Atoms and Nuclei Formula Sheet
All key Atoms & Nuclei Physics formulas - Bohr model, hydrogen spectrum, mass defect, binding energy, radioactivity, fission & fusion for JEE quick revision.
Every must-know formula, constant, and decay rule for the Atoms and Nuclei chapter, grouped sub-topic by sub-topic for last-minute revision.
Rutherford Model and Alpha Scattering
Distance of closest approach (head-on collision, all KE converts to PE):
$$\boxed{r_0 = \frac{2kZe^2}{K} = \frac{2.88\,Z}{K(\text{MeV})}\text{ fm}}$$Impact parameter vs scattering angle:
$$\boxed{b = r_0 \cot\left(\frac{\theta}{2}\right) = \frac{k(2Ze^2)}{K}\cot\left(\frac{\theta}{2}\right)}$$Coulomb force on alpha particle ($+2e$) from nucleus ($+Ze$): $\;F = \dfrac{k(2e)(Ze)}{r^2} = \dfrac{2kZe^2}{r^2}$
Rutherford scattering formula:
$$\frac{dN}{d\Omega} = \frac{N_i\,t\,n}{16r^2}\left(\frac{2kZe^2}{K}\right)^2 \csc^4\left(\frac{\theta}{2}\right) \qquad N(\theta) \propto \frac{Z^2}{K^2 \sin^4(\theta/2)}$$| Quantity | Relation | Notes |
|---|---|---|
| Closest approach | $r_0 \propto Z/K$ and $\propto q^2$ | $q = 2e$ for $\alpha$, $e$ for proton |
| Impact parameter | $b = r_0\cot(\theta/2)$ | $b \to 0 \Rightarrow \theta \to 180^\circ$; $b\to\infty \Rightarrow \theta\to 0^\circ$ |
| Scattering count | $N \propto Z^2,\; \propto 1/K^2,\; \propto 1/\sin^4(\theta/2),\; \propto t$ | All verified experimentally |
| Nuclear density | $\sim 10^{17}$ kg/m³ | $r_{\text{atom}}/r_{\text{nucleus}} \sim 10^5$ |
Bohr Model (Hydrogen-like Atoms)
Postulate 1 - quantized angular momentum:
$$\boxed{L = mvr = \frac{nh}{2\pi} = n\hbar}$$Postulate 2 - transition energy: $\;E_{\text{photon}} = E_i - E_f = h\nu$
Radius of nth orbit:
$$\boxed{r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m Z e^2} = \frac{n^2 a_0}{Z} = 0.529\times\frac{n^2}{Z}\text{ Å}}$$Velocity in nth orbit:
$$\boxed{v_n = \frac{Ze^2}{2\varepsilon_0 nh} = \frac{Z}{n}\cdot\frac{c}{137} = 2.19\times10^6\times\frac{Z}{n}\text{ m/s}}$$Energy of nth level:
$$\boxed{E_n = -\frac{13.6\,Z^2}{n^2}\text{ eV}}$$n and Z Dependence
| Quantity | Formula (H-like) | n-dependence | Z-dependence |
|---|---|---|---|
| Radius | $r_n = n^2 a_0/Z$ | $\propto n^2$ | $\propto 1/Z$ |
| Velocity | $v_n = Zke^2/n\hbar$ | $\propto 1/n$ | $\propto Z$ |
| Energy | $E_n = -13.6Z^2/n^2$ eV | $\propto 1/n^2$ | $\propto Z^2$ |
| Period | $T_n = 2\pi r_n/v_n$ | $\propto n^3$ | $\propto 1/Z^2$ |
