Atoms and Nuclei revision notes

Alpha-Particle Scattering & Rutherford's Model

J.J. Thomson’s Plum Pudding Model: The first atomic model proposed that positive charge is uniformly distributed throughout the volume, with negatively charged electrons embedded in it like seeds in a watermelon. This was discarded as it could not explain the large-angle scattering of particles.

Geiger-Marsden Alpha Scattering Experiment: A beam of 5.5 MeV $\alpha$-particles from a $^{214}_{83}\text{Bi}$ radioactive source was directed at a thin gold foil (thickness $2.1 \times 10^{-7}$ m),. The scattered particles were observed using a rotatable zinc sulphide screen and microscope.

• Observations: Most $\alpha$-particles passed undeviated, meaning most of the atom is empty space,. Only ~0.14% scattered by more than $1^\circ$, and about 1 in 8000 deflected by $>90^\circ$.
• Rutherford's Conclusion: The entire positive charge and most of the mass are tightly concentrated at the center in a nucleus of size $10^{-15}$ m to $10^{-14}$ m (atom size is $\sim 10^{-10}$ m).
• Distance of Closest Approach ($d$): At closest approach, the $\alpha$-particle momentarily stops, and its initial kinetic energy ($K$) converts entirely into electrostatic potential energy,. $d = \frac{1}{4\pi\epsilon_0} \frac{2Ze^2}{K}$ → [JEE TIP] Trap - Closest Approach: The distance of closest approach is an upper limit for the nuclear radius. The $\alpha$-particle reverses direction without actually touching the nucleus because $d$ is larger than the sum of their radii,.
• Impact Parameter ($b$): The perpendicular distance of the initial velocity vector of the $\alpha$-particle from the central axis of the nucleus. → [JEE TIP] Trap - Head-on Collision: For a head-on collision, $b = minimum$ ($b \approx 0$), resulting in a scattering angle of $\theta \approx 180^\circ$ (rebounds back). For large $b$, the particle goes nearly undeviated ($\theta \approx 0^\circ$).

Bohr Model of the Hydrogen Atom

Niels Bohr combined classical mechanics with early quantum concepts to propose a new atomic model for hydrogenic (single-electron) atoms.

• Postulate 1 (Stationary States): An electron revolves in certain stable, allowed orbits without emitting radiant energy, contrary to classical electromagnetic theory.
• Postulate 2 (Quantization of Angular Momentum): The electron revolves only in orbits where its angular momentum ($L$) is an integral multiple of $h/2\pi$. $L = mvr = \frac{nh}{2\pi}$
• Postulate 3 (Energy Transitions): A photon is emitted or absorbed when an electron transitions between a higher energy state ($E_i$) and a lower energy state ($E_f$),. $h\nu = E_i - E_f$

Radius, Velocity, and Energy in the $n$-th Orbit: By equating electrostatic force with centripetal force ($\frac{1}{4\pi\epsilon_0}\frac{e^2}{r^2} = \frac{mv^2}{r}$) and applying Postulate 2, we get:

• Radius: $r_n = \left(\frac{\epsilon_0 h^2}{\pi m e^2}\right) n^2 = a_0 n^2$. (For Hydrogen-like ions: $r_n \propto \frac{n^2}{Z}$)
  • Bohr radius ($a_0$) for $n=1$ in Hydrogen is $5.3 \times 10^{-11}$ m.
• Velocity: $v_n = \frac{e^2}{2\epsilon_0 h} \cdot \frac{1}{n}$. (For Hydrogen-like ions: $v_n \propto \frac{Z}{n}$)
• Energy: Total energy $E = K + U = \frac{mv^2}{2} - \frac{e^2}{4\pi\epsilon_0 r} = -\frac{e^2}{8\pi\epsilon_0 r}$,. $E_n = -\frac{me^4}{8\epsilon_0^2 h^2} \left(\frac{1}{n^2}\right) = -\frac{13.6 \text{ eV}}{n^2}$ → [JEE TIP] Trap - Energy Relations: Kinetic energy $K = -E$ (positive), Potential energy $U = 2E$ (negative),. The negative sign signifies that the electron is bound to the nucleus,.
• Ionization Energy: The energy required to free the electron from the ground state ($n=1$). For Hydrogen, this is $13.6 \text{ eV}$.
• Reduced Mass Concept (JEE Advanced Topic): If the mass of the nucleus ($M$) is comparable to the electron mass ($m$), replace $m$ with the reduced mass $\mu = \frac{mM}{m+M}$. All energy levels scale proportionally with $\mu$.

