Nuclear Physics | Atoms and Nuclei Class 12, JEE and NEET

In the current NCERT Class 12 Physics book, the topic is in two chapters: Chapter 12, Atoms, and Chapter 13, Nuclei. Together they make Unit VIII, Atoms and Nuclei, in the CBSE syllabus, and they carry 12 marks with Dual Nature of Radiation and Matter (Chapters 11 to 13) in the 2026-27 scheme. In the JEE Main syllabus, they are one unit, called Atoms and Nuclei. These notes cover both chapters in the order of the NCERT book, with Nuclear Physics from the Nuclei chapter in the second half.

Atoms and Nuclei Class 12 Notes: Chapter Overview and Syllabus

Check the latest CBSE, NTA (JEE Main), JEE Advanced and NMC (NEET) syllabi for the exact lists. The number of questions changes every year, so use previous years’ papers to see the trend.

Tip from the eSaral Physics team: Learn the Bohr formulas as ratios: $r \propto n^2/Z$, $v \propto Z/n$ and $E \propto Z^2/n^2$. Most numerical questions are then a one-line ratio, and not a long calculation.

Atoms and Nuclei Class 12 Notes for IIT JEE and NEET

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In the current NCERT Class 12 Physics book, the topic is in two chapters: Chapter 12, Atoms, and Chapter 13, Nuclei. Together they make Unit VIII, Atoms and Nuclei, in the CBSE syllabus, and they carry 12 marks with Dual Nature of Radiation and Matter (Chapters 11 to 13) in the 2026-27 scheme. In the JEE Main syllabus, they are one unit, called Atoms and Nuclei. These notes cover both chapters in the order of the NCERT book, with Nuclear Physics from the Nuclei chapter in the second half.

Atoms and Nuclei Class 12 Notes: Chapter Overview and Syllabus

Check the latest CBSE, NTA (JEE Main), JEE Advanced and NMC (NEET) syllabi for the exact lists. The number of questions changes every year, so use previous years’ papers to see the trend.

Tip from the eSaral Physics team: Learn the Bohr formulas as ratios: $r \propto n^2/Z$, $v \propto Z/n$ and $E \propto Z^2/n^2$. Most numerical questions are then a one-line ratio, and not a long calculation.

Rutherford’s Model and Alpha-Particle Scattering in Atoms and Nuclei Class 12 Notes

In the experiment of Geiger and Marsden, a beam of alpha particles was directed at a thin gold foil. Most particles went straight through, a few were deflected through large angles, and about one in $8000$ was scattered back. This showed that the positive charge and almost all the mass of an atom are in a very small nucleus (about $10^{-15}$ m, compared with about $10^{-10}$ m for the atom).

For a head-on collision, the alpha particle of kinetic energy $K$ stops at the distance of closest approach:

$$r_0 = \frac{1}{4\pi\varepsilon_0}\,\frac{2Ze^{2}}{K}$$

The impact parameter $b$ and the scattering angle $\theta$ are related by $b = \dfrac{Ze^{2}\cot(\theta/2)}{4\pi\varepsilon_0 K}$, so a small $b$ gives a large angle of scattering.

Limits of Rutherford’s model: it cannot explain why an electron in an orbit does not radiate and spiral into the nucleus, and it cannot explain the line spectrum of an atom.

Bohr Model of the Hydrogen Atom in Atoms and Nuclei Class 12 Notes

Postulates of Bohr:

1. An electron moves in certain stationary orbits without radiating energy.
2. The orbits that are allowed have an angular momentum that is an integral multiple of $\dfrac{h}{2\pi}$:

$$m v r = \frac{n h}{2\pi}, \qquad n = 1, 2, 3, \ldots$$

3. When an electron jumps from an orbit of energy $E_i$ to an orbit of energy $E_f$, a photon is emitted or absorbed, with $h\nu = \lvert E_i - E_f\rvert$.

For an atom with one electron and the atomic number $Z$ (hydrogen, $\mathrm{He^{+}}$, $\mathrm{Li^{2+}}$):

The de Broglie wave explains the quantisation: $2\pi r_n = n\lambda$, so a stationary orbit holds a whole number of wavelengths.

