Maharashtra State Board · Class 12 · Physics · Chapter 15

Structure of Atoms and Nuclei — Formula Sheet

Board Formulas

19 formulas

1.Bohr's Quantisation Condition
$L = m v r = \frac{nh}{2 π}, n = 1, 2, 3, \dots$
$m$ : Mass of electron = 9.1×10⁻³¹ kg · $v$ : Speed of electron in the orbit (m s⁻¹) · $r$ : Radius of the orbit (m) · $h$ : Planck's constant = 6.63×10⁻³⁴ J s
Bohr's second postulate: only those orbits are allowed in which the angular momentum of the electron is an integral multiple of h/2π. n is the principal quantum number.
2.Radius of the nth Bohr Orbit★
$r_{n} = \frac{ε _{0} h ^{2} n ^{2}}{π m Z e ^{2}}, r_{1} \approx 0.53 \times 1 0^{- 10} m (hydrogen)$
$ε_{0}$ : Permittivity of free space = 8.85×10⁻¹² C² N⁻¹ m⁻² · $Z$ : Atomic number (Z = 1 for hydrogen) · $e$ : Electronic charge = 1.6×10⁻¹⁹ C
r ∝ n²/Z. For hydrogen (Z = 1), r_n = 0.53 n² Å, so the 2nd and 3rd orbits are 4 and 9 times larger than the first.
3.Speed of the Electron in the nth Orbit
$v_{n} = \frac{Z e ^{2}}{2 ε _{0} nh}, v_{1} \approx 2.18 \times 1 0^{6} m s^{- 1} (hydrogen)$
v ∝ Z/n: the electron moves more slowly in outer orbits. Obtained by substituting r_n into v = nh/(2πmr).
4.Energy of the Electron in the nth Orbit★
$E_{n} = - \frac{m Z ^{2} e ^{4}}{8 ε _{0}^{2} h ^{2} n ^{2}} = - \frac{13.6 Z ^{2}}{n ^{2}} eV$
Negative because the electron is bound. E ∝ −Z²/n². The ionisation energy of hydrogen from the ground state is 13.6 eV.
5.Kinetic, Potential and Total Energy
$K = - E, U = 2 E, U = - 2 K$
In any Bohr orbit the kinetic energy equals the magnitude of the total energy, and the potential energy is twice the total energy. For hydrogen in the ground state: K = 13.6 eV, U = −27.2 eV, E = −13.6 eV.
6.Bohr's Frequency Condition
$h ν = E_{n_{2}} - E_{n_{1}} (n_{2} > n_{1})$
Bohr's third postulate: a photon is emitted when the electron jumps from a higher orbit n₂ to a lower orbit n₁. Absorption is the reverse jump.
7.Wavelength of a Spectral Line (Rydberg Formula)★
$\frac{1}{λ} = R Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}), R = \frac{m e ^{4}}{8 ε _{0}^{2} h ^{3} c} = 1.097 \times 1 0^{7} m^{- 1}$
$R$ : Rydberg constant = 1.097×10⁷ m⁻¹ · $n_{1}$ : Lower energy level · $n_{2}$ : Higher energy level (n₂ = n₁ + 1, n₁ + 2, …)
1/λ is the wave number (m⁻¹). n₁ is the lower level, n₂ the higher (the textbook writes the same formula as 1/λ = R_H Z²(1/n² − 1/m²), with n the lower and m the higher level). For hydrogen Z = 1. Series limit: put n₂ → ∞, giving 1/λ = R/n₁².
8.Spectral Series of Hydrogen
$Lyman: n_{1} = 1; Balmer: n_{1} = 2; Paschen: n_{1} = 3; Brackett: n_{1} = 4; Pfund: n_{1} = 5$
Lyman lies in the ultraviolet, Balmer in the visible, and Paschen, Brackett and Pfund in the infrared. The longest wavelength of a series uses n₂ = n₁ + 1.
9.Nuclear Radius
