ATOMS & NUCLEI | Physics | Class XII | CBSE / JEE / NEET | SELF STUDY

 ATOMS & NUCLEI

A COMPREHENSIVE CHAPTER ANALYSIS

Physics | Class XII | CBSE / JEE / NEET

 

From Atomic Models to Nuclear Reactions — Everything You Need to Master

1. INTRODUCTION TO ATOMIC STRUCTURE

Matter, in its most fundamental form, is composed of atoms — extraordinarily minute entities that constitute the building blocks of every substance in the known universe. The word 'atom' originates from the ancient Greek 'atomos,' meaning indivisible. Ironically, centuries of scientific inquiry have revealed that atoms are anything but indivisible; they are themselves intricate systems of subatomic particles, each governed by the laws of quantum mechanics and electrodynamics.

The atom consists of a centrally located, positively charged nucleus — an almost incomprehensibly dense core — surrounded by a cloud of negatively charged electrons that occupy probabilistic regions of space known as orbitals. The nucleus itself is composed of two types of particles: protons, which carry a positive electric charge, and neutrons, which are electrically neutral. Together, protons and neutrons are referred to as nucleons.

The radius of a typical atom is on the order of 10⁻¹⁰ metres (one Angstrom), whereas the nuclear radius is approximately 10⁻¹⁵ metres (one Femtometre or Fermi). This implies that the nucleus occupies a fantastically small fraction of the atom's total volume — roughly 10⁻¹⁵ of it — yet it contains nearly 99.95% of the atom's mass. If an atom were magnified to the size of a football stadium, the nucleus would be approximately the size of a marble at the centre.

Key Insight: An atom is predominantly empty space. The electrons, despite defining the atom's chemistry and size, are incredibly light compared to the nucleus.

1.1 Fundamental Subatomic Particles

Particle	Symbol	Charge	Mass (kg)	Location
Proton	p	+1.6 × 10⁻¹⁹ C	1.673 × 10⁻²⁷ kg	Nucleus
Neutron	n	0 (Neutral)	1.675 × 10⁻²⁷ kg	Nucleus
Electron	e⁻	−1.6 × 10⁻¹⁹ C	9.109 × 10⁻³¹ kg	Orbitals (shells)

 

2. HISTORICAL DEVELOPMENT OF ATOMIC MODELS

The conception of the atom has undergone a remarkable and progressive transformation over the past two centuries. Each successive atomic model was constructed to reconcile the discrepancies exposed by the experiments of its era, resulting in an increasingly sophisticated and accurate description of atomic architecture.

2.1 Dalton's Atomic Theory (1803)

John Dalton, an English chemist and physicist, proposed the first scientific atomic theory. According to Dalton, matter consists of indivisible, indestructible particles called atoms. He postulated that all atoms of a given element are identical in mass and properties, that atoms of different elements differ in mass, and that chemical reactions involve the rearrangement of atoms. While fundamentally correct in many respects, Dalton's model had no provision for subatomic structure and could not explain electrical phenomena.

2.2 Thomson's Plum Pudding Model (1897)

J.J. Thomson's discovery of the electron in 1897 shattered Dalton's notion of atomic indivisibility. Thomson proposed that the atom is a sphere of uniformly distributed positive charge, within which negatively charged electrons are embedded like plums in a pudding. This model, though ingenious for its time, was incapable of explaining the nuclear scattering results that would soon emerge from Rutherford's laboratory.

2.3 Rutherford's Nuclear Model (1911)

Ernest Rutherford's landmark gold foil experiment, conducted with Hans Geiger and Ernest Marsden, revolutionised the understanding of atomic structure. A beam of alpha particles was directed at an extremely thin gold foil. While most alpha particles passed straight through — consistent with the diffuse positive charge in Thomson's model — a small fraction deflected at enormous angles, and some even bounced nearly directly backwards.

Rutherford correctly interpreted these results as evidence for a dense, positively charged nucleus at the atom's centre. Most of the atom, he concluded, was vacant space, through which alpha particles passed unimpeded. The rare large-angle deflections occurred when alpha particles passed close to, or nearly collided with, the tiny but massive nucleus. Rutherford estimated the nuclear radius to be at least 10,000 times smaller than the atomic radius.

