Atoms formulas

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

38 formulas6 sectionsClass 12 · Chapter 122 of 6 sections free

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

Most used formulasOther formulas and cases

1Rutherford α-scattering

Most α pass undeviated → atom mostly empty. Few deflect by large angles, ~1 in 8000 by ≈ 180° → positive charge and almost all mass in a tiny nucleus (~$10^{-15}$ m vs atom ~$10^{-10}$ m).

$$r_0=\frac{1}{4\pi\varepsilon_0}\,\frac{2Ze^2}{K}=\frac{1}{4\pi\varepsilon_0}\,\frac{4Ze^2}{mv^2}$$

Distance of closest approach (head-on, $b=0$): all KE becomes PE. $r_0\propto Z/K$; for a general projectile of charge $q$: $r_0=\dfrac{kqZe}{K}$.

$$b=\frac{1}{4\pi\varepsilon_0}\,\frac{Ze^2\cot(\theta/2)}{K}=\frac{r_0}{2}\cot\frac\theta2$$

Impact parameter $b$ for scattering angle $\theta$. $b=0\Rightarrow\theta=180^\circ$; larger $b$ → smaller $\theta$.

$$N(\theta)\propto\frac{Z^2\,t}{K^2\sin^4(\theta/2)}$$

Number scattered at angle $\theta$: falls very steeply with $\theta$; $\propto$ foil thickness $t$, $\propto Z^2$, $\propto1/v^4$.

2Bohr model: orbit values

$$\frac{kZe^2}{r^2}=\frac{mv^2}{r},\qquad mvr=\frac{nh}{2\pi},\qquad h\nu=E_{n_2}-E_{n_1}$$

$$\begin{array}{l}\displaystyle \frac{kZe^2}{r^2}=\frac{mv^2}{r}\\[6pt]\displaystyle mvr=\frac{nh}{2\pi}\\[6pt]\displaystyle h\nu=E_{n_2}-E_{n_1}\end{array}$$

Postulates: Coulomb force gives centripetal force; angular momentum quantised; no radiation in stationary orbits, photon on a jump. ($k=1/4\pi\varepsilon_0$)

$$r_n=\frac{\varepsilon_0h^2}{\pi me^2}\frac{n^2}{Z}=0.529\,\frac{n^2}{Z}\ \text{Å}$$

Bohr radius $a_0=0.529$ Å. $r\propto n^2/Z$; $r\propto1/m$ (muonic atoms are smaller).

$$v_n=\frac{e^2}{2\varepsilon_0h}\frac Zn=\frac{c}{137}\frac Zn=2.19\times10^6\frac Zn\ \text{m/s}$$

Independent of $m$. $\dfrac{v_1}{c}=\alpha=\dfrac1{137}$ (fine-structure constant).

Quantity	∝	Quantity	∝
radius $r$	$n^2/Z$	speed $v$	$Z/n$
energy $E$, KE	$Z^2/n^2$	angular speed $\omega$, frequency $f$	$Z^2/n^3$
time period $T$	$n^3/Z^2$	momentum $p=mv$	$Z/n$
current $i=ef$	$Z^2/n^3$	field at nucleus $B=\mu_0i/2r$	$Z^3/n^5$
acceleration $v^2/r$, force	$Z^3/n^4$	angular momentum $L$	$n$ (only)

$$T_1=1.52\times10^{-16}\ \text{s},\quad f_1=6.6\times10^{15}\ \text{Hz},\quad i_1\approx1.05\ \text{mA}$$

$$\begin{array}{l}\displaystyle T_1=1.52\times10^{-16}\ \text{s}\\[6pt]\displaystyle f_1=6.6\times10^{15}\ \text{Hz}\\[6pt]\displaystyle i_1\approx1.05\ \text{mA}\end{array}$$

Values for H ground state. Orbit holds $n$ de Broglie waves: $2\pi r_n=n\lambda$, so $\lambda_n=2\pi a_0\,n/Z$.

$$\mu=iA=\frac{evr}{2}=n\,\frac{eh}{4\pi m},\qquad \frac{\mu}{L}=\frac{e}{2m}$$

Orbital magnetic moment; Bohr magneton $\mu_B=\dfrac{eh}{4\pi m}=9.27\times10^{-24}$ A m². Ratio $\mu/L$ is the same in every orbit.

4 more sections and 25 formulas in the full chapter

3Energy levels5 formulas · 1 case table · 1 diagram
4Spectrum of hydrogen6 formulas · 1 case table · 1 diagram
5Special cases9 formulas
6Discharge tube, e/m and Millikan5 formulas

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