Bohr's Model of Atom: Comprehensive NEET Physics Notes

1. Bohr's Model of Atom

1.1 Introduction to Bohr’s Model

Niels Bohr introduced his model of the atom in 1913, combining Rutherford's nuclear atom concept with Planck's quantum theory. Bohr's model provided a breakthrough in understanding atomic structure and explained how electrons could occupy fixed energy levels without radiating energy and collapsing into the nucleus.

Key Postulate: Electrons revolve around the nucleus in fixed circular orbits known as "stationary orbits," with each orbit corresponding to a specific energy level.

1.2 Key Postulates of Bohr's Model

Stationary Orbits: Electrons revolve around the nucleus in certain stable orbits without radiating energy.
Quantization of Angular Momentum: The angular momentum of an electron in any orbit is quantized and given by: $m v r = n \frac{h}{2 π}$ where:
- $m$ = mass of the electron
- $v$ = velocity of the electron
- $r$ = radius of the orbit
- $h$ = Planck's constant
- $n$ = Principal quantum number (n = 1, 2, 3, …)
Energy Emission and Absorption: When an electron jumps from one orbit to another, energy is emitted or absorbed in the form of photons. The energy difference between the two levels is given by: $E_{2} - E_{1} = h ν$ where:
- $E_{2}$ and $E_{1}$ are the energies of the final and initial orbits, respectively
- $ν$ = frequency of the emitted/absorbed radiation

1.3 Derivation of Energy Levels in Bohr’s Model

1.3.1 Centripetal Force and Electrostatic Force Balance

For an electron moving in a circular orbit around the nucleus, the electrostatic force provides the necessary centripetal force. According to Coulomb’s law: $\frac{1}{4 π ϵ _{0}} \frac{Z e ^{2}}{r ^{2}} = \frac{m v ^{2}}{r}$ Where:

$Z$ = Atomic number
$e$ = Charge of the electron
$ϵ_{0}$ = Permittivity of free space
$r$ = Radius of the electron's orbit

By combining this with Bohr’s quantization of angular momentum, the radius and velocity of the nth orbit can be expressed as: $r_{n} = \frac{n ^{2} h ^{2} ϵ _{0}}{πm Z e ^{2}}$

1.3.2 Energy of an Electron in a Bohr Orbit

The total energy of an electron in the nth orbit is the sum of its kinetic and potential energies: $E_{n} = - \frac{Z ^{2} m e ^{4}}{8 ϵ _{0}^{2} h ^{2} n ^{2}}$ The negative sign indicates that the electron is bound to the nucleus.

NEET Tip: Remember that energy levels are negative, indicating that electrons are bound to the atom. The most negative (lowest) energy state is the ground state (n = 1).

1.4 Hydrogen Atom Spectrum

Bohr's model accurately explains the spectral lines of the hydrogen atom. When an electron transitions from a higher energy level (e.g., n = 3, 4, etc.) to a lower energy level (e.g., n = 2), it emits light, producing distinct spectral lines. The main spectral series include:

Lyman series: Ultraviolet region (transitions to $n = 1$ )
Balmer series: Visible region (transitions to $n = 2$ )
Paschen series: Infrared region (transitions to $n = 3$ )

Visual Aid: A diagram illustrating the different electron transitions and corresponding spectral series would enhance understanding, especially for visual learners.

Mnemonic:
"Lazy Boys Play Hard": Lyman, Balmer, Paschen, and higher series correspond to transitions to $n = 1, 2, 3$ respectively.

1.5 Application of Bohr's Model to Multi-Electron Atoms

Although Bohr’s model was primarily developed for hydrogen-like atoms (single-electron systems), it has been adapted for multi-electron atoms by incorporating the concept of effective nuclear charge. This helps explain their spectral lines, although more advanced models are needed for complete accuracy.

1.6 Limitations of Bohr’s Model

Could not explain spectra of atoms with more than one electron (multi-electron systems).
Did not account for finer details (fine structure) and the splitting of spectral lines (Zeeman effect and Stark effect).
Fails to explain the wave nature of electrons as proposed by de Broglie.

Quick Recap

Bohr's model explains that electrons revolve in specific energy levels or "stationary orbits."
Angular momentum is quantized: $m v r = n \frac{h}{2 π}$ .
Energy levels are given by: $E_{n} = - \frac{Z ^{2} m e ^{4}}{8 ϵ _{0}^{2} h ^{2} n ^{2}}$ .
The model successfully explains the hydrogen atom spectrum (Lyman, Balmer, Paschen series).

Concept Connection

Chemistry Link

The concept of energy levels directly links to the periodic table and explains the electron configurations of elements, affecting their chemical properties.

NEET Problem-Solving Strategy

Use Bohr's formulae directly for hydrogen-like atoms: $E_{n} \propto \frac{1}{n ^{2}}$ .
For spectral line calculations, use the energy difference formula: $E = h ν$ .

Practice Questions

Calculate the radius of the second orbit (n = 2) for a hydrogen atom using Bohr's model.
Solution: Using the formula: $r_{n} = \frac{n ^{2} h ^{2} ϵ _{0}}{πm Z e ^{2}}$ , substituting $n = 2$ , $Z = 1$ , $h = 6.626 \times 1 0^{- 34}$ Js, $ϵ_{0} = 8.854 \times 1 0^{- 12}$ F/m, $m = 9.109 \times 1 0^{- 31}$ kg, and $e = 1.602 \times 1 0^{- 19}$ C,
$r_{2} \approx 2.12 \times 1 0^{- 10} m$ .
Calculate the energy of the electron in the ground state (n = 1) of a hydrogen atom.
Solution: Using $E_{n} = - \frac{Z ^{2} m e ^{4}}{8 ϵ _{0}^{2} h ^{2} n ^{2}}$ for $n =$