Limits of the Bohr model: it works for hydrogen-like atoms (one electron) only, it does not explain the relative intensities of the lines or the fine structure, and it cannot explain the splitting of lines in a magnetic field.

Hydrogen Spectrum and Energy Levels in Atoms and Nuclei Class 12 Notes

When an electron falls from the level $n_2$ to the level $n_1$ ($n_2 > n_1$), the wavelength of the photon is given by:

$$\frac{1}{\lambda} = R\,Z^{2}\left(\frac{1}{n_1^{2}} - \frac{1}{n_2^{2}}\right), \qquad R = 1.097\times10^{7}\ \text{m}^{-1}$$

The wavelength in angstrom and the photon energy in eV are related by $\lambda\,(\text{\AA}) = \dfrac{12400}{E\,(\text{eV})}$.

• The ionisation energy of hydrogen (from the ground state) is $13.6$ eV, and the energy of the first excited state is $-3.4$ eV.
• The number of different lines that an atom in the level $n$ can emit is $\dfrac{n(n - 1)}{2}$.
• The longest wavelength of a series comes from the transition between two adjacent levels ($n_2 = n_1 + 1$).

Composition, Size and Mass of the Nucleus in Atoms and Nuclei Class 12 Notes

A nucleus of the element X has $Z$ protons and $N = A - Z$ neutrons, and it is written as $^{A}_{Z}\mathrm{X}$, where $A$ is the mass number.

$$R = R_0A^{1/3}, \qquad R_0 \approx 1.2\ \text{fm}$$

Since the volume of the nucleus is proportional to $A$, the density of nuclear matter is the same for all nuclei, about $2.3\times10^{17}$ kg m$^{-3}$. The atomic mass unit is $1\,\text{u} = 1.66\times10^{-27}$ kg, which is equivalent to $931.5$ MeV$/c^{2}$.

Mass Defect and Binding Energy in Atoms and Nuclei Class 12 Notes

The mass of a nucleus is less than the sum of the masses of its separate nucleons. The difference is the mass defect, and it appears as the binding energy:

$$\Delta m = \big[Zm_p + (A - Z)m_n\big] - M, \qquad E_b = \Delta m\,c^{2} = \Delta m \times 931.5\ \text{MeV}$$

where $\Delta m$ is in u. If you use the masses of atoms, use the mass of the hydrogen atom for $m_p$, and the electron masses cancel. The binding energy per nucleon is $\dfrac{E_b}{A}$, and it measures the stability of the nucleus.

The binding energy curve:

• $E_b/A$ rises quickly for the light nuclei, and it has peaks at $^{4}\mathrm{He}$, $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$.
• It is nearly constant (about $8$ MeV) for $30 < A < 170$, with its maximum of about $8.7$ to $8.8$ MeV in the iron and nickel region ($A \approx 56$ to $62$).
• It falls slowly for the heavy nuclei (about $7.6$ MeV for $^{238}\mathrm{U}$).

So energy is released when heavy nuclei split (fission) and when light nuclei join (fusion), because both change the nucleus into one with a higher binding energy per nucleon.

Nuclear Force in Atoms and Nuclei Class 12 Notes

• It is the strongest of the basic forces, much stronger than the electric force at a distance of a few femtometres.
• It is short-range (a few fm), and it is nearly zero beyond about 2.5 to 3 fm.
• It is attractive at a separation of about 0.8 fm to a few fm, and it becomes repulsive below about 0.8 fm.
• It is charge-independent (the force between two protons, two neutrons or a proton and a neutron is about the same) and it does not depend on the charge.
• It shows saturation, which means that a nucleon interacts only with its nearest neighbours.

Radioactivity and the Decay Law in Atoms and Nuclei Class 12 Notes

Decay law: the rate of decay is proportional to the number of nuclei that are present:

$$\frac{dN}{dt} = -\lambda N, \qquad N = N_0e^{-\lambda t}$$

The unit of activity is the becquerel (1 Bq = 1 decay per second), and $1$ curie $= 3.7\times10^{10}$ Bq. The half-life and the decay constant depend only on the nucleus, and not on the amount of the sample or on the temperature.

