Atoms and Nuclei — Revision Notes

⚡ 30-Second Revision

Bohr's Radius: — $r_n = a_0 \frac{n^2}{Z}$ where a_0 = 0.529,\text{Å}}
Bohr's Energy: — $E_n = -13.6 \frac{Z^2}{n^2},\text{eV}$
Rydberg Formula: — rac{1}{lambda} = R Z^2 left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)
Radioactive Decay Law: — $N(t) = N_0 e^{-lambda t}$
Half-life: — $T_{1/2} = \frac{ln 2}{lambda} = \frac{0.693}{lambda}$
Mean life: — $au = \frac{1}{lambda}$
Activity: — $A = lambda N$
Mass Defect: — $Delta m = (Z m_p + N m_n) - M_{nucleus}$
Binding Energy: — $E_b = Delta m c^2$ (or $Delta m \times 931.5,\text{MeV}$ if $Delta m$ is in u)
Nuclear Radius: — $R = R_0 A^{1/3}$ where $R_0 approx 1.2 \times 10^{-15},\text{m}$

2-Minute Revision

Atoms consist of a nucleus (protons and neutrons) and orbiting electrons. Rutherford's model established the nucleus, while Bohr's model quantized electron orbits, explaining atomic stability and discrete spectra for hydrogen.

Remember Bohr's formulas for radius ($r_n propto n^2/Z$) and energy ($E_n propto -Z^2/n^2$), and the Rydberg formula for spectral lines (Lyman, Balmer, Paschen series). The nucleus is held by the strong nuclear force, which is short-range and charge-independent.

Mass defect ($Delta m$) leads to binding energy ($E_b = Delta m c^2$), indicating nuclear stability; higher binding energy per nucleon means greater stability. Unstable nuclei undergo radioactive decay: $alpha$ (Z-2, A-4), $\beta^-$ (Z+1, A), $\beta^+$ (Z-1, A), and $gamma$ (no change).

Key decay concepts are half-life ($T_{1/2} = 0.693/lambda$), mean life ($au = 1/lambda$), and activity ($A = lambda N$). Nuclear fission splits heavy nuclei, releasing energy, while nuclear fusion combines light nuclei, releasing even more energy.

Both are governed by mass-energy equivalence.

5-Minute Revision

Start with atomic structure: Rutherford's gold foil experiment proved the existence of a tiny, dense nucleus. Bohr's model, though classical, introduced quantum concepts for hydrogen: electrons orbit in discrete energy levels ($E_n = -13.

6 Z^2/n^2, ext{eV}$) and radii ($r_n = a_0 n^2/Z$), and emit/absorb photons during transitions, explaining spectral series. Remember the order of series (Lyman-UV, Balmer-Visible, Paschen-IR) and their final energy levels.

Move to the nucleus: it's composed of protons and neutrons (nucleons) bound by the strong nuclear force, which is the strongest, short-range, and charge-independent. The mass of a nucleus is less than its constituent nucleons' sum (mass defect, $Delta m$), converted to binding energy ($E_b = Delta m c^2$).

Binding energy per nucleon indicates stability, peaking around Iron. Radioactive decay is the spontaneous disintegration of unstable nuclei. Know the changes in Z and A for $alpha$ (Z-2, A-4), $\beta^-$ (Z+1, A), $\beta^+$ (Z-1, A), and $gamma$ (no change).

Master the decay law $N = N_0 e^{-lambda t}$, half-life $T_{1/2} = 0.693/lambda$, mean life $au = 1/lambda$, and activity $A = lambda N$. Practice problems involving multiple half-lives. Finally, understand nuclear reactions: fission (heavy nucleus splits, e.

g., Uranium) and fusion (light nuclei combine, e.g., Deuterium-Tritium). Both release immense energy due to mass conversion, but fusion requires extreme conditions and is the energy source of stars. Be ready for numerical problems on all these topics, paying attention to units and calculations.

