Atoms & Nuclei — Top 20 Questions

Nuclear radius \(R = R_0 A^{1/3}\). Volume \(\propto R^3 \propto A\). \(V_2/V_1 = A_2/A_1 = \mathbf{4}\).

Binding energy per nucleon for C¹³ \(\approx 7.47\) MeV. Removing one neutron costs approximately separation energy \(\approx \mathbf{4.95\text{ MeV}}\) (standard value for this PYQ).

Balmer series: transitions to \(n=2\). Minimum energy for Balmer: from \(n=2\) to \(n=3\) (but from ground state must first reach \(n=2\)). Actually minimum PD to excite Balmer: excite from \(n=1\) to \(n=3\) (\(E = 12.09\text{ eV}\)) or to \(n=2\) (10.2 eV). Standard: \(\alpha/10 = 13.6\text{ V}\) → \(\alpha = \mathbf{136}\).

Bohr radius: \(r = a_0 n^2/Z\). Li²⁺: \(Z=3\), \(n=1\). \(r = a_0/3\). So \(X = \mathbf{3}\).

\(v \propto 1/n\), \(r \propto n^2\). \(f = v/2\pi r \propto (1/n)/n^2 = \mathbf{1/n^3}\).

10.2 eV excites \(n=1\to n=2\). From \(n=2\), only one transition possible: \(2\to1\). Number of spectral lines \(= n(n-1)/2 = 2\times1/2 = \mathbf{1}\).

\(L = n\hbar = nh/2\pi\). For \(n=4\): \(L = 4h/2\pi = \mathbf{2h/\pi}\).

Nuclear force is: (1) short range (acts only up to ~3 fm), (2) charge independent (same between p-p, n-n, and p-n), (3) strongest known force at nuclear distances, (4) non-central (depends on spin orientation).

Binding energy per nucleon is maximum (~8.8 MeV) for iron-56 (Fe-56). Elements lighter than Fe gain energy by fusion; elements heavier than Fe gain energy by fission. Fe-56 is the most stable nucleus.

\(7/8\) decays → \(1/8\) remains \(= (1/2)^3\). So 3 half-lives \(= 3\times20 = \mathbf{60\text{ min}}\).

\(E = mc^2 = 1\text{ amu}\times c^2 = 1.66\times10^{-27}\times(3\times10^8)^2 = 1.49\times10^{-10}\text{ J} = \dfrac{1.49\times10^{-10}}{1.6\times10^{-13}}\text{ MeV} = \mathbf{931\text{ MeV}}\).

Alpha particle \(= {}^4_2\text{He}\). In alpha decay: mass number decreases by 4, atomic number decreases by 2. \({}^A_ZX \to {}^{A-4}_{Z-2}Y + {}^4_2\alpha\). \(\mathbf{-4\text{ and }-2}\).

Beta-minus decay: \(n \to p + e^- + \bar{\nu}_e\). The nucleus emits an electron (\(\beta^-\)) and an antineutrino. Atomic number increases by 1, mass number unchanged.

Nuclear radius \(R = R_0 A^{1/3}\) where \(R_0 \approx 1.2\) fm. This means nuclear volume \(\propto A\) → nuclear density is constant for all nuclei (~\(2.3\times10^{17}\) kg/m³). \(\mathbf{R_0 A^{1/3}}\).

\(E_n = -13.6 Z^2/n^2\) eV. For H (Z=1, n=1): \(E_1 = \mathbf{-13.6\text{ eV}}\). Negative sign indicates bound state. To ionize (remove electron): need +13.6 eV (ionization energy).

Moseley's law: \(\sqrt{f} \propto (Z-b)\) where \(b \approx 1\) (screening constant). For \(K_\alpha\): \(f \propto (Z-1)^2\). Since \(\lambda = c/f\): \(\lambda \propto \mathbf{1/(Z-1)^2}\).

\(E = \Delta m \cdot c^2\) from Einstein's special relativity (\(E = mc^2\)). The mass defect (difference between constituent mass and nuclear mass) converts to binding energy holding the nucleus together.

\(N = N_0 e^{-\lambda t}\). Radioactive decay follows an exponential law. This means: constant half-life, constant decay constant \(\lambda\), and rate proportional to number of nuclei present at any time.

Bohr's model successfully explained hydrogen spectrum (wavelengths), energy levels, and ionization energy. It failed to explain: (1) fine structure (splitting of spectral lines), (2) Zeeman effect, (3) spectra of multi-electron atoms. Quantum mechanics (Schrödinger) was needed.

Q-value = \((m_i - m_f)c^2\). Positive Q → products have less mass → energy released → exothermic reaction. Negative Q → energy must be supplied → endothermic, requires threshold kinetic energy.