Atomic Structure & Bohr-Sommerfeld Model: Master Notes

Chapter Overview

Atomic physics models the quantized energy states of hydrogenic atoms and characteristic X-ray emissions. In IIT-JEE (Advanced), core problems center on Bohr parameter scaling ($r_n\propto n^2/Z,v_n\propto Z/n,E_n\propto -Z^2/n^2$), reduced mass corrections for exotic atoms (Muonic Hydrogen & Positronium), spectral line series transitions, and Moseley's Law for characteristic X-rays.

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1. Bohr Postulates & Hydrogen-Like Atom Scaling

For any single-electron atom of nuclear charge $+Ze$ ($\text{H},{\text{He}}^{+},{\text{Li}}^{2+},{\text{Be}}^{3+}$):

1. Quantization of Angular Momentum:$$L=m{v}_n{r}_n=n\hslash =n\frac{h}{2\pi }(n=1,2,3,\dots )$$
2. Electrostatic Centripetal Balance:$$\frac{m{v}_n^2}{r_n}=\frac{1}{4\pi {\epsilon }_0}\frac{Z{e}^2}{r_n^2}$$

1.1 Master Scaling Laws

Variable	General Formula	Scaling with $(n,Z)$	Value for Hydrogen ($Z=1,n=1$)
Orbital Radius ($r_n$)	$r_n=\frac{{\epsilon }_0{h}^2}{\pi m{e}^2}\frac{n^2}{Z}$	$\boxed{{\displaystyle r_n\propto \frac{n^2}{Z}}}$	$r_1=0.529\text{AA}$
Orbital Velocity ($v_n$)	$v_n=\frac{e^2}{2{\epsilon }_{0}h}\frac{Z}{n}$	$\boxed{{\displaystyle v_n\propto \frac{Z}{n}}}$	$v_1=\frac{c}{137}\approx 2.18\times {10}^6\text{ m/s}$
Kinetic Energy ($K_n$)	$K_n=\frac{1}{2}m{v}_n^2$	$\boxed{{\displaystyle K_n\propto \frac{Z^2}{n^2}}}$	$K_1=+13.6\text{ eV}$
Potential Energy ($U_n$)	$U_n=-\frac{1}{4\pi {\epsilon }_0}\frac{Z{e}^2}{r_n}$	$\boxed{{\displaystyle U_n\propto -\frac{Z^2}{n^2}}}$	$U_1=-27.2\text{ eV}=-2K_1$
Total Energy ($E_n$)	$E_n=K_n+U_n=-K_n=\frac{U_n}{2}$	$\boxed{{\displaystyle E_n=-13.6\frac{Z^2}{n^2}\text{ eV}}}$	$E_1=-13.6\text{ eV}$
Orbital Time Period ($T_n$)	$T_n=\frac{2\pi r_n}{v_n}$	$\boxed{{\displaystyle T_n\propto \frac{n^3}{Z^2}}}$	$T_1\approx 1.52\times {10}^{-16}\text{ s}$

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2. Hydrogen Spectral Series & Rydberg Formula

$$\boxed{{\displaystyle \frac{1}{\lambda }=R_{\mathrm{\infty }}{Z}^2(\frac{1}{n_1^2}-\frac{1}{n_2^2})}}$$

(where $R_{\mathrm{\infty }}=\frac{m{e}^4}{8{\epsilon }_0^2{h}^{3}c}\approx 1.097\times {10}^7{\text{ m}}^{-1}$ is the Rydberg Constant).

Multi-Mode DiagramAtomic Structure: Bohr Quantized Energy Ladder

Option 1: Publication-Grade Scientific Vector SVG

Hydrogen quantized energy levels E_n = -13.6 / n^2 eV with Lyman, Balmer, and Paschen spectral emission series.

Bohr Energy: $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$

Rydberg Formula: $\frac{1}{\lambda} = R Z^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$

Option 2: Interactive 60 FPS Canvas Physics Engine

Simulate photon emission during electron quantum transition.

Initial Level n₂: 3

Photon Wavelength ($\lambda$):102.6 nm

Option 3: Interactive 3D Spatial Viewport

3D Bohr orbital shells with quantized electron wave.

+X

+Y

+Z

👆 Click and drag anywhere to rotate in 3D space

Azimuth Angle (ψ):35°

Elevation Angle (θ):-22°

Bohr Quantization:$L = n \hbar$

Option 5: Reactive Parametric Physics with Live Calculus

Energy transition & wavelength calculator.