Atomic Spectra & De Broglie's Explanation

When a rarefied gas is excited, it emits a discrete line spectrum,.

• Spectral Series: When transitioning from $n_2$ to $n_1$, the wavelength $\lambda$ is given by the Rydberg formula: $\frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ → [JEE TIP] Trap - Number of Spectral Lines: When a gas of atoms is excited to the $n$-th state, the maximum possible number of emission lines is $\frac{n(n-1)}{2}$. If it's a single atom, the maximum is $(n-1)$.

De Broglie's Explanation of Quantization: Louis de Broglie explained Bohr's second postulate by treating the orbiting electron as a particle wave. For an orbit to be stable, the wave must form a standing wave, meaning the orbit's circumference must be an integral multiple of the de Broglie wavelength ($\lambda = h/mv$),. $2\pi r_n = n\lambda = n\left(\frac{h}{mv}\right) \implies mvr_n = \frac{nh}{2\pi}$

X-Rays & Moseley's Law (JEE Advanced Topic)

When high-speed electrons strike a heavy metal target, X-rays are produced.

• Continuous X-rays (Bremsstrahlung): Produced due to deceleration of electrons. The minimum wavelength (cutoff) is $\lambda_{min} = \frac{hc}{eV}$, where $V$ is the accelerating voltage.
• Characteristic X-rays: Produced when an inner-shell electron is knocked out and an outer electron falls into the vacancy.
• Moseley's Law: The frequency ($\nu$) of characteristic X-rays relates to atomic number $Z$: $\sqrt{\nu} = a(Z - b)$ Here, $b$ is the screening constant ($b \approx 1$ for $K_\alpha$ lines).

Nucleus Composition & Size

• Atomic Mass Unit (u): Defined as $1/12$th of the mass of the $^{12}\text{C}$ atom. $1 \text{ u} = 1.660539 \times 10^{-27} \text{ kg} \approx 931.5 \text{ MeV/c}^2$,.
• Composition: A nucleus contains protons ($Z$, atomic number) and neutrons ($N = A - Z$, where $A$ is mass number). Protons and neutrons are collectively called nucleons.
• Discovery of Neutron: Discovered by James Chadwick (1932) by bombarding Beryllium with $\alpha$-particles. Applying conservation of energy and momentum showed the neutral radiation was particles, not photons. A free neutron is unstable (mean life ~ 1000 s) and decays into a proton, electron, and antineutrino, but inside a nucleus, it is stable. Mass of neutron $m_n = 1.00866 \text{ u}$.
• Size & Density: Nuclear radius follows the empirical formula $R = R_0 A^{1/3}$, where $R_0 = 1.2 \times 10^{-15} \text{ m} = 1.2 \text{ fm}$.
  • Nuclear volume $V \propto R^3 \propto A$,.
  • Nuclear density is roughly constant for all nuclei ($\approx 2.3 \times 10^{17} \text{ kg/m}^3$), independent of mass number $A$.