| Frequency | $f = 1/T$ | $\propto 1/n^3$ | $\propto Z^2$ |
| Kinetic energy | $K = -E_n$ | $\propto 1/n^2$ | $\propto Z^2$ |
| Potential energy | $U = 2E_n$ | $\propto 1/n^2$ | $\propto Z^2$ |
Virial theorem (energy split): $\;E = -K = \dfrac{U}{2}, \qquad K = \dfrac{ke^2}{2r}, \quad U = -\dfrac{ke^2}{r}$
Hydrogen Spectrum
Generalized Rydberg formula ($n_i > n_f$ for emission):
$$\boxed{\frac{1}{\lambda} = RZ^2\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)}, \qquad R = 1.097\times10^7\text{ m}^{-1}$$Photon energy form: $\;E_{\text{photon}} = 13.6\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$ eV
Series limit ($n_i = \infty$, shortest wavelength of a series):
$$\boxed{\lambda_{\text{limit}} = \frac{n_f^2}{R}}$$Number of spectral lines from level $n$ (all cascades down):
$$\boxed{N = \frac{n(n-1)}{2}}$$Spectral Series Reference
| Series | $n_f$ | Region | First line / limit |
|---|---|---|---|
| Lyman | 1 | UV | L$\alpha$ 121.6 nm; limit 91.2 nm |
| Balmer | 2 | Visible + near UV | H$\alpha$ 656.3 nm; limit 364.6 nm |
| Paschen | 3 | Near IR | Pa$\alpha$ 1875 nm; limit 820 nm |
| Brackett | 4 | IR | 4051 nm; limit 1458 nm |
| Pfund | 5 | Far IR | 7460 nm; limit 2279 nm |
Balmer (visible) lines: H$\alpha$ 656.3 nm (red), H$\beta$ 486.1 nm (cyan), H$\gamma$ 434.0 nm (blue), H$\delta$ 410.2 nm (violet).
Nuclear Structure and Mass-Energy
Nuclear notation: $^A_Z X_N$ with $\;N = A - Z$ (nucleons = protons + neutrons).
Nuclear radius (volume $\propto A$, so density is constant):
$$\boxed{R = R_0 A^{1/3}, \qquad R_0 = 1.2\text{ fm}}$$Nuclear density: $\;\rho = \dfrac{3 m_u}{4\pi R_0^3} \approx 2.3\times10^{17}\text{ kg/m}^3$ (same for all nuclei).
Radius ratio: $\;\dfrac{R_1}{R_2} = \left(\dfrac{A_1}{A_2}\right)^{1/3}$
Mass-energy unit:
$$\boxed{1\text{ u} = 1.66054\times10^{-27}\text{ kg} = 931.5\text{ MeV}/c^2}$$Mass defect (mass that becomes binding energy):
$$\boxed{\Delta m = [Z m_p + (A-Z)m_n] - M_{\text{nucleus}} = [Z m_p + N m_n + Z m_e] - M_{\text{atom}}}$$Particle Masses
| Particle | Mass (u) | Mass (MeV/$c^2$) | Charge |
|---|---|---|---|
| Proton | 1.007276 | 938.27 | $+e$ |
| Neutron | 1.008665 | 939.57 | 0 |
| Electron | 0.000549 | 0.511 | $-e$ |
Useful: $m_n > m_p > m_e$; $\;m_n - m_p \approx 1.3$ MeV/$c^2$; $\;m_p/m_e = 1836$; $\;m_n/m_e = 1839$.
Packing fraction: $\;f = \dfrac{M - A}{A}$ (negative $f$ $\Rightarrow$ more tightly bound).