Figure 1: Rutherford's Gold Foil Experiment — Revealing the Dense Atomic Nucleus

However, Rutherford's planetary model suffered a critical flaw: according to classical electromagnetic theory, an electron orbiting a nucleus would continuously radiate energy, spiralling inward and causing the atom to collapse within nanoseconds. This contradiction demanded a radical new framework — quantum theory.

2.4 Bohr's Quantised Model (1913)

Niels Bohr resolved the stability problem by introducing a revolutionary postulate: electrons occupy fixed, quantised orbits (also called stationary states or energy levels) and do not radiate energy while remaining in these orbits. Energy is emitted or absorbed only when an electron transitions between orbits. The energy of emitted or absorbed radiation corresponds precisely to the energy difference between the initial and final orbits, and is quantised as a photon of energy E = hf, where h is Planck's constant and f is the frequency.

Bohr's postulates may be stated as follows. First, electrons revolve around the nucleus in specific circular orbits without radiating electromagnetic energy. Second, only those orbits are permitted for which the angular momentum of the electron is an integral multiple of h/2π, i.e., mvr = nℏ, where n is the principal quantum number (n = 1, 2, 3, ...). Third, when an electron transitions from a higher energy orbit (nᵢ) to a lower energy orbit (nf), it emits a photon whose energy equals the energy difference between the two orbits.

Figure 2: Bohr's Model — Quantised Electron Orbits Around the Nucleus

Bohr's Key Formulae for Hydrogen Atom

Radius of nth orbit:  rₙ = n² × a₀  where  a₀ = 0.529 Å  (Bohr radius)

Energy of nth level:  Eₙ = −13.6 / n²  eV

Frequency of emitted photon:  hf = Eᵢ − Ef  =  13.6 × (1/nf² − 1/nᵢ²)  eV

The Bohr model successfully explained the discrete spectral lines of hydrogen. The Lyman series (transitions to n=1) lies in the ultraviolet region, the Balmer series (transitions to n=2) falls in the visible spectrum, and the Paschen, Brackett, and Pfund series lie in the infrared region.

Spectral Series	Final Level (nf)	Region of Spectrum	Formula
Lyman	n = 1	Ultraviolet	1/λ = R(1/1² − 1/nᵢ²)
Balmer	n = 2	Visible	1/λ = R(1/2² − 1/nᵢ²)
Paschen	n = 3	Infrared (near)	1/λ = R(1/3² − 1/nᵢ²)
Brackett	n = 4	Infrared	1/λ = R(1/4² − 1/nᵢ²)
Pfund	n = 5	Far Infrared	1/λ = R(1/5² − 1/nᵢ²)

 

Here, R is the Rydberg constant = 1.097 × 10⁷ m⁻¹. The Bohr model, while a magnificent achievement, was limited to single-electron atoms and could not address multi-electron atoms or the fine structure of spectral lines. It was eventually superseded by the full quantum mechanical (wave-mechanical) model developed by Schrödinger, Heisenberg, and Dirac.

3. THE ATOMIC NUCLEUS — STRUCTURE AND PROPERTIES

The nucleus is the heart of the atom — an extraordinarily compact and massive entity that defines the identity of an element and stores an immense reservoir of energy. Despite occupying a volume roughly 10⁻¹⁵ of the total atomic volume, the nucleus accounts for 99.95% of the atom's mass, making it one of the densest known structures in the observable universe.

3.1 Nuclear Terminology and Notation

The composition of a nucleus is described by two fundamental quantum numbers. The atomic number Z denotes the number of protons within the nucleus and uniquely identifies the chemical element. The mass number A is the total count of nucleons (protons plus neutrons) within the nucleus. Hence, the number of neutrons N = A − Z. A specific nuclide is represented in standard notation as ᴬ_Z X, where X is the chemical symbol of the element.