Mass-Energy Equivalence & Binding Energy

• Mass-Energy Equivalence: Einstein's relation $E = mc^2$ shows mass and energy are interchangeable.
• Mass Defect ($\Delta M$): The mass of a nucleus is always less than the sum of the masses of its constituent free nucleons,. $\Delta M = [Z m_p + (A-Z) m_n] - M_{nucleus}$
• Binding Energy ($E_b$): The energy equivalent to the mass defect, representing the energy required to break the nucleus into its individual constituent nucleons,. $E_b = \Delta M \cdot c^2 \quad \text{or} \quad E_b \text{ (in MeV)} = \Delta M \text{ (in u)} \times 931.5$
• Binding Energy per Nucleon ($E_{bn}$): $E_{bn} = E_b / A$. This is the true indicator of nuclear stability.
  • $E_{bn}$ is practically constant ($\sim 8.0 \text{ MeV}$) for intermediate nuclei ($30 < A < 170$).
  • The curve has a maximum of about $8.75 \text{ MeV}$ at $^{56}\text{Fe}$.
  • $E_{bn}$ is lower for light nuclei ($A<30$) and heavy nuclei ($A>170$), making them susceptible to fusion and fission respectively.

Nuclear Forces

The strong nuclear force binds protons and neutrons together against massive Coulombic repulsion.

• Key Properties:
  1. Much stronger than Coulomb and gravitational forces.
  2. Short-ranged: Falls rapidly to zero past a few femtometers.
  3. Charge-independent: $n-n$, $p-p$, and $n-p$ forces are approximately the same magnitude.
  4. Saturation Effect: A nucleon only interacts with its nearest neighbors. This explains why $E_{bn}$ is constant for intermediate nuclei.
  5. Attractive and Repulsive nature: The force is strongly repulsive for separation $r < 0.8 \text{ fm}$, and attractive for $r > 0.8 \text{ fm}$ (peaking at $0.8 \text{ fm}$ where potential energy is minimum),.

Radioactivity & Decay Laws

Discovered by A.H. Becquerel in 1896 by accidentally exposing a photographic plate to uranium-potassium sulphate.

• $\alpha$-decay: Emission of a helium nucleus ($^4_2\text{He}$).
  • → [JEE TIP] Kinematics of $\alpha$-decay: Using momentum conservation, the kinetic energy of the $\alpha$-particle is $K_\alpha = \frac{A-4}{A} Q$, where $A$ is the parent mass number.
• $\beta$-decay: Emission of electrons or positrons (particles with electron mass but opposite charge).
• $\gamma$-decay: High-energy photons emitted when an excited nucleus transitions to a lower energy state.
• Radioactive Decay Law (JEE Topics): The rate of decay is proportional to the number of active nuclei present. $\frac{dN}{dt} = -\lambda N \implies N(t) = N_0 e^{-\lambda t}$.
• Activity ($R$ or $A$): The number of disintegrations per second. Unit is Becquerel (1 Bq = 1 decay/sec).
• Half-Life ($T_{1/2}$): Time taken for half the active nuclei to decay. $T_{1/2} = \frac{\ln 2}{\lambda}$.
• Mean Life ($\tau$): Time at which the number of nuclei reduces to $1/e$ of its initial value. $\tau = \frac{1}{\lambda}$.

Nuclear Reactions: Fission & Fusion

• $Q$-value: The disintegration energy. $Q = ( \Sigma m_{initial} - \Sigma m_{final} ) c^2 = K_{final} - K_{initial}$.
• Nuclear Fission: A heavy nucleus ($A>170$) breaks into two intermediate-mass fragments.
  • Example: $^{235}_{92}\text{U} + ^1_0\text{n} \rightarrow \ ^{144}_{56}\text{Ba} + ^{89}_{36}\text{Kr} + 3 ^1_0\text{n}$.
  • Example: $^{235}_{92}\text{U} + ^1_0\text{n} \rightarrow \ ^{133}_{51}\text{Sb} + ^{99}_{41}\text{Nb} + 4 ^1_0\text{n}$.
  • Energy released is roughly $200 \text{ MeV}$ per fission of U-235,. The gain in binding energy per nucleon is roughly $0.9 \text{ MeV}$.
• Nuclear Fusion: Two light nuclei fuse to form a heavier, more tightly bound nucleus,.
  • Requires extremely high temperature ($T \sim 3 \times 10^9 \text{ K}$) to overcome the Coulomb repulsive barrier ($\sim 400 \text{ keV}$ for protons),.
  • Energy Generation in Stars: The sun's core ($1.5 \times 10^7 \text{ K}$) uses high-energy tail protons. The proton-proton (p-p) cycle fuses four hydrogen atoms into one helium atom: $4 ^1_1\text{H} \rightarrow \ ^4_2\text{He} + 2e^+ + 2\nu + 2\gamma + 26.7 \text{ MeV}$.

