Question

The spectral series of $H-$ atom gives ${\overline{v}}_{\text{min}}$ and ${\overline{v}}_{\text{max}}$ depending on the types of transition. Which of the following is/are true?

• A. $\begin{array}{ccc}\text{Series name}& {\overline{v}}_{\text{max}}& {\overline{v}}_{\text{min}}\\ \text{Lyman}& {\overline{R}}_H& \frac{3}{4}{\overline{R}}_H\end{array}$
• B. $\begin{array}{ccc}\text{Series name}& {\overline{v}}_{\text{max}}& {\overline{v}}_{\text{min}}\\ \text{Balmer}& \frac{{\overline{R}}_H}{4}& \frac{5}{36}{\overline{R}}_H\end{array}$
• C. $\begin{array}{ccc}\text{Series name}& {\overline{v}}_{\text{max}}& {\overline{v}}_{\text{min}}\\ \text{Paschen}& \frac{{\overline{R}}_H}{9}& \frac{7}{144}{\overline{R}}_H\end{array}$
• D. $\begin{array}{ccc}\text{Series name}& {\overline{v}}_{\text{max}}& {\overline{v}}_{\text{min}}\\ \text{Brackett}& \frac{{\overline{R}}_H}{16}& \frac{9}{400}{\overline{R}}_H\end{array}$

Answer 1

Answer: (A), (B), (C), (D)

$\begin{array}{ccc}\text{Series name}& {\overline{v}}_{\text{max}}& {\overline{v}}_{\text{min}}\\ \text{Brackett}& \frac{{\overline{R}}_H}{16}& \frac{9}{400}{\overline{R}}_H\end{array}$

$\overline{v}={\overline{R}}_H[\frac{1}{n_1^2}-\frac{1}{n_2^2}]$
${\overline{v}}_{\text{max}}=\frac{{\overline{R}}_H}{n_1^2}$ (when $n_2\to \infty$)
$={\overline{R}}_H$ for $n_1=1\text{(Lyman)}=\frac{R_H}{4}$ for $n_1=2\text{(Balmer)}$
$=\frac{{\overline{R}}_H}{9}$ for $n_1=3\text{(Paschen)}=\frac{R_H}{16}$ for $n_1=4\text{(Brackett)}$
$\overline{v_{\text{min}}}$ when $n_2=(n_1+1)$
Thus, ${\overline{v}}_{\text{min}}=\frac{3}{4}{\overline{R}}_H\text{for Lyman}$
$=\frac{5}{36}{\overline{R}}_H\text{for Balmer}$
$=\frac{7}{144}{\overline{R}}_H\text{for Paschen}$
$=\frac{9}{400}{\overline{R}}_H\text{for Brackett}$