Same-Family Nuclei
| Type | Same | Different | Example |
|---|---|---|---|
| Isotopes | $Z$ | $A, N$ | $^{12}$C, $^{14}$C |
| Isobars | $A$ | $Z, N$ | $^{14}$C, $^{14}$N |
| Isotones | $N$ | $Z, A$ | $^{14}$C, $^{15}$N |
| Isomers | $Z, A, N$ | energy state | $^{99m}$Tc |
Binding Energy
Binding energy (energy to fully disassemble the nucleus):
$$\boxed{BE = \Delta m\, c^2 = \Delta m(\text{u})\times 931.5\text{ MeV}}$$Binding energy per nucleon (stability measure):
$$\boxed{\frac{BE}{A}} \qquad \text{maximum at } ^{56}\text{Fe} \approx 8.8\text{ MeV/nucleon}$$Q-value via binding energies: $\;Q = BE_{\text{products}} - BE_{\text{reactants}}$
| Nucleus | $A$ | $BE/A$ (MeV) | Note |
|---|---|---|---|
| $^2$H | 2 | 1.11 | weakly bound |
| $^4$He | 4 | 7.07 | doubly magic, very stable |
| $^{12}$C | 12 | 7.68 | |
| $^{16}$O | 16 | 7.97 | |
| $^{56}$Fe | 56 | 8.790 | peak / most stable |
| $^{235}$U | 235 | 7.59 | fissile |
| $^{238}$U | 238 | 7.57 | fertile |
Semi-empirical (Weizsäcker) mass formula (concept only, not detailed for JEE):
$$BE = a_v A - a_s A^{2/3} - a_c\frac{Z^2}{A^{1/3}} - a_a\frac{(A-2Z)^2}{A} + \delta(A,Z)$$with $a_v \approx 15.8$, $a_s \approx 18.3$, $a_c \approx 0.714$, $a_a \approx 23.2$ MeV.
Radioactivity and Decay Laws
Decay law:
$$\boxed{N(t) = N_0\, e^{-\lambda t}}$$Activity (decay rate):
$$\boxed{A(t) = -\frac{dN}{dt} = \lambda N = A_0\, e^{-\lambda t}}$$Units: 1 Bq = 1 decay/s; $\;1$ Ci $= 3.7\times10^{10}$ Bq.
Half-life:
$$\boxed{T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}}$$After $n$ half-lives ($n = t/T_{1/2}$): $\;N = N_0\left(\dfrac{1}{2}\right)^n$
Mean life:
$$\boxed{\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln 2} = 1.44\,T_{1/2}}$$Carbon-14 dating ($T_{1/2} = 5730$ y): $\;t = \dfrac{T_{1/2}}{\ln 2}\ln\!\left(\dfrac{N_0}{N}\right) = \dfrac{T_{1/2}}{\ln 2}\ln\!\left(\dfrac{A_0}{A}\right)$
Multiple decay modes: $\;\lambda_{\text{eff}} = \lambda_1 + \lambda_2 + \cdots, \qquad T_{\text{eff}} = \dfrac{\ln 2}{\lambda_1 + \lambda_2}$
Secular equilibrium ($T_p \gg T_d$): $\;\lambda_p N_p = \lambda_d N_d$ (equal activities).
Decay Types
| Decay | Reaction | $\Delta A$ | $\Delta Z$ |
|---|---|---|---|
| $\alpha$ | $^A_Z X \to\ ^{A-4}_{Z-2}Y + \ ^4_2\text{He}$ | $-4$ | $-2$ |
| $\beta^-$ | $^A_Z X \to\ ^{A}_{Z+1}Y + e^- + \bar{\nu}_e$ | 0 | $+1$ |
| $\beta^+$ | $^A_Z X \to\ ^{A}_{Z-1}Y + e^+ + \nu_e$ | 0 | $-1$ |
| Electron capture | $^A_Z X + e^- \to\ ^{A}_{Z-1}Y + \nu_e$ | 0 | $-1$ |
| $\gamma$ | $^A_Z X^* \to\ ^A_Z X + \gamma$ | 0 | 0 |
Underlying nucleon conversions: $\;n \to p + e^- + \bar{\nu}_e$ (for $\beta^-$); $\;p \to n + e^+ + \nu_e$ (for $\beta^+$). $\beta^+$ requires $M_{\text{parent}} > M_{\text{daughter}} + 2m_e$.
Penetration: $\alpha < \beta < \gamma$. Ionization: $\alpha > \beta > \gamma$.
Nuclear Reactions, Fission and Fusion
Conservation laws: mass-energy, charge, nucleon number $A$, momentum, and angular momentum are all conserved (check both $A$ and $Z$).
Q-value:
$$\boxed{Q = (M_{\text{reactants}} - M_{\text{products}})c^2 = \Delta m(\text{u})\times 931.5\text{ MeV}}$$$Q > 0$ exothermic (energy released); $Q < 0$ endothermic (energy absorbed).