Key Concepts & Definitions

Impact Parameter (bbb):

The perpendicular distance of the initial velocity vector of an incoming α\alphaα-particle from the central axis of the target nucleus.

Stationary States:

Non-radiating, stable orbits allowed for electrons in the Bohr model.

Ionization Energy:

Energy required to remove an electron from its ground state to infinity (13.6 eV for Hydrogen).

Isotopes:

Nuclides with the same atomic number (ZZZ) but different neutron numbers (NNN). They have identical chemical behavior,,.

Isobars:

Nuclides with the same mass number (AAA), e.g., 13H^3_1\text{H}13​H and 23He^3_2\text{He}23​He.

Isotones:

Nuclides with the same neutron number (NNN), e.g., 80198Hg^{198}_{80}\text{Hg}80198​Hg and 79197Au^{197}_{79}\text{Au}79197​Au.

Conditions & Limitations

• Bohr Model Applicability: Strictly applicable ONLY to hydrogenic (single-electron) atoms/ions. It fails for multi-electron atoms because it cannot account for electrical forces (repulsion) between electrons,. It also cannot explain the relative intensities of specific spectral lines.
• Rutherford Scattering Formula: Assumes the heavy target nucleus remains perfectly stationary during the collision (gold is 50x heavier than $\alpha$),. Valid only for Coulomb interaction where the distance of closest approach is greater than the nuclear radius,.

Important Graphs & Diagrams

• Binding Energy per Nucleon ($E_{bn}$) vs Mass Number ($A$):
  • Shape: Starts low, rises steeply, shows local peaks at even-even nuclei indicative of shell structure, reaches a flat maximum (~8.75 MeV) near $A=56$ (Iron), and then slowly declines to ~7.6 MeV for heavy nuclei like Uranium.
  • Significance: The flat region ($30 < A < 170$) proves the short-range saturation of strong nuclear forces. The declining tails prove that fusion (light nuclei) and fission (heavy nuclei) are exothermic processes because they result in more tightly bound states,.
• Nuclear Potential Energy vs Nucleon Separation ($r$):
  • Shape: The potential energy is highly positive (repulsive) for $r < 0.8 \text{ fm}$, dips to a minimum (most stable, strongly attractive) at $r_0 \approx 0.8 \text{ fm}$, and asymptotes to zero for $r > \text{a few fm}$.
  • Significance: Visually demonstrates the short-range nature and the hard-core repulsion of the strong nuclear force,.
• Standing Wave in Bohr Orbit:
  • A standing particle wave formed on a circular orbit where $2\pi r = 4\lambda$ (for $n=4$) illustrates De Broglie's hypothesis that only resonant standing waves can persist, avoiding destructive interference.

Previous Year JEE Topics

• Bohr Model Ratios: Directly calculating proportions between $r, v, T,$ and $E$ across different orbits ($n$) and atoms ($Z$).
• Atomic Spectra Transitions: Identifying the series (Lyman, Balmer, etc.) and calculating max/min wavelengths for specific transitions.
• Alpha Decay Kinematics: Finding the recoil velocity of the daughter nucleus and the exact kinetic energy of the emitted $\alpha$-particle using momentum conservation.
• Binding Energy Calculations: Evaluating the $Q$-value of a given nuclear fission or fusion reaction by subtracting the total initial binding energy from the total final binding energy.
• Radioactivity Mathematics: Evaluating parallel decay paths, time taken for a specific fraction of a sample to decay, and instantaneous activity calculations.

