Atoms and Nuclei - Important Formulas Table

 Atoms and Nuclei - Important Formulas Table

S.No.	Concept / Quantity	Formula	Key Notes
1	Radius of nth Bohr Orbit (Hydrogen)	$r_n = 0.529 \times n^2$ Å	$r_1 = 0.529$ Å
2	Velocity of Electron in nth Orbit	$v_n = \frac{2.18 \times 10^6}{n}$ m/s	For Hydrogen
3	Energy of Electron in nth Orbit (Hydrogen)	$E_n = -\frac{13.6}{n^2}$ eV	Ground state = -13.6 eV
4	Energy Difference (Transition)	$\Delta E = 13.6 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ eV	Emission/absorption
5	Rydberg Formula (Wavelength)	$\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$	R = Rydberg constant
6	Rydberg Constant	$R = 1.097 \times 10^7$ m⁻¹	-
7	Frequency of Radiation	$\nu = \frac{\Delta E}{h}$	-
8	Bohr’s Quantization Condition	$mvr = \frac{nh}{2\pi}$	Angular momentum
9	Ionization Energy (Hydrogen)	13.6 eV	From n=1 to ∞
10	Excitation Energy (n=1 to n=2)	10.2 eV	-
11	Radius of Nucleus	$R = R_0 A^{1/3}$	$R_0 = 1.2 \times 10^{-15}$ m
12	Nuclear Volume	$V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi R_0^3 A$	-
13	Nuclear Density	$\rho = \frac{3m}{4\pi R_0^3} \approx 2.3 \times 10^{17}$ kg/m³	Constant for all nuclei
14	Mass Defect	$\Delta m = [Z m_p + (A-Z)m_n - M]$	-
15	Binding Energy	$BE = \Delta m \cdot c^2$	In MeV or Joule
16	Binding Energy per Nucleon	$BE/A$	Peak at Fe-56 (~8.79 MeV)
17	Einstein Mass-Energy Relation	$E = \Delta m \cdot c^2$	-
18	Radioactive Decay Law	$N = N_0 e^{-\lambda t}$	-
19	Activity / Decay Rate	$A = \lambda N = A_0 e^{-\lambda t}$	Unit: Becquerel (Bq)
20	Half-Life	$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$	-
21	Mean Life / Average Life	$\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{0.693}$	-
22	Relation between Half-life and Mean life	$T_{1/2} = 0.693 \tau$	-
23	Alpha Decay	${}_Z^A X \rightarrow {}_{Z-2}^{A-4} Y + {}_2^4 He$	-
24	Beta Minus Decay	${}_Z^A X \rightarrow {}_{Z+1}^A Y + e^- + \bar{\nu}$	-
25	Gamma Decay	${}_Z^A X^* \rightarrow {}_Z^A X + \gamma$	No change in A or Z
26	Number of Nuclei after n Half-lives	$N = N_0 \left( \frac{1}{2} \right)^n$	-
27	Energy Released in Nuclear Reaction	$Q = (\Sigma m_{\text{reactant}} - \Sigma m_{\text{product}}) c^2$	-
28	Fission vs Fusion Energy	Fusion releases more energy per nucleon for light nuclei	-
29	Bohr Magneton	$\mu_B = \frac{e h}{4\pi m_e} = 9.27 \times 10^{-24}$ J/T	-
30	de Broglie Wavelength in Bohr Orbit	$2\pi r_n = n \lambda$	-

Important Constants

• Rydberg constant $R = 1.097 \times 10^7$ m⁻¹
• Bohr radius $a_0 = 0.529$ Å
• $hc = 12400$ eV Å
• 1 u = 931.5 MeV/c²
• Avogadro number $N_A = 6.023 \times 10^{23}$

Note: This chapter is highly important for both theory (Bohr model, radioactivity) and numericals (energy calculation, half-life, binding energy) in Class 12 Board Exams, JEE Main, JEE Advanced, and NEET.