Initial level n₂: 3

: 45

Energy Released ($\Delta E$):$12.09 eV$

Active Mode:Option 1 (Print & Derivation Vector SVG)Switch modes using the tabs above to explore different visualizations.

• Number of Spectral Lines emitted from state $n$ to ground state:$$N_{\text{lines}}=\frac{n(n-1)}{2}$$

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3. Reduced Mass Corrections (Finite Nuclear Mass $M$)

When nuclear mass $M$ is not infinitely greater than orbiting particle mass $m$:

$$\boxed{{\displaystyle \mu =\frac{mM}{m+M}}}$$

• Replace $m\to \mu$ in all Bohr energy and Rydberg formulas:$$E_n^{\prime }=\left(\frac{\mu }{m_e}\right)E_n,R^{\prime }=\left(\frac{\mu }{m_e}\right)R_{\mathrm{\infty }}$$
• Positronium (${\text{e}}^{-}+{\text{e}}^{+}$): $\mu =\frac{m_e\cdot m_e}{m_e+m_e}=\frac{m_e}{2}⟹E_1=\frac{-13.6}{2}=-6.8\text{ eV}$.

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4. X-Rays & Moseley's Law

4.1 Continuous X-Rays (Duane-Hunt Law)

$$\boxed{{\displaystyle {\lambda }_{min}=\frac{hc}{eV}\approx \frac{12400}{V(\text{volts})}\text{AA}}}$$

4.2 Characteristic X-Rays & Moseley's Law

$$\boxed{{\displaystyle \sqrt{\nu }=a(Z-b)}}$$

• For $K_{\alpha }$ transition ($n=2\to n=1$): Screening constant $b\approx 1$, $a=\sqrt{\frac{3}{4}Rc}$.
• For $L_{\alpha }$ transition ($n=3\to n=2$): Screening constant $b\approx 7.4$.

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5. Authentic Previous Years Questions (PYQs)

PYQ 1: JEE Advanced 2021 (Paper 2) — Muonic Hydrogen Rydberg Wavelength

Question:
A muonic hydrogen atom consists of a muon of mass $m_{\mu }=207m_e$ orbiting a proton ($m_p=1836m_e$). Find the ground state binding energy $E_1$ of muonic hydrogen in $\text{eV}$.

Step-by-Step Solution:

1. Reduced Mass of Muonic Hydrogen:$$\mu =\frac{m_{\mu }{m}_p}{m_{\mu }+m_p}=\frac{(207m_e)(1836m_e)}{207m_e+1836m_e}=\frac{380052}{2043}{m}_e\approx 186.03m_e$$
2. Ground State Energy:$$E_1=\left(\frac{\mu }{m_e}\right)E_{\text{hydrogen, 1}}=186.03\times (-13.6\text{ eV})=\boxed{{\displaystyle -2530\text{ eV}=-2.53\text{ keV}}}$$

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PYQ 2: JEE Advanced 2019 — Moseley's Law & Atomic Number Determination

Question:
The $K_{\alpha }$ X-ray wavelength of element $A$ ($Z_A=43$) is ${\lambda }_A=0.70\text{AA}$. Find the atomic number $Z_B$ of an element $B$ whose $K_{\alpha }$ wavelength is ${\lambda }_B=0.54\text{AA}$. (Use screening constant $b=1$).

Step-by-Step Solution:

From Moseley's Law for $K_{\alpha }$ emission:

$$\frac{c}{\lambda }=\nu =a^2(Z-1{)}^2⟹\frac{1}{\sqrt{\lambda }}\propto (Z-1)$$$$\frac{Z_B-1}{Z_A-1}=\sqrt{\frac{{\lambda }_A}{{\lambda }_B}}=\sqrt{\frac{0.70}{0.54}}=\sqrt{\frac{35}{27}}\approx \sqrt{1.296}=1.138$$$$Z_B-1=(43-1)\times 1.138=42\times 1.138\approx 47.8⟹\boxed{{\displaystyle Z_B=49(\text{Indium})}}$$

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6. High-Yield Formula Sheet

Entity	Formula	Notes
Bohr Radius	$r_n=0.529\frac{n^2}{Z}\text{AA}$	Hydrogenic ions
Total Energy	$E_n=-13.6\frac{Z^2}{n^2}\text{ eV}$	Bound state $E<0$
X-Ray Cutoff	${\lambda }_{min}=\frac{12400}{V}\text{AA}$	Duane-Hunt Law
Moseley's Law	$\sqrt{\nu }=a(Z-1)$	For $K_{\alpha }$ emission
Reduced Mass	$\mu =\frac{m_1{m}_2}{m_1+m_2}$	Finite nuclear mass