Reaction shorthand: $X(a,b)Y$, e.g. first artificial transmutation $^{14}_7\text{N} + \ ^4_2\text{He} \to\ ^{17}_8\text{O} + \ ^1_1\text{H}$, i.e. $^{14}\text{N}(\alpha,p)^{17}\text{O}$.
Fission
$$^{235}_{92}\text{U} + \ ^1_0 n \to\ ^{236}_{92}\text{U}^* \to\ ^{144}\text{Ba} + \ ^{89}\text{Kr} + 3n + \sim200\text{ MeV}$$Neutron multiplication factor: $\;k = \dfrac{\text{neutrons in generation } (n+1)}{\text{neutrons in generation } n}$
- $k < 1$ subcritical (dies out); $\;k = 1$ critical (steady); $\;k > 1$ supercritical (grows).
| Quantity | Value |
|---|---|
| Energy per U-235 fission | $\sim200$ MeV |
| Energy per kg U-235 | $8.2\times10^{13}$ J |
| Neutrons per fission | 2-3 (avg 2.5) |
| Critical mass U-235 (bare sphere) | $\sim52$ kg |
| Critical mass Pu-239 (bare sphere) | $\sim10$ kg |
Breeding: $\;^{238}\text{U} + n \to\ ^{239}\text{U} \to\ ^{239}\text{Np} + \beta^- \to\ ^{239}\text{Pu} + \beta^-$. Fissile: $^{233}$U, $^{235}$U, $^{239}$Pu. Fertile: $^{232}$Th, $^{238}$U.
Fusion
$$^2_1\text{H} + \ ^3_1\text{H} \to\ ^4_2\text{He}\,(3.5\text{ MeV}) + n\,(14.1\text{ MeV}) = 17.6\text{ MeV}$$Stellar proton-proton chain: $\;4(^1_1\text{H}) \to\ ^4_2\text{He} + 2e^+ + 2\nu_e + 26.7\text{ MeV}$
Coulomb barrier: $\;E_{\text{barrier}} = \dfrac{kZ_1 Z_2 e^2}{r}$ ($\approx 0.4$ MeV for D-T at contact).
Lawson criterion (net energy gain): $\;n\tau > 10^{20}$ s/m³.
| Fusion reaction | Q-value (MeV) |
|---|---|
| D + T $\to$ He + n | 17.6 |
| D + D $\to$ $^3$He + n | 3.27 |
| D + D $\to$ T + p | 4.03 |
| p + p $\to$ D + $e^+$ + $\nu$ | 0.42 |
Energy Conversions
| Energy | Equivalent |
|---|---|
| 1 kg matter ($E=mc^2$) | $9\times10^{16}$ J |
| 1 MeV | $1.602\times10^{-13}$ J |
| 1 kg TNT | $4.6\times10^6$ J |
| 1 kg U-235 fission | $\sim17$ kilotons TNT |
Key Constants at a Glance
| Constant | Symbol | Value |
|---|---|---|
| Bohr radius | $a_0$ | 0.529 Å $= 5.29\times10^{-11}$ m |
| Rydberg energy | $E_0$ | 13.6 eV $= 2.18\times10^{-18}$ J |
| Rydberg constant | $R$ | $1.097\times10^7$ m$^{-1}$ |
| Ground-state velocity | $v_1$ | $c/137 = 2.19\times10^6$ m/s |
| Reduced Planck constant | $\hbar$ | $1.055\times10^{-34}$ J·s |
| Fine-structure constant | $\alpha$ | $1/137 \approx 0.0073$ |
| Coulomb constant | $k$ | $9\times10^9$ N·m²/C² $= 1.44$ MeV·fm |
| Atomic mass unit | u | $1.66054\times10^{-27}$ kg $= 931.5$ MeV/$c^2$ |
| Nuclear radius constant | $R_0$ | 1.2 fm |
| $hc$ (handy) | - | 1240 eV·nm |