Question

Consider one $H{e}^{+}$ ion in excited state $(n=5)$. Which of the following observations will hold true as per the Bohr’s model

• A. $10$ emission spectral lines will be seen
• B. The ionisation energy needed is less than $2eV$
• C. The longest emitted wavelength is less than $10/R$ (R = Rydberg’s constant)
• D. The electronic separation from the centre of nucleus is more than $6\mathring{A}$

Answer 1

Answer: (D)

A multitude of $H{e}^{+}$ will produce $10$ possible emission lines. A single $H{e}^{+}$ can produce maximum $4$ lines (not more)
$5\to 4\to 3\to 2\to 1$
$E_n=-13.6\frac{z^2}{n^2}=-13.6\times \frac{2^2}{5^2}$
$=-13.6\times \frac{4}{25}=-2.176eV$
$\therefore IE=+2.176eV>2eV$
$\frac{1}{\lambda }=R{Z}^2\left[\frac{1}{n_1^2}-\frac{1}{n_2^2}\right]$
$\frac{1}{\lambda }=R.4\left[\frac{1}{4^2}-\frac{1}{5^2}\right]$
$\frac{1}{\lambda }=4R\left[\frac{9}{16\times 25}\right]$
$\frac{1}{\lambda }=\frac{16\times 25}{4\times 9\times R}=\frac{100}{9R}>\frac{10}{R}$
$r_n=0.53\times \frac{n^2}{Z}=0.53\times \frac{5^2}{Z}$
$=6.625\mathring{A}>6^{\circ }A$