Nuclei formulas

Class 12 physics formula sheet for NEET and JEE: the key equations of NCERT chapter 13, the special cases questions are built on, and diagrams where they help.

34 formulas5 sectionsClass 12 · Chapter 132 of 5 sections free

By Sreeraj P, M.Sc Physics · 10+ years teaching NEET and JEE

Most used formulasOther formulas and cases

1Nucleus: composition and size

$$A=Z+N,\qquad R=R_0A^{1/3},\quad R_0\approx1.2\ \text{fm}$$

$$\begin{array}{l}\displaystyle A=Z+N\\[6pt]\displaystyle R=R_0A^{1/3},\quad R_0\approx1.2\ \text{fm}\end{array}$$

1 fm $=10^{-15}$ m. $R\propto A^{1/3}$, surface $\propto A^{2/3}$, volume $\propto A$. Nucleus ~$10^{-15}$ m, atom ~$10^{-10}$ m.

$$\rho=\frac{3m_u}{4\pi R_0^3}\approx2.3\times10^{17}\ \text{kg/m}^3$$

Nuclear density is the same for all nuclei (independent of $A$); ratio of densities of any two nuclei = 1.

Name	Same	Example
Isotopes	$Z$	¹H, ²H, ³H
Isobars	$A$	⁴⁰Ar, ⁴⁰Ca
Isotones	$N=A-Z$	³⁷Cl, ³⁹K
Isodiaphers	$N-Z=A-2Z$	²³Na, ²⁷Al
Isomers	$Z$ and $A$, different energy state	⁸⁰ᵐBr, ⁸⁰Br

$$d=\frac{1}{4\pi\varepsilon_0}\frac{qQ}{K}$$

Closest approach of a charge $q$ with KE $K$ fired at nucleus $Q$ (upper limit on nuclear size).

Nuclear force: strongest (relative $F_g:F_e:F_n\approx1:10^{36}:10^{38}$), short range (~2–3 fm; repulsive below ~0.8 fm), charge independent ($F_{pp}=F_{nn}=F_{pn}$), saturating, spin-dependent, non-central, exchange (pion) force.

2Mass–energy and binding energy

$$E=mc^2,\qquad 1\ \text{u}=1.6605\times10^{-27}\ \text{kg}=931.5\ \text{MeV}/c^2$$

$$\begin{array}{l}\displaystyle E=mc^2\\[6pt]\displaystyle 1\ \text{u}=1.6605\times10^{-27}\ \text{kg}=931.5\ \text{MeV}/c^2\end{array}$$

$m_p=1.00728$ u (938.3 MeV), $m_n=1.00866$ u (939.6 MeV), $m_e=0.00055$ u (0.511 MeV). 1 kg ≡ $9\times10^{16}$ J.

$$\begin{array}{l}\displaystyle \Delta m=Zm_p+(A-Z)m_n-M_{\text{nucleus}}\\[5pt]\displaystyle BE=\Delta m\,c^2=\Delta m(\text{u})\times931.5\ \text{MeV}\end{array}$$

With atomic masses use $Zm_H$ in place of $Zm_p$ (electron masses cancel). BE = energy needed to pull the nucleus apart into free nucleons.

$$b=\frac{BE}{A},\qquad f=\frac{M-A}{A}$$

$b$ = binding energy per nucleon, $f$ = packing fraction. Higher $BE/A$ → more stable. Packing fraction negative → stable; zero for ¹²C.

$BE/A$: ²H 1.1 MeV (lowest), ⁴He 7.07 (peak among light nuclei), ⁵⁶Fe ≈ 8.8 MeV (maximum), U ≈ 7.6 MeV. Nearly flat (~8.5) for $30<A<170$ → saturation of nuclear force.
Heavy nucleus splits (fission) or light nuclei join (fusion) → both move towards the peak → energy released.

$$\begin{array}{l}\displaystyle Q=(\textstyle\sum m_{\text{reactants}}-\sum m_{\text{products}})c^2\\[5pt]\displaystyle =\textstyle\sum BE_{\text{products}}-\sum BE_{\text{reactants}}\end{array}$$

$Q>0$ exoergic (energy released), $Q<0$ endoergic. With $BE/A$ values: $Q=\sum A_i b_i(\text{products})-\sum A_ib_i(\text{reactants})$. Conserved: $A$, charge, energy, momentum, angular momentum.

3 more sections and 21 formulas in the full chapter

3Radioactive decays6 formulas · 1 case table
4Decay law9 formulas · 1 case table · 1 diagram
5Fission and fusion6 formulas

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