Atomic Structure
Bohr's Model of Atom: Formulas, Concepts & Revision Chart
Neils Bohr's model is a cornerstone of quantum mechanics for JEE and NEET. It successfully explains the stability of atoms and the line spectra of hydrogen-like species (ions with one electron).
1. Key Postulates of Bohr's Theory
- Stationary Orbits: Electrons revolve around the nucleus in fixed circular paths called orbits or shells ($K, L, M, N...$) without radiating energy.
- Quantization Condition: An electron can move only in those orbits where its angular momentum ($L$) is an integral multiple of $\frac{h}{2\pi}$.
$$ mvr = \frac{nh}{2\pi} $$
- Energy Transitions: Energy is emitted or absorbed only when an electron jumps from one orbit to another.
$$ \Delta E = E_2 - E_1 = h\nu $$
2. Important Formulas for Hydrogen-like Atoms
For a species with atomic number $Z$ and orbit number $n$, the following relations hold true. Memorizing these values saves crucial time in exams.
Radius of Orbit ($r_n$)
$$ r_n = 0.529 \frac{n^2}{Z} \text{ \AA} $$
Velocity of Electron ($v_n$)
$$ v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} $$
Total Energy ($E_n$)
$$ E_n = -13.6 \frac{Z^2}{n^2} \text{ eV/atom} $$
Note: $K.E. = -E_n$ and $P.E. = 2E_n$
Exam Cheat Sheet: Proportionality Trends
- Radius ($r$): Directly proportional to $n^2$ and inversely to $Z$. ($r \propto n^2/Z$)
- Velocity ($v$): Directly proportional to $Z$ and inversely to $n$. ($v \propto Z/n$)
- Energy ($E$): Directly proportional to $Z^2$ and inversely to $n^2$. ($E \propto Z^2/n^2$)
- Time Period ($T$): Proportional to $n^3/Z^2$.
Figure: Summary of Bohr's Postulates and Mathematical Relations
3. Hydrogen Spectrum
When an electron jumps from a higher orbit ($n_2$) to a lower orbit ($n_1$), energy is emitted as radiation. The wavelength ($\lambda$) is given by the Rydberg Formula:
$$ \frac{1}{\lambda} = \bar{\nu} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) $$
Spectral Series
- Lyman Series ($n_1=1$): Ultraviolet Region (UV).
- Balmer Series ($n_1=2$): Visible Region (Only series visible to naked eye).
- Paschen ($n_1=3$), Brackett ($n_1=4$), Pfund ($n_1=5$): Infrared Region (IR).
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