CBSE · Class 12 · Physics · Chapter 11

Dual Nature of Radiation and Matter / Atoms & Nuclei — Formula Sheet

Board Formulas

14 formulas

1.Photon Energy★
$E = h ν = \frac{h c}{λ}$
$h$ : Planck's constant = 6.626×10⁻³⁴ J·s · $ν$ : Frequency (Hz) · $λ$ : Wavelength (m) · $c$ : Speed of light = 3×10⁸ m/s
Each photon carries a fixed quantum of energy proportional to its frequency.
2.Einstein's Photoelectric Equation★
$K E_{ma x} = h ν - ϕ = h (ν - ν_{0})$
$ϕ$ : Work function of metal (J or eV) · $ν_{0}$ : Threshold frequency (Hz)
Photon energy = work function + kinetic energy of ejected electron. Explains threshold frequency.
3.Stopping Potential
$e V_{0} = K E_{ma x} = h (ν - ν_{0})$
Reverse potential that just stops the fastest photoelectrons. Independent of intensity.
4.de Broglie Wavelength★
$λ = \frac{h}{p} = \frac{h}{m v}$
Every moving particle has a matter wave. Wavelength inversely proportional to momentum.
5.de Broglie Wavelength of Electron via Voltage
$λ = \frac{h}{2 m e V} \approx \frac{1.227}{V} nm$
Convenient form for electron accelerated through potential V (in volts).
6.Bohr's Quantisation of Angular Momentum★
$m v r = \frac{nh}{2 π}$
Angular momentum in nth orbit is an integer multiple of ħ. Bohr's second postulate.
7.Radius of nth Bohr Orbit
$r_{n} = \frac{n ^{2} h ^{2} ε _{0}}{π m e ^{2}} = 0.529 n^{2} A ˚$
First Bohr radius (n = 1) ≈ 0.529 Å = 5.29×10⁻¹¹ m.
8.Energy of nth Bohr Level★
$E_{n} = - \frac{13.6}{n ^{2}} eV$
Negative because electron is bound. Ground state (n = 1) has E = −13.6 eV.
9.Photon Frequency for Transition
$h ν = E_{n_{2}} - E_{n_{1}}$
Emission when n_2 > n_1; absorption otherwise.
10.Rydberg Formula
$\frac{1}{λ} = R (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$
Wavelength of spectral line in hydrogen. R = 1.097×10⁷ m⁻¹. Lyman: n_1=1, Balmer: n_1=2, Paschen: n_1=3.
11.Mass–Energy Equivalence
$E = m c^{2}$
Rest-mass energy. 1 u ≈ 931.5 MeV.
12.Radioactive Decay Law
$N = N_{0} e^{- λ t}$
Number of surviving nuclei decreases exponentially. λ is decay constant, not wavelength.
13.Half-Life and Mean Life
$T_{1/2} = \frac{l n 2}{λ} = \frac{0.693}{λ}, τ = \frac{1}{λ}$
T_(1/2) < τ. Ratio: T_(1/2) = 0.693 τ.
14.Activity
$A = λ N = A_{0} e^{- λ t}$
Number of disintegrations per second. SI unit: becquerel (Bq). 1 Ci = 3.7×10¹⁰ Bq.