Notation:  ᴬ_Z X   where   A = Mass Number,   Z = Atomic Number,   N = A − Z

Isotopes are atoms of the same element (same Z) but with different mass numbers (different A), hence differing in neutron count. Examples include Carbon-12 (⁶¹²C) and Carbon-14 (⁶¹⁴C), or Hydrogen's three isotopes: Protium (¹H), Deuterium (²H), and Tritium (³H). Isobars are nuclides with the same mass number A but different atomic numbers. Isotones share the same neutron number N but differ in A and Z.

3.2 Nuclear Size and Density

Experimental evidence from nuclear scattering experiments has established that the nuclear radius R follows a systematic relationship with mass number A:

R = R₀ × A^(1/3)   where   R₀ ≈ 1.2 × 10⁻¹⁵ m  (Femtometre)

This relationship implies that the nuclear volume V ∝ A, and therefore nuclear density is approximately constant across all nuclides — a remarkable empirical fact. The nuclear density is calculated to be approximately 2.3 × 10¹⁷ kg/m³, which is about 10¹⁴ times the density of ordinary matter. A teaspoonful of nuclear matter would weigh approximately 2 billion tonnes.

3.3 Forces Within the Nucleus

The nucleus presents a profound puzzle from the standpoint of classical physics: how can positively charged protons, which repel each other via the Coulomb electrostatic force, remain bound together in such an extremely small volume? The answer lies in the strong nuclear force — one of the four fundamental forces of nature.

•       Strong Nuclear Force: An attractive force acting between all nucleons (proton-proton, neutron-neutron, and proton-neutron). It is the strongest of all known forces, approximately 137 times stronger than the electromagnetic force at nuclear distances. Crucially, it is an extremely short-range force, effective only up to distances of about 1–3 Femtometres.

•       Electromagnetic Force (Coulomb Repulsion): Acts between protons over unlimited range. For small nuclei, the strong nuclear force dominates. For very large nuclei, the accumulation of Coulomb repulsion between many protons can overcome the strong force, leading to instability.

•       Weak Nuclear Force: Responsible for beta decay processes. It facilitates the transformation of neutrons into protons (and vice versa) within the nucleus, playing a critical role in radioactive transmutation.

Key Fact: The strong nuclear force is charge-independent — it acts equally between any pair of nucleons regardless of their charge. It is also spin-dependent and has a repulsive core at very short distances (<0.4 fm), preventing nucleons from collapsing into each other.

4. MASS DEFECT AND NUCLEAR BINDING ENERGY

One of the most profound insights of modern physics is the equivalence of mass and energy, encapsulated in Einstein's iconic equation E = mc². This principle finds one of its most dramatic applications in the concept of nuclear binding energy.

4.1 Mass Defect (Δm)

When nucleons bind together to form a nucleus, the mass of the resulting nucleus is measurably less than the sum of the masses of its constituent free protons and neutrons. This discrepancy is known as the mass defect (Δm). It is not a measurement error but a fundamental physical reality: the missing mass has been converted into energy, which was released when the nucleus formed and which binds the nucleus together.

Δm = Z·mₚ + N·mₙ − M_nucleus

Where mₚ = mass of a proton = 1.00728 u, mₙ = mass of a neutron = 1.00866 u, M_nucleus is the actual measured mass of the nucleus, and u = unified atomic mass unit = 1.66054 × 10⁻²⁷ kg.

4.2 Nuclear Binding Energy (Eₔ)

The binding energy is the energy equivalent of the mass defect, calculated using Einstein's mass-energy relation. It represents the minimum energy that must be supplied to completely disassemble the nucleus into its constituent free nucleons.

Eₔ = Δm · c²   (in Joules)

Eₔ (in MeV) = Δm (in u) × 931.5 MeV/u

The binding energy per nucleon, Eₔ/A, is a crucial measure of nuclear stability. It represents how tightly each nucleon is bound within the nucleus. A higher binding energy per nucleon signifies a more stable nucleus.

Figure 3: Binding Energy per Nucleon vs. Mass Number — The Stability Curve of the Nucleus

As illustrated in the graph, the binding energy per nucleon rises steeply for light nuclei, reaches a broad maximum of approximately 8.79 MeV per nucleon at Iron-56 (⁵⁶Fe), and then gradually decreases for heavier nuclei. This curve has profound consequences for nuclear energy:

•       Fusion (light elements joining): Moving up the curve from left toward the iron peak releases energy, because the products are more tightly bound than the reactants.