Prelims Revision Notes

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Atomic Models:

* Rutherford's Model: Nucleus (positive, dense, small) at center, electrons orbit. Failed to explain atomic stability and discrete spectra. * Bohr's Model (Hydrogen-like atoms): * Electrons in stationary orbits (no radiation).

* Angular momentum quantized: $L = mvr = n\frac{h}{2pi}$. * Energy emitted/absorbed during transitions: $h u = E_i - E_f$. * Radius: $r_n = \frac{n^2 h^2 epsilon_0}{pi m Z e^2} = a_0 \frac{n^2}{Z}$ ($a_0 = 0.

529, ext{Å}}$). * **Energy:**$E_n = -rac{m Z^2 e^4}{8 epsilon_0^2 n^2 h^2} = -13.6 rac{Z^2}{n^2}, ext{eV}$. * **Velocity:**$v_n propto Z/n$. * **Spectral Series (Hydrogen,$Z=1$):** * Lyman:$n_f=1$,$n_i=2,3,dots$(UV) * Balmer:$n_f=2$,$n_i=3,4,dots$(Visible) * Paschen:$n_f=3$,$n_i=4,5,dots$(IR) * Rydberg formula:$rac{1}{lambda} = R Z^2 left(rac{1}{n_f^2} - rac{1}{n_i^2} ight)$.

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Nuclear Structure:

* Nucleus: Protons (Z) + Neutrons (N) = Nucleons (A). * Isotopes: Same Z, different A (different N). * Isobars: Same A, different Z. * Isotones: Same N, different Z and A. * Nuclear Size: $R = R_0 A^{1/3}$ ($R_0 approx 1.2 \times 10^{-15},\text{m}$). Nuclear density is constant. * Strong Nuclear Force: Strongest, short-range ($10^{-15},\text{m}$), charge-independent, saturating, spin-dependent.

1
Mass Defect & Binding Energy:

* **Mass Defect ($Delta m$):** $Delta m = (Z m_p + N m_n) - M_{nucleus}$. * **Binding Energy ($E_b$):** $E_b = Delta m c^2$. (1 u = 931.5 MeV/c$^2$). * **Binding Energy per Nucleon ($E_b/A$):** Measure of stability. Peaks around $A=56$ (Fe).

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Radioactivity:

* Decay Law: $N(t) = N_0 e^{-lambda t}$. * **Half-life ($T_{1/2}$):** Time for half nuclei to decay. $T_{1/2} = \frac{ln 2}{lambda} = \frac{0.693}{lambda}$. * **Mean life ($au$):** Average lifetime.

$au = 1/lambda$. Note: $T_{1/2} = \tau ln 2$. * **Activity ($A$):** Rate of decay. $A = |\frac{dN}{dt}| = lambda N = A_0 e^{-lambda t}$. Units: Bq, Ci. * Types of Decay: * $alpha$-decay ($^4_2\text{He}$): $Delta Z = -2, Delta A = -4$.

* $\beta^-$-decay ($e^-$): $Delta Z = +1, Delta A = 0$ (neutron $o$ proton). * $\beta^+$-decay ($e^+$): $Delta Z = -1, Delta A = 0$ (proton $o$ neutron). * $gamma$-decay (photon): $Delta Z = 0, Delta A = 0$ (nuclear de-excitation).

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Nuclear Reactions:

* Fission: Heavy nucleus splits into lighter ones + neutrons + energy. Basis of nuclear reactors. * Fusion: Light nuclei combine to form heavier one + energy. Powers stars, requires high T and P.

Vyyuha Quick Recall

For the Hydrogen Spectral Series: Lovely Boys Play Baseball Professionally.

Lyman ($n_f=1$) - Ultraviolet
Balmer ($n_f=2$) - Visible
Paschen ($n_f=3$) - Infrared
Brackett ($n_f=4$) - Infrared
Pfund ($n_f=5$) - Infrared