Nuclear Fission and Fusion in Atoms and Nuclei Class 12 Notes

Nuclear Fission

A heavy nucleus splits into two lighter nuclei, with neutrons and a large amount of energy:

$$^{235}_{92}\mathrm{U} + {}^{1}_{0}n \rightarrow {}^{141}_{56}\mathrm{Ba} + {}^{92}_{36}\mathrm{Kr} + 3\,{}^{1}_{0}n + Q, \qquad Q \approx 200\ \text{MeV}$$

The neutrons that are released can cause more fissions, and this is a chain reaction. In a nuclear reactor, the reaction is controlled: moderators (heavy water or graphite) slow the neutrons down, control rods (cadmium or boron) absorb neutrons, and a coolant removes the heat. An uncontrolled chain reaction is the principle of the atom bomb.

Nuclear Fusion

Light nuclei join to form a heavier nucleus. In the Sun, the proton–proton cycle changes four protons into one helium nucleus:

$$4\,{}^{1}_{1}\mathrm{H} \rightarrow {}^{4}_{2}\mathrm{He} + 2e^{+} + 2\nu + 26.7\ \text{MeV}$$ $$^{2}_{1}\mathrm{H} + {}^{3}_{1}\mathrm{H} \rightarrow {}^{4}_{2}\mathrm{He} + {}^{1}_{0}n + 17.6\ \text{MeV}$$

Fusion needs a temperature of the order of $10^{7}$ to $10^{8}$ K, so that the nuclei have enough kinetic energy to overcome the Coulomb repulsion. For the same mass, fusion releases more energy than fission.

The energy released in a nuclear reaction is the Q value: $Q = \big(\sum m_{\text{reactants}} - \sum m_{\text{products}}\big)c^{2}$.

IIT JEE and NEET Problem Tips in Atoms and Nuclei Class 12 Notes

• Bohr problems: write the ratios $r \propto n^{2}/Z$, $v \propto Z/n$, $E \propto Z^{2}/n^{2}$, and the time period $T \propto n^{3}/Z^{2}$.
• Wavelength of a line: find the energy difference in eV, and use $\lambda\,(\text{\AA}) = 12400/E$.
• Half-life problems: find the number of half-lives $n = t/T_{1/2}$ first, and then use $N = N_0/2^{n}$. After 10 half-lives, about $0.1\%$ is left.
• Counting $\alpha$ and $\beta$ particles: the fall in $A$ is $4\times$ the number of $\alpha$ particles. Then use the change in $Z$ to find the number of $\beta^-$ particles.
• Binding energy: $\Delta m$ in u $\times\,931.5$ gives the energy in MeV. Do not forget to subtract the electron masses if you use the masses of nuclei.

Key Formulas in Atoms and Nuclei Class 12 Notes

Solved Examples from Atoms and Nuclei Class 12 Notes

Example 1 (Bohr energy): Find the energy of the electron of a hydrogen atom in the level $n = 3$.

$E_3 = -\dfrac{13.6}{3^{2}} \approx -1.51$ eV.

Example 2 (radius): Find the radius of the second orbit of $\mathrm{Li^{2+}}$.

$r_2 = 0.529 \times \dfrac{2^{2}}{3} \approx 0.705$ Å.

Example 3 (Balmer series): Find the wavelength of the first line of the Balmer series ($n_2 = 3 \to n_1 = 2$).

$\dfrac{1}{\lambda} = 1.097\times10^{7}\left(\dfrac{1}{4} - \dfrac{1}{9}\right) = 1.097\times10^{7}\times\dfrac{5}{36} \approx 1.524\times10^{6}$ m$^{-1}$, so $\lambda \approx 656$ nm.

Example 4 (number of lines): How many lines can a hydrogen atom emit from the level $n = 4$?

$\dfrac{n(n - 1)}{2} = \dfrac{4 \times 3}{2} = 6$ lines.