•       Fission (heavy elements splitting): Moving down the curve from right toward the iron peak also releases energy, for the same reason — the fission products are more stable (higher Eₔ/A) than the parent nucleus.

•       Iron is the most stable element in the universe. It is the endpoint of stellar nuclear synthesis through fusion, and further fusion or fission of iron actually requires energy input rather than releasing it.

Historical Note: The development of nuclear weapons and power plants in the 20th century was made possible by the profound insight that tiny mass defects — millionths of a kilogram — translate, via E=mc², into enormous energy releases of millions of electron-volts per reaction.

5. RADIOACTIVITY — SPONTANEOUS NUCLEAR TRANSFORMATION

Radioactivity is the spontaneous emission of radiation from an unstable nucleus as it seeks a more stable configuration. Discovered by Henri Becquerel in 1896 and extensively studied by Marie and Pierre Curie, radioactivity revealed that certain atomic nuclei are inherently unstable and undergo spontaneous transformation, emitting energetic particles or electromagnetic radiation in the process.

The phenomenon is entirely spontaneous and cannot be accelerated, retarded, or controlled by any chemical or physical means — temperature, pressure, magnetic or electric fields have absolutely no effect on the rate of radioactive decay. This underscores that radioactivity is a purely nuclear phenomenon, governed by quantum mechanical probability.

5.1 Types of Radioactive Decay

Figure 4: The Three Types of Radioactive Decay and Their Relative Penetrating Powers

Alpha (α) Decay

In alpha decay, the nucleus emits an alpha particle — a tightly bound cluster of two protons and two neutrons (equivalent to a helium-4 nucleus, ⁴₂He). This occurs predominantly in heavy nuclei (A > 200) where the ratio of protons to neutrons makes the nucleus unstable against proton-proton Coulomb repulsion.

ᴬ_Z X  →  ᴬ⁻⁴_(Z-2) Y  +  ⁴₂He  +  Energy

Alpha particles possess high kinetic energy (typically 4–9 MeV) but have very low penetrating power due to their large charge (+2e) and relatively large mass. They are stopped by a few centimetres of air or a single sheet of paper. Despite low penetration, they are extremely ionising and pose serious radiological hazard if alpha-emitting materials are ingested or inhaled.

Beta (β) Decay

Beta decay occurs in two varieties. In beta-minus (β⁻) decay, a neutron within the nucleus transforms into a proton, simultaneously emitting an electron (β⁻ particle) and an electron antineutrino (ν̄ₑ). The atomic number increases by one while the mass number remains unchanged.

ᴬ_Z X  →  ᴬ_(Z+1) Y  +  e⁻  +  ν̄ₑ

In beta-plus (β⁺) decay, a proton transforms into a neutron, emitting a positron (the antimatter counterpart of the electron) and an electron neutrino (νₑ). The atomic number decreases by one.

ᴬ_Z X  →  ᴬ_(Z-1) Y  +  e⁺  +  νₑ

The neutrinos are essentially massless, electrically neutral particles that interact extremely weakly with matter and pass through virtually everything. The continuous energy spectrum of beta particles (rather than discrete energies) was a major puzzle until Pauli proposed the existence of the neutrino in 1930 to conserve energy and momentum.

Gamma (γ) Radiation

Gamma radiation consists of high-energy photons emitted when a nucleus transitions from an excited energy state to a lower energy state. Gamma emission often accompanies alpha or beta decay, as the daughter nucleus may initially be left in an excited state. Unlike alpha and beta decay, gamma emission does not change the atomic number or mass number of the nucleus.

ᴬ_Z X*  →  ᴬ_Z X  +  γ  (photon)

Gamma rays have extremely high penetrating power — they require several centimetres of lead or metres of concrete to be effectively attenuated — but are far less ionising per unit path length than alpha or beta particles.

