's model using quantum concepts for hydrogen and hydrogen-like atoms ($Z = 1, 2, 3...$).
- Stationary Orbits: Electrons revolve around the nucleus only in certain stable, non-radiating circular orbits called stationary orbits where total energy remains constant.
- Quantization of Angular Momentum: Electrons revolve only in those orbits where their orbital angular momentum ($L$) is an integral multiple of $\frac{h}{2\pi}$. $$L = mvr = \frac{nh}{2\pi} \quad (n = 1, 2, 3, ...)$$ (where $n$ is the principal quantum number)
- Frequency Condition (Transition): An electron transitions from a higher energy state ($E_2$) to a lower energy state ($E_1$) by emitting a photon of energy: $$h\nu = E_2 - E_1$$
### 4. Key Radii, Velocities, and Energies in Bohr's Model (Hydrogen Atom)
For an orbit of principal quantum number $n$:
-
Radius of the $n$-th Orbit ($r_n$): $$r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2 Z} = 0.529 \frac{n^2}{Z} , \text{Å}$$ (For hydrogen, $Z=1$, $r_n \propto n^2$)
-
Velocity of Electron in the $n$-th Orbit ($v_n$): $$v_n = \frac{e^2}{2\varepsilon_0 n h} = 2.18 \times 10^6 \frac{Z}{n} , \text{m/s}$$ (Note: $v_n \propto \frac{Z}{n}$)
-
Frequency of Revolution ($\nu$): $$\nu = \frac{v}{2\pi r} \propto \frac{Z^2}{n^3}$$
-
Potential Energy ($U$): $$U = -\frac{1}{4\pi\varepsilon_0} \frac{e^2}{r_n} = -2 \text{ (Kinetic Energy)}$$
-
Kinetic Energy ($K$): $$K = \frac{1}{2}mv^2 = \frac{1}{4\pi\varepsilon_0} \frac{e^2}{2r_n} = - \frac{1}{2} (\text{Potential Energy})$$
-
Total Energy ($E_n$): $$E_n = K + U = -\frac{1}{4\pi\varepsilon_0} \frac{e^2}{2r_n}$$ $$E_n = -13.6 \frac{Z^2}{n^2} , \text{eV}$$ (For hydrogen atom, ground state energy $E_1 = -13.6 , \text{eV}$)
### 5. Hydrogen Spectrum (Spectral Series)
When an electron falls from a higher energy level ($n_2$) to a lower energy level ($n_1$), spectral lines are emitted.
- Rydberg Formula for Wavelength ($\lambda$): $$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$$ (where $R$ is the Rydberg Constant, $R \approx 1.097 \times 10^7 , \text{m}^{-1}$)
Classification of Spectral Series:
| Series | Lower State ($n_1$) | Upper State ($n_2$) | Spectral Region |
|---|---|---|---|
| Lyman Series | $1$ | $2, 3, 4, ...$ | Ultraviolet (UV) |
| Balmer Series | $2$ | $3, 4, 5, ...$ | Visible |
| Paschen Series | $3$ | $4, 5, 6, ...$ | Infrared (IR) |
| Brackett Series | $4$ | $5, 6, 7, ...$ | Infrared (IR) |
| Pfund Series | $5$ | $6, 7, 8, ...$ | Far Infrared (FIR) |
- Maximum wavelength ($\lambda_{\max}$): Obtained for transition between consecutive levels ($n_2 = n_1 + 1$).
- Minimum wavelength ($\lambda_{\min}$): Obtained for series limit transition ($n_2 = \infty$).
### 6. Limitations of Bohr’s Model
- It is applicable only to hydrogen-like single-electron systems (e.g., $\text{He}^+, \text{Li}^{2+}$) and fails for multi-electron atoms.
- It cannot explain the relative intensities of spectral lines.
- It does not explain the Zeeman effect (splitting of spectral lines in a magnetic field) and Stark effect (splitting in an electric field).
- It treats electron orbits as planar, whereas actual atoms have 3D wave-mechanical structures.
### Important Constants for Quick Reference
- Planck's Constant ($h$): $6.626 \times 10^{-34} , \text{J}\cdot\text{s}$
- Charge of Electron ($e$): $1.6 \times 10^{-19} , \text{C}$
- Permittivity of Free Space ($\varepsilon_0$): $8.854 \times 10^{-12} , \text{C}^2\text{N}^{-1}\text{m}^{-2}$
- Bohr Radius ($r_1$ for $H$): $0.529 , \text{Å} = 5.29 \times 10^{-11} , \text{m}$
- Ground state energy of H-atom: $-13.6 , \text{eV}$