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Atoms and Nuclei JEE Notes
JEE typically asks 2–3 questions (worth 8–12 marks) from Atoms & Nuclei every year. The weightage is evenly split between atomic structure (Bohr model, spectra) and nuclear physics (binding energy, radioactivity). Topics are mostly theory-based but involve quick numerical calculations—ideal for scoring if formulas are on your fingertips. In this note set we group the syllabus into two logical clusters, highlight all high-frequency results and provide a rapid-fire revision sheet.
Atomic Structure and Bohr Model
1. Key Postulates of Bohr Model
- Electrons revolve in discrete circular orbits with quantised angular momentum $$mvr = n\hbar$$ where $$n = 1,2,3,\dots$$
- Total energy in the nth orbit of hydrogen-like species: $$E_n = - \frac{13.6 Z^2}{n^2}$$ eV.
- Radius of nth orbit: $$r_n = a_0 \frac{n^2}{Z}$$; Bohr radius $$a_0 = 0.529$$ Å.
- Frequency of emitted/absorbed photon on transition $$n_2 \rightarrow n_1$$: $$\nu = \frac{E_{n_2} - E_{n_1}}{h}$$.
2. Hydrogen Spectral Series
| Series | Transition Ends at | Spectral Region |
|---|---|---|
| Lyman | $$n_1 = 1$$ | Ultraviolet |
| Balmer | $$n_1 = 2$$ | Visible |
| Paschen | $$n_1 = 3$$ | Infra-red |
| Bracket | $$n_1 = 4$$ | Infra-red |
| Pfund | $$n_1 = 5$$ | Far IR |
3. Rydberg Formula
The wavelength of light emitted/absorbed is given by $$\frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$$ with $$R = 1.097 \times 10^{7}$$ m-1.
4. Worked Example
Q. Find the wavelength of the photon emitted when an electron in hydrogen jumps from $$n = 3$$ to $$n = 2$$.
Solution.
- Use the Rydberg formula with $$Z = 1, n_1 = 2, n_2 = 3$$.
- $$\frac{1}{\lambda} = R \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = 1.097 \times 10^{7} \left( \frac{1}{4} - \frac{1}{9} \right)$$.
- Bracket value: $$\frac{1}{4} - \frac{1}{9} = \frac{5}{36}$$.
- $$\frac{1}{\lambda} = 1.097 \times 10^{7} \times \frac{5}{36} = 1.522 \times 10^{6} \text{ m}^{-1}$$.
- $$\lambda = 6.57 \times 10^{-7} \text{ m} = 657 \text{ nm}$$ (red line of Balmer series).
Such numericals are common in JEE Questions.
Nuclear Properties and Radioactivity
Nuclear Size, Mass and Binding Energy
- Nuclear radius: $$R = R_0 A^{1/3}$$ where $$R_0 \approx 1.1 \times 10^{-15}$$ m.
- Mass defect: $$\Delta m = Z m_p + (A-Z) m_n - M_{\text{nucleus}}$$.
- Binding energy: $$B = \Delta m c^2$$; Binding energy per nucleon indicates stability. Peaks around Iron (A ≈ 56).
Radioactive Decay Law
- Nucleus count at time t: $$N = N_0 e^{-\lambda t}$$.
- Activity: $$A = -\frac{dN}{dt} = \lambda N$$. Unit: Becquerel (Bq).
- Half-life: $$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$$.
- Mean life: $$\tau = \frac{1}{\lambda}$$.
Types of Radioactive Emissions
| Emission | Symbol | Change in A, Z | Typical Energy (MeV) |
|---|---|---|---|
| Alpha | $$^{4}_{2}\!He$$ | $$A-4,\, Z-2$$ | 4–8 |
| Beta− | $$e^-$$ | $$A,\, Z+1$$ | 0.1–3 |
| Beta+ | $$e^+$$ | $$A,\, Z-1$$ | 0.1–3 |
| Gamma | $$\gamma$$ | No change | <3 |
Nuclear Reactions and Q-value
For a reaction $$a + X \rightarrow Y + b$$,
$$Q = \left( m_a + m_X - m_Y - m_b \right) c^2$$. If $$Q \gt 0$$, reaction is exothermic (releases energy); if $$Q \lt 0$$, external energy is required.
Sample Numerical
Q. C14 has a half-life of 5730 years. What fraction of the original nuclei will remain after 17,190 years?
Solution. 17,190 years = 3 half-lives. Each half-life leaves half of the sample. So fraction remaining $$= \left( \frac{1}{2} \right)^3 = \frac{1}{8} = 0.125$$.
Practice more on JEE Mains Mock Test to master such quick calculations.
Important Formulas and Results at a Glance
| Concept | Formula / Value |
|---|---|
| Bohr radius | $$a_0 = 0.529$$ Å |
| Energy of nth orbit | $$E_n = - \frac{13.6 Z^2}{n^2}$$ eV |
| Orbital velocity (hydrogen) | $$v_n = \frac{2.18 \times 10^{6}}{n}$$ m s-1 |
| Rydberg constant | $$R = 1.097 \times 10^{7}$$ m-1 |
| Nuclear radius | $$R = 1.1 \times 10^{-15} A^{1/3}$$ m |
| Mass-energy relation | $$E = mc^2$$ (1 u = 931 MeV) |
| Radioactive decay | $$N = N_0 e^{-\lambda t}$$ |
| Half-life | $$T_{1/2} = 0.693/\lambda$$ |
| Activity | $$A = \lambda N$$ |
| Binding energy | $$B = \Delta m c^2$$ |
Download consolidated JEE Formula Sheets for print-friendly PDFs of these formulas.
JEE Important Points, Common Mistakes and Quick Revision
High-yield Points for the Exam
- Questions on hydrogen atom often combine de-Broglie wavelength with Bohr postulates—be ready to switch units quickly.
- Graphical questions: know the exponential decay curve and energy level diagram thoroughly.
- In radioactivity, JEE loves half-life chaining problems where multiple decays run in series; write decay equations systematically.
- Remember that gamma emission does not change A or Z; many aspirants confuse it with beta.
Common Mistakes to Avoid
- Using wrong units (eV vs joule). Always multiply by $$1.6 \times 10^{-19}$$ when necessary.
- For Rydberg formula, mixing up $$n_1$$ and $$n_2$$. Keep $$n_2 \gt n_1$$.
- Substituting $$T_{1/2}$$ directly in place of mean life $$\tau$$; they differ by factor $$1/\ln 2$$.
- Ignoring recoil energy in alpha decay numericals; while small, it can be asked in Advanced level.
Rapid Revision in 60 Seconds
- Write down the six key formulas: $$E_n,\; r_n,\; R,\; N=N_0e^{-\lambda t},\; T_{1/2},\; B=\Delta mc^2$$.
- Recall spectral series order: Lyman-Balmer-Paschen-Bracket-Pfund.
- Visualise binding energy per nucleon curve—valley at Iron.
- Refresh decay symbols α (4,2), β− (0,−1), β+ (0,+1), γ (0,0).
- Glance through one solved question from each cluster.
Now attempt past questions in JEE Mains Previous Papers and JEE Advanced Previous Papers to lock these points in memory. If you need structured guidance, explore JEE Mains Online Coaching.
Atoms and Nuclei JEE Notes: Conclusion
Atoms and Nuclei is a compact yet scoring chapter. Master the Bohr model, hydrogen spectra, binding energy calculations and decay law. Keep all key formulas handy, practise quick unit conversions and avoid common sign errors. With consistent practice from past papers and mocks, you can comfortably secure full marks from this section.