Property	Alpha (α)	Beta (β)	Gamma (γ)
Nature	He-4 nucleus (²p + ²n)	Electron or Positron	High-energy photon
Charge	+2e	±e	0
Mass	~6.64 × 10⁻²⁷ kg	~9.1 × 10⁻³¹ kg	0 (massless)
Speed	~5–7% of c	Up to ~99% of c	c (speed of light)
Ionising Power	Highest (~10,000×)	Moderate (~100×)	Lowest (1×)
Penetration	~4 cm air / paper	~3 mm aluminium	Several cm lead

 

5.2 The Law of Radioactive Decay

Radioactive decay is a statistical, probabilistic process governed by quantum mechanics. Each nucleus of an unstable species has a fixed probability of decaying per unit time, independent of its history, the presence of other nuclei, or any external conditions. This leads to exponential decay behaviour.

If N₀ is the initial number of radioactive nuclei at time t = 0, then the number of undecayed nuclei at time t is given by the fundamental decay law:

N(t) = N₀ · e^(−λt)

Here, λ is the decay constant, which represents the probability of decay per nucleus per unit time. It is characteristic of each particular radioactive isotope.

Figure 5: Exponential Radioactive Decay — The Half-Life Concept Illustrated

Half-Life (T₁/₂)

The half-life T₁/₂ is the time required for exactly half the original number of radioactive nuclei to decay. It is the most commonly cited measure of a radioisotope's rate of decay and is related to the decay constant by:

T₁/₂ = ln(2) / λ  ≈  0.693 / λ

After each successive half-life, half of the remaining nuclei decay. Thus, after n half-lives, the fraction remaining is (1/2)ⁿ. Half-lives span an extraordinary range — from fractions of a microsecond for the most unstable nuclei to billions of years for nearly stable ones. For example, Carbon-14 has a half-life of 5,730 years (used in radiocarbon dating), while Uranium-238 has a half-life of 4.47 × 10⁹ years.

Activity (A)

The activity of a radioactive sample is the number of disintegrations per second. It is proportional to the number of undecayed nuclei:

A = λN = λN₀ · e^(−λt) = A₀ · e^(−λt)

The SI unit of activity is the Becquerel (Bq), defined as one disintegration per second. The older unit is the Curie (Ci), where 1 Ci = 3.7 × 10¹⁰ Bq.

6. NUCLEAR REACTIONS — FISSION AND FUSION

Nuclear reactions involve transformations of nuclei, accompanied by the release or absorption of energy. The two most energetically significant nuclear reactions are fission — the splitting of heavy nuclei — and fusion — the joining of light nuclei. Both derive their enormous energy output from the mass-energy equivalence principle and from the shape of the binding energy per nucleon curve.

6.1 Nuclear Fission

Nuclear fission is the process by which a heavy nucleus (typically Uranium-235 or Plutonium-239) absorbs a slow (thermal) neutron and splits into two intermediate-mass daughter nuclei (fission fragments), releasing two or three additional neutrons and a substantial quantity of energy — approximately 200 MeV per fission event. This energy is primarily in the kinetic energy of the fission fragments and neutrons, which is ultimately converted to heat.

²³⁵₉₂U  +  ¹₀n  →  [²³⁶₉₂U]*  →  ⁹²₃₆Kr  +  ¹⁴¹₅₆Ba  +  3 ¹₀n  +  Energy (~200 MeV)

Figure 6: Nuclear Fission Chain Reaction — The Mechanism Behind Nuclear Reactors and Weapons

The critical feature of nuclear fission is the chain reaction: the neutrons released in each fission event can induce further fission in neighbouring nuclei, potentially leading to an exponentially growing cascade of fissions. If the mass of fissile material exceeds the critical mass — the minimum mass required to sustain a self-perpetuating chain reaction — an uncontrolled chain reaction produces a nuclear explosion. In a nuclear reactor, the chain reaction is controlled by moderators (to slow neutrons) and control rods (to absorb excess neutrons), maintaining a steady, sustainable reaction rate.

Applications: Nuclear fission powers approximately 10% of the world's electricity through nuclear power plants. It also represents the destructive force of fission nuclear weapons (atomic bombs), first detonated in 1945.

6.2 Nuclear Fusion

Nuclear fusion is the process by which two light nuclei collide and merge to form a heavier nucleus, releasing energy. Fusion powers every star in the universe, including our Sun. The primary fusion reaction in the Sun is the proton-proton chain, which ultimately fuses four hydrogen nuclei (protons) into one helium-4 nucleus, releasing about 26.7 MeV of energy per cycle.

