Important/Critical Points to Remember

1. The wavelength of limiting line \(=\frac{\mathrm{n}_{1}^{2}}{\mathrm{R}_{\mathrm{H}}}\) for all series. So, for lyman series \(=\frac{1}{\mathrm{R}_{\mathrm{H}}}\)

2. Spectral Series

(i) Lyman Series

Region

(ii) Balmer Series

UV

(iii) Paschen Series

(iv) Brackett Series \(\quad 4\)

(v) Pfund Series \(\quad 5\)

\(\begin{array}{llll}\text { (vi) Humphrey } & 6 & 7,8 \ldots & \text { Far IR }\end{array}\)

3. In terms of time and energy Heisenberg's uncertainty principle may be given as \(\Delta \mathrm{E} \Delta \mathrm{t} \geq \frac{\mathrm{h}}{4 \pi}\) (for energy and time)

4. \((\mathrm{n}+l)\) rule

This rule states that electrons are filled in orbitals according to their increasing values of \(n+\ell\). When \((\mathrm{n}+\ell)\) is same for sub energy levels, the electrons first occupy the sublevels with lowest "n" value. Thus, order of filling up of orbitals is as follows:

\(1 s<2 s<2 p<3 s<3 p<4 s<3 d<4 p<5 s<4 d<5 p<6 s<4 f<5 d\)

5. Orbital \& Spin Angular Momentum

\(\frac{\mathrm{h}}{2 \pi} \sqrt{\ell(\ell+1),} \frac{\mathrm{h}}{2 \pi} \sqrt{\mathrm{s}(\mathrm{s}+1)}\)

\(1 / 2 \mathrm{~m} v^{2}=\mathrm{h} v-\mathrm{h} v^{0}(\mathrm{w})\left(\right.\) work function or B.E.) ; \(\quad v^{0}=\) Threshold frequency \(\mathrm{W}=h v^{0}=\frac{h c}{\lambda_{0}}\)

1. Radii of Orbits

\section{Tips to Problem Solving}

\(r=\frac{\left(4 \pi \varepsilon_{0}\right) n^{2} h^{2}}{4 \pi^{2} \mathrm{mZe}^{2}}\)

By putting value of constants \(r=0.529 \times \frac{n^{2}}{Z} \stackrel{0}{\mathrm{~A}} ; r=0.529 \times \frac{n^{2}}{Z} \times 10^{-10} \mathrm{~m}\);

\(\mathrm{r}=0.529 \times 10^{-8} \times \frac{\mathrm{n}^{2}}{\mathrm{Z}} \mathrm{cm}\)

For H-like atoms. ; Thus \(r_{n}=r_{1} \times n^{2}\)

2. Speed of Electron

\(\mathrm{v}=\frac{2 \pi \mathrm{Ze} \mathrm{e}^{2}}{\left(4 \pi \varepsilon_{0}\right) \mathrm{nh}}=2.188 \times 10^{6} \times \frac{\mathrm{Z}}{\mathrm{n}} \mathrm{m} / \mathrm{sec}\)

\section{The Energy of Electron}

Total energy (E) = K.E. + P.E. \(E_{n}=-\frac{2 \pi^{2} Z^{2} m e^{4}}{\left(4 \pi \epsilon_{0}\right)^{2} n^{2} h^{2}}\)

where \(=\) orbit number

\(E_{n}=E_{1} \times \frac{Z^{2}}{n^{2}}\) for \(H\)-like atoms

\(\mathrm{E}_{\mathrm{n}}=-\frac{21.79 \times 10^{-19} \mathrm{Z}^{2}}{\mathrm{n}^{2}} \mathrm{~J} /\) atom \(=-\frac{13.6}{\mathrm{n}^{2}} \mathrm{Z}^{2}\) eV per atom

\(=-\frac{313.6 \times \mathrm{Z}^{2}}{\mathrm{n}^{2}} \mathrm{kcal} / \mathrm{mol}\)

4. Relation Between Potential energy (P.E), Kinetic energy

(K.E) \& Total energy

T.E. \(=\frac{\text { P.E. }}{2}=-\) K.E.

P.E. \(=-\frac{\mathrm{Ze}^{2}}{\mathrm{r}}\), K.E. \(=\frac{1}{2} \frac{\mathrm{Ze}^{2}}{\mathrm{r}}\), T.E. \(=-\frac{1}{2} \frac{\mathrm{Ze}^{2}}{\mathrm{r}}\)

\section{Rydberg Equation} Rydberg

The wavelength \((\lambda)\), wave number \((\bar{v})\) for the electromagnetic radiation can be calculated by equation.

\(\bar{v}=\frac{1}{\lambda}=\mathrm{R} \times \mathrm{Z}^{2}\left[\frac{1}{\mathrm{n}_{1}^{2}}-\frac{1}{\mathrm{n}_{2}^{2}}\right]\)

\(\mathrm{Z}=\) atomic number

\(\mathrm{R}=109677 \mathrm{~cm}^{-1}-\) Rydberg constant

\(\mathrm{n}_{2}=\) higher orbit

\(\mathrm{n}_{1}=\) lower orbit

5. Total number of spectral lines

(i) \(\frac{\mathrm{n}(\mathrm{n}-1)}{2} \rightarrow\) when electron jumps from nth level to ground level

(ii) \(\frac{\left(\mathrm{n}_{2}-\mathrm{n}_{1}\right)\left(\mathrm{n}_{2}-\mathrm{n}_{1}+1\right)}{2} \rightarrow\) when electron returns from \(\mathrm{n}_{2}\) to \(\mathrm{n}_{1}\).