Question

Consider one $He^{+}$ 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 $He^{+}$ will produce $10$ possible emission lines. A single $He^{+}$ can produce maximum $4$ lines (not more)
$5\rightarrow4\rightarrow3\rightarrow2\rightarrow1$
$E_{n}=-13.6 \frac{z^{2}}{n^{2}}=-13.6 \times\frac{2^{2}}{5^{2}}$
$=-13.6 \times\frac{4}{25} =-2.176 eV $
$\therefore IE =+2.176 eV>2eV $
$\frac{1}{\lambda}=RZ^{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] $
$\frac{1}{\lambda}=4R \left[\frac{9}{16\times25}\right]$
$\frac{1}{\lambda}=\frac{16\times25}{4\times9\times R}=\frac{100}{9R}>\frac{10}{R} $
$r_{n}=0.53\times\frac{n^{2}}{Z}=0.53\times\frac{5^{2}}{Z} $
$=6.625 \mathring{A} > 6^{\circ} A$