²₁H  +  ³₁H  →  ⁴₂He  +  ¹₀n  +  17.6 MeV  (Deuterium-Tritium fusion)

Fusion reactions require the reactant nuclei to approach each other to within the range of the strong nuclear force (~1 fm), overcoming the enormous electrostatic repulsion between their positive charges. This requires temperatures of the order of 10⁷ to 10⁸ Kelvin, at which matter exists as a plasma (fully ionised gas). These conditions exist in the cores of stars and in thermonuclear weapons (hydrogen bombs).

Controlled nuclear fusion for peaceful power generation is the subject of intensive global research. If achieved, it promises virtually limitless clean energy — the fuels (deuterium and lithium) are abundantly available, and the primary by-product (helium) is non-radioactive. Major experimental facilities like ITER (International Thermonuclear Experimental Reactor) in France are working toward achieving sustained fusion reactions.

Feature	Nuclear Fission	Nuclear Fusion
Process	Heavy nucleus splits	Light nuclei merge
Energy per reaction	~200 MeV (U-235)	~17.6 MeV (D-T)
Energy per kg of fuel	~8.2 × 10¹³ J	~3.4 × 10¹⁴ J (>4× fission)
Fuel	U-235, Pu-239 (rare)	D & T from hydrogen/lithium (abundant)
Radioactive waste	Long-lived fission fragments	Minimal (mainly He)
Condition required	Critical mass, slow neutrons	~10⁸ K plasma (very difficult)
Current status	Widely used in power plants	Experimental (ITER, NIF)

 

7. NUCLEAR ENERGY AND ITS APPLICATIONS

The extraction of nuclear energy — whether from fission or fusion — represents perhaps the most transformative technological application of modern physics. The energy yield from nuclear reactions dwarfs that of any chemical process by a factor of millions. Just one kilogram of Uranium-235 undergoing complete fission releases energy equivalent to burning approximately 3,000 tonnes of coal.

7.1 Nuclear Reactor

A nuclear reactor is a device in which a controlled, self-sustaining nuclear fission chain reaction is maintained to produce heat, which is then used to generate electricity via steam turbines. The essential components include: the fuel (enriched uranium or plutonium), the moderator (heavy water, graphite, or light water — used to slow neutrons to thermal speeds optimal for fission), control rods (boron or cadmium rods that absorb neutrons to regulate the reaction rate), the coolant (water, heavy water, or liquid sodium — removes heat from the reactor core), and the reactor vessel and biological shielding (to contain radiation).

7.2 Applications of Radioactivity

•       Radiocarbon Dating (¹⁴C): Carbon-14 is produced in the upper atmosphere and absorbed by living organisms. When an organism dies, ¹⁴C is no longer replenished and decays with a half-life of 5,730 years. By measuring the ratio of ¹⁴C to ¹²C, archaeologists can date organic materials up to ~50,000 years old with remarkable precision.

•       Medical Diagnosis (Nuclear Medicine): Radioisotopes such as Technetium-99m are used in PET (Positron Emission Tomography) and SPECT scans to image internal organs, detect tumours, and assess blood flow without invasive procedures.

•       Radiation Therapy: High-energy gamma rays from Cobalt-60 or focused beams of particles are used to destroy cancerous tumours with precision, minimising damage to surrounding healthy tissue.

•       Industrial Applications: Gamma radiography is used to detect flaws in metal castings, welds, and pipelines (non-destructive testing). Thickness gauges use beta particle absorption to control manufacturing tolerances.

•       Smoke Detectors: Americium-241 ionises air inside the detector. Smoke particles disrupt the ion current, triggering the alarm — a ubiquitous everyday application of radioactivity.

