Nuclei — JEE Mains Physics

Formulas (2)

• Nuclear radius and constant density · Nuclear radius
  $$R = R_0A^{1/3}, \qquad \rho = \frac{3m}{4\pi R_0^{3}}\ \text{(same for all A)}$$
• Mass defect and binding energy · Binding energy
  $$BE = \left[Zm_p + (A - Z)m_n - M\right]c^{2}, \qquad 1\ \text{u}\,c^{2} = 931.5\ \text{MeV}$$

Reference tables (1)

Watch out for (9)

• Any statement that orders nuclear densities is false
  → Nuclear radius and constant density
• Cube the radius ratio, do not cube-root it twice
  → Nuclear radius and constant density
• Absorbed electrons do not change A
  → Nuclear radius and constant density
• Atom mass or nucleus mass
  → Mass defect and binding energy
• Bound mass is the smaller one
  → Mass defect and binding energy
• Per nucleon or in total
  → Mass defect and binding energy
• Heavier is not always more tightly bound
  → The binding-energy curve and the nuclear force
• Isobars share A, isotopes share Z
  → The binding-energy curve and the nuclear force
• Stability is judged per nucleon
  → The binding-energy curve and the nuclear force

Formulas (3)

• Q from binding energies per nucleon · Q-value from binding energy
  $$Q = \sum_{\text{products}} A\,b - \sum_{\text{reactants}} A\,b$$
• Q from masses, and how it is shared · Q-value from masses; alpha's share
  $$Q = \left(\sum m_i - \sum m_f\right)c^{2}, \qquad K_\alpha = Q\,\frac{A - 4}{A}$$
• Energy from a sample, and power · Energy from a sample
  $$E = \frac{m}{M}N_A\,Q, \qquad \text{rate} = \frac{P}{Q}$$

Watch out for (9)

• Multiply by A before subtracting
  → Q from binding energies per nucleon
• Products minus reactants, for binding energy
  → Q from binding energies per nucleon
• Count every product nucleus
  → Q from binding energies per nucleon
• The alpha does not get all of Q
  → Q from masses, and how it is shared
• Before minus after, for masses
  → Q from masses, and how it is shared
• Keep the electrons balanced
  → Q from masses, and how it is shared
• Grams over grams per mole
  → Energy from a sample, and power
• MeV is not joules
  → Energy from a sample, and power
• Several nuclei per reaction
  → Energy from a sample, and power

Formulas (1)

• Counting half-lives · Half-life law
  $$N = N_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}$$

Reference tables (1)

Watch out for (6)

• Count the alphas first
  → Alpha, beta and gamma decay: what changes
• Beta-minus raises Z
  → Alpha, beta and gamma decay: what changes
• Neutrino or antineutrino
  → Alpha, beta and gamma decay: what changes
• Decayed is not left
  → Counting half-lives
• Halve, do not subtract halves
  → Counting half-lives
• Find n from a ratio, then the time
  → Counting half-lives

Formulas (2)

• Decay constant, mean life and activity · Decay law
  $$N = N_0e^{-\lambda t}, \qquad T_{1/2} = \frac{\ln 2}{\lambda}, \qquad \tau = \frac{1}{\lambda}, \qquad A = \lambda N$$
• Two isotopes, two routes and a decay chain · Two routes add their decay constants
  $$\lambda = \lambda_1 + \lambda_2, \qquad T = \frac{T_1T_2}{T_1 + T_2}$$

Watch out for (6)

• λ must be in per second for becquerel
  → Decay constant, mean life and activity
• Mean life is longer than half-life
  → Decay constant, mean life and activity
• Decay ignores conditions
  → Decay constant, mean life and activity
• Add decay constants, not half-lives
  → Two isotopes, two routes and a decay chain
• Equal masses are not equal numbers
  → Two isotopes, two routes and a decay chain
• Read where B starts
  → Two isotopes, two routes and a decay chain