Top 10 JEE MCQ Traps

• Trap 1 - Closest Approach:
  • Misconception $\rightarrow$ The distance of closest approach equals the radius of the target nucleus.
  • Correct Understanding $\rightarrow$ It is strictly an upper limit for the nuclear radius. The $\alpha$-particle reverses direction well before actually touching the nucleus because $d >$ sum of their radii,.
• Trap 2 - Nuclear Density Variation:
  • Misconception $\rightarrow$ Heavier nuclei have a much greater mass density than lighter nuclei.
  • Correct Understanding $\rightarrow$ Nuclear density is approximately constant ($\sim 2.3 \times 10^{17} \text{ kg/m}^3$) for all nuclei, independent of mass number $A$,.
• Trap 3 - Scope of Bohr Model:
  • Misconception $\rightarrow$ The Bohr model correctly predicts energy states for all atoms.
  • Correct Understanding $\rightarrow$ The Bohr model strictly applies ONLY to hydrogenic (single-electron) atoms/ions like H, He$^+$, Li$^{2+}$,. It fails when electron-electron repulsion is introduced,.
• Trap 4 - Orbital vs Spectral Frequency:
  • Misconception $\rightarrow$ The frequency of the emitted spectral line equals the frequency of the revolving electron.
  • Correct Understanding $\rightarrow$ The emitted photon frequency is derived from the energy difference divided by $h$ ($\nu = \Delta E/h$) and does not equal the mechanical frequency, except as an approximation at very high quantum numbers (Correspondence Principle).
• Trap 5 - Sign of Binding Energy & Mass Defect:
  • Misconception $\rightarrow$ The mass of a nucleus is equal to the sum of the masses of its constituent protons and neutrons.
  • Correct Understanding $\rightarrow$ The nuclear mass is always less than the sum of its constituents. The difference (mass defect) provides the positive binding energy holding the nucleus together.
• Trap 6 - Stability and Total Binding Energy:
  • Misconception $\rightarrow$ A nucleus with a higher total binding energy is always more stable.
  • Correct Understanding $\rightarrow$ It is the Binding Energy per Nucleon ($E_{bn}$), not total BE, that determines true nuclear stability.
• Trap 7 - Charge Dependence of Nuclear Force:
  • Misconception $\rightarrow$ The strong nuclear force depends on the electrical charges of the nucleons.
  • Correct Understanding $\rightarrow$ The strong nuclear force is perfectly charge-independent; the $n-n$, $p-p$, and $n-p$ strong forces are roughly identical in magnitude.
• Trap 8 - Free vs Bound Neutrons:
  • Misconception $\rightarrow$ Neutrons are eternally stable particles everywhere.
  • Correct Understanding $\rightarrow$ A free neutron is actually unstable (mean life ~1000s) and decays into a proton, electron, and antineutrino. It is only completely stable when bound inside a nucleus.
• Trap 9 - Number of Expected Fission Products:
  • Misconception $\rightarrow$ Fission of U-235 always produces the exact same daughter nuclei (e.g., Ba and Kr).
  • Correct Understanding $\rightarrow$ Fission can produce many different pairs of intermediate mass fragments (e.g., Ba+Kr, Sb+Nb, Xe+Sr),.
• Trap 10 - Conservation Laws in Reactions:
  • Misconception $\rightarrow$ Mass and energy are conserved entirely separately in nuclear reactions.
  • Correct Understanding $\rightarrow$ Mass and energy are not separately conserved; rather, mass-energy interconversion occurs. The total "mass-energy" is what is strictly conserved ($Q$-value logic),.