8. IMPORTANT FORMULAE AND QUICK REFERENCE

Quantity / Concept	Formula	Units / Remarks
Bohr radius (n=1)	a₀ = 0.529 Å	Radius of H ground state orbit
Radius of nth orbit	rₙ = n² a₀	H atom only
Energy of nth level (H)	Eₙ = −13.6/n² eV	Ground state: n=1, E₁=−13.6 eV
Nuclear radius	R = R₀ A^(1/3)	R₀ = 1.2 fm = 1.2×10⁻¹⁵ m
Mass defect	Δm = Zmₚ + Nmₙ − M	In unified atomic mass units (u)
Binding energy	Eᵦ = Δm × 931.5 MeV	Per nucleus; 1 u = 931.5 MeV/c²
Radioactive decay law	N(t) = N₀ e^(−λt)	λ = decay constant (s⁻¹)
Half-life	T½ = 0.693/λ	Time for half the nuclei to decay
Activity	A = λN	Unit: Becquerel (Bq) = 1 decay/s
Mass-energy equivalence	E = mc²	c = 3×10⁸ m/s
Energy-momentum relation	E² = (pc)² + (m₀c²)²	For relativistic particles
Rydberg formula	1/λ = R(1/nf² − 1/nᵢ²)	R = 1.097×10⁷ m⁻¹

 

9. SOLVED EXAMPLES

Example 1: Calculate the binding energy of ⁴₂He (Alpha Particle)

Given: Mass of ⁴₂He nucleus = 4.00150 u, mₚ = 1.00728 u, mₙ = 1.00866 u

Δm = 2(1.00728) + 2(1.00866) − 4.00150 = 2.01456 + 2.01732 − 4.00150 = 0.03038 u

Eᵦ = 0.03038 × 931.5 = 28.30 MeV

Binding energy per nucleon = 28.30 / 4 = 7.07 MeV/nucleon

Example 2: Half-Life Calculation

A radioactive sample has an initial activity of 4000 Bq. After 30 minutes, the activity is 500 Bq. Find the half-life.

A/A₀ = (1/2)^n  →  500/4000 = (1/2)^n  →  1/8 = (1/2)³  →  n = 3

Therefore, 3 half-lives = 30 min  →  T½ = 10 minutes

Example 3: Energy Released in Fission

If 1 gram of U-235 undergoes complete fission, each releasing 200 MeV, calculate the total energy released.

Number of atoms = (1/235) × 6.022×10²³ = 2.56×10²¹ atoms

Total energy = 2.56×10²¹ × 200 MeV = 5.12×10²³ MeV = 8.2×10¹⁰ J ≈ 82 GJ

10. CHAPTER SUMMARY AND KEY TAKEAWAYS

The chapter on Atoms and Nuclei represents one of the most intellectually profound journeys in all of physics — from the philosophical concept of the indivisible atom to the practical harnessing of nuclear energy that powers cities and enables life-saving medical technologies.

•       Atomic models evolved from Dalton (solid indivisible spheres) through Thomson (plum pudding), Rutherford (nuclear planetary model), and Bohr (quantised orbits) to the modern quantum mechanical orbital model. Each model corrected specific flaws of its predecessor.

•       The nucleus is an extraordinarily dense, positively charged core whose radius scales as R = R₀A^(1/3). It is held together by the short-range, charge-independent strong nuclear force, which overcomes Coulomb repulsion.

•       Mass defect (Δm) and binding energy (Eᵦ = Δmc²) explain the remarkable stability of nuclei. The binding energy per nucleon peaks at Iron-56, which dictates that both fission of heavy nuclei and fusion of light nuclei release energy.

•       Radioactive decay — alpha, beta, and gamma — are spontaneous, quantum mechanical processes following the exponential decay law N = N₀e^(−λt). The half-life is an intrinsic property of each nuclide.

•       Nuclear fission (splitting of U-235/Pu-239) releases ~200 MeV per event and sustains chain reactions used in nuclear reactors and weapons. Nuclear fusion (merging of H isotopes) releases even more energy per unit mass and powers the Sun.

•       Applications of nuclear physics permeate modern life: power generation, medical imaging and cancer therapy, archaeological dating, industrial non-destructive testing, and smoke detection.

 

"The energy available from nuclear reactions dwarfs that of all chemical reactions combined.

It is the energy that forges stars and, in our hands, transforms civilisation."

—  Class XII Physics | Atoms & Nuclei Chapter Analysis