Example 5 (photon energy): Find the energy and the wavelength of the photon emitted when an electron of hydrogen goes from $n = 3$ to $n = 1$.

$E = 13.6\left(1 - \dfrac{1}{9}\right) \approx 12.09$ eV, and $\lambda = \dfrac{12400}{12.09} \approx 1026$ Å (ultraviolet, Lyman series).

Example 6 (distance of closest approach): A 5 MeV alpha particle is fired at a gold nucleus ($Z = 79$). Find the distance of closest approach.

$r_0 = \dfrac{9\times10^{9}\times 2\times 79\times(1.6\times10^{-19})^{2}}{5\times1.6\times10^{-13}} \approx 4.5\times10^{-14}$ m, which is about 45 fm.

Example 7 (nuclear radius): Find the radius of the nucleus of $^{27}\mathrm{Al}$.

$R = 1.2\times 27^{1/3} = 1.2\times 3 = 3.6$ fm.

Example 8 (binding energy): Find the binding energy and the binding energy per nucleon of $^{4}\mathrm{He}$ (use $m_H = 1.007825$ u, $m_n = 1.008665$ u and $M_{He} = 4.002603$ u).

$\Delta m = 2(1.007825) + 2(1.008665) - 4.002603 = 0.030377$ u. $E_b = 0.030377\times931.5 \approx 28.3$ MeV, so $E_b/A \approx 7.07$ MeV.

Example 9 (half-life): 64 g of a radioactive substance has a half-life of 5 days. How much is left after 20 days?

$n = \dfrac{20}{5} = 4$ half-lives, so $N = \dfrac{64}{2^{4}} = 4$ g.

Example 10 (decay constant and mean life): The half-life of a sample is 10 minutes. Find the decay constant and the mean life.

$\lambda = \dfrac{0.693}{600} \approx 1.155\times10^{-3}$ s$^{-1}$, and $\tau = \dfrac{1}{\lambda} \approx 866$ s, which is about 14.4 minutes.

**Example 11 (counting $\alpha$ and $\beta$ particles):** $^{238}_{92}\mathrm{U}$ changes into $^{206}_{82}\mathrm{Pb}$ through a series of decays. Find the number of $\alpha$ and $\beta^-$ particles.

The fall in $A$ is $238 - 206 = 32$, so the number of $\alpha$ particles is $\dfrac{32}{4} = 8$. These reduce $Z$ by $16$, to $76$. To reach $82$, we need $82 - 76 = 6$ $\beta^-$ particles.

Example 12 (Q value of fusion): Find the energy released in $^{2}\mathrm{H} + {}^{3}\mathrm{H} \rightarrow {}^{4}\mathrm{He} + n$ (masses: $2.014102$, $3.016049$, $4.002603$ and $1.008665$ u).

$\Delta m = (2.014102 + 3.016049) - (4.002603 + 1.008665) = 0.018883$ u, so $Q = 0.018883\times931.5 \approx 17.6$ MeV.

eSaral brings you detailed Class 11th Physics study material.  eSaral provides a series of detailed chapter-wise notes for all the Subjects of class 11th and 12th.  These notes will also help you in your IIT JEE & NEET preparations. We hope these Physics Notes for Class 12 will help you understand the important topics and remember the key points from an exam point of view.

Common Mistakes in Atoms and Nuclei Class 12 Problems

How to Study Atoms and Nuclei Class 12 Notes

1. Read both NCERT chapters first (Atoms and Nuclei), and download the textbook from eSaral’s NCERT Class 12 books page.
2. Learn the Bohr formulas as ratios, and the series table of the hydrogen spectrum. Then learn the binding energy curve and the decay law.
3. Practise one problem each on the Bohr model, spectral lines, mass defect, half-life, counting of $\alpha$ and $\beta$ particles and the Q value, using the tips on this page.
4. Use eSaral’s Complete Class 12 Physics notes for all the chapters, and the free Physics revision series on YouTube for quick formula revision.
5. Test yourself with eSaral’s JEE Main chapter-wise PYQs, JEE Advanced chapter-wise PYQs and NEET chapter-wise PYQs.