$R = R_{0} A^{1/3}, R_{0} \approx 1.2 \times 1 0^{- 15} m$
$R$ : Radius of the nucleus (m) · $A$ : Mass number
Nuclear volume is proportional to the mass number A. 1 fermi (fm) = 10⁻¹⁵ m.
10.Nuclear Density
$ρ = \frac{A m _{N}}{\frac{4}{3} π R _{0}^{3} A} = \frac{3 m _{N}}{4 π R _{0}^{3}} \approx 2.3 \times 1 0^{17} kg m^{- 3}$
A cancels, so all nuclei have roughly the same density, about 10¹⁴ times the density of water. m_N ≈ 1.67×10⁻²⁷ kg is the mass of a nucleon.
11.Mass Defect
$Δ m = [Z m_{p} + (A - Z) m_{n}] - M$
$m_{p}$ : Mass of proton = 1.007276 u · $m_{n}$ : Mass of neutron = 1.008665 u · $M$ : Mass of the nucleus (u)
The nucleus weighs less than its separated protons and neutrons. M is the nuclear mass. If atomic masses are given, use the mass of a hydrogen atom (m_H = 1.007825 u) in place of m_p so that the electron masses cancel.
12.Binding Energy★
$B . E . = Δ m c^{2}, 1 u \equiv 931.5 MeV$
Energy needed to separate the nucleus into its free nucleons (equally, the energy released when it forms). With Δm in u, B.E. in MeV = Δm × 931.5. 1 u = 1.66×10⁻²⁷ kg.
13.Binding Energy per Nucleon
$B ˉ = \frac{B . E .}{A}$
A measure of stability. It peaks at about 8.8 MeV near A ≈ 56 (iron) and is lower for very light and very heavy nuclei. This is why fusion of light nuclei and fission of heavy nuclei both release energy.
14.Law of Radioactive Decay★
$\frac{d N}{d t} = - λ N, N = N_{0} e^{- λ t}$
$N_{0}$ : Number of nuclei at t = 0 · $N$ : Number of undecayed nuclei at time t · $λ$ : Decay constant (s⁻¹)
The rate of decay is proportional to the number of undecayed nuclei present. Decay is random and is not affected by temperature, pressure or chemical state.
15.Half-Life★
$T_{1/2} = \frac{l n 2}{λ} = \frac{0.693}{λ}, N = N_{0} (\frac{1}{2})^{t / T_{1/2}}$
Time in which half of the nuclei present decay. After n half-lives, N = N₀/2ⁿ.
16.Mean Life
$τ = \frac{1}{λ} = \frac{T _{1/2}}{0.693} \approx 1.44 T_{1/2}$
Average lifetime of a nucleus. In one mean life the number of nuclei falls to 1/e (about 37%) of its initial value.
17.Activity
$A = \frac{d N}{d t} = λ N = A_{0} e^{- λ t}$
Number of decays per second. SI unit: becquerel (1 Bq = 1 decay s⁻¹). 1 curie (Ci) = 3.7×10¹⁰ Bq. Activity falls with the same half-life as N.
18.Alpha and Beta Decay
$_{Z}^{A} X \to_{Z - 2}^{A - 4} Y +_{2}^{4} He,_{Z}^{A} X \to_{Z + 1}^{A} Y + e^{-} + ν ˉ$
α decay lowers A by 4 and Z by 2. β⁻ decay keeps A, raises Z by 1 (a neutron becomes a proton) and emits an antineutrino. In β⁺ decay Z falls by 1 and a neutrino is emitted. γ decay changes neither A nor Z.
19.Energy Released in a Nuclear Reaction (Q-value)
$Q = (\sum m_{r e a c t an t s} - \sum m_{p r o d u c t s}) c^{2}$
Positive Q means energy is released (as in fission and fusion). Fission of one U-235 nucleus releases about 200 MeV.