Hydrogen \(({ }_1 \mathrm{H}^1)\), Deuterium \(({ }_1 \mathrm{H}^2)\), singly ionised Helium \(({ }_2 \mathrm{He}^4)^+\) and doubly ionised lithium \(({ }_3 \mathrm{Li}^6)^{++}\) all have one electron around the nucleus. Consider and electron transition from \(n=2\) to \(n=1\). If the wavelengths of emitted radiation are \(\lambda_1,\lambda_2,\lambda_3\) and \(\lambda_4\) respectively then approximately which one of the following is correct?
1. \( \lambda_1=2 \lambda_2=2 \lambda_3=\lambda_4 \)
2. \( \lambda_1=\lambda_2=4 \lambda_3=9 \lambda_4 \)
3. \( \lambda_1=2 \lambda_2=3 \lambda_3=4 \lambda_4 \)
4. \( 4 \lambda_1=2 \lambda_2=2 \lambda_3=\lambda_4\)

Some energy levels of a molecule are shown in the figure. The ratio of the wavelength \(r=\frac{\lambda_1}{\lambda_2}\), is given by:

                
1. \( r=\frac{4}{3} \)
2. \( r=\frac{2}{3} \)
3. \( r=\frac{3}{4} \)
4. \( r=\frac{1}{3}\)

An electron from various excited states of hydrogen atom emit radiation to come to the ground state. Let \(\lambda_n,\lambda_g\) be the de Broglie wavelength of the electron in the \(n^{\text{th}}\) state and the ground state respectively. Let \(\Lambda_n\) be the wavelength of the emitted photon in the transition from the \(n^{\text{th}}\) state to the ground state. For large \(n\), (\(A,B\) are constants)
1. \( \Lambda_{{n}} \approx {A}+\frac{{B}}{\lambda_{{n}}^2} \)
2. \( \Lambda_{{n}} \approx {A}+{B} \lambda_{{n}} \)
3. \( \Lambda_{{n}}{ }^2 \approx {A}+{B} \lambda_{{n}}{ }^2 \)
4. \(\Lambda_{{n}}{ }^2 \approx \lambda \)

If the series limit frequency of the Lyman series is \(\nu_L\), then the series limit frequency of the Pfund series is:
1. \(25\nu_L\)
2. \(16\nu_L\)
3. \(\nu_L/16\)
4. \(\nu_L/25\)

Radiation coming from transitions \(n=2\) to \(n=1\) of hydrogen atoms fall on \(He^+\) ions in \(n=1\) and \(n=2\) states. The possible transition of helium ions as they absorb energy from the radiation is:
1. \(n=2\rightarrow n=5\)
2. \(n=2\rightarrow n=3\)
3. \(n=1\rightarrow n=4\)
4. \(n=2\rightarrow n=4\)
 

Taking the wavelength of first Balmer line in hydrogen spectrum \((n=3~\text{to}~n=2)\) as \(660~\text{nm}\), the wavelength of the \(2^{nd}\) Balmer line \((n=4~\text{to}~n=2)\) will be:
1. \(889.2~\text{nm}\)
2. \(388.9~\text{nm}\)
3. \(488.9~\text{nm}\)
4. \(642.7~\text{nm}\)

In \(\mathrm{Li^{++}}\), electron in first Bohr orbit is excited to a level by a radiation of wavelength \(\lambda\). When the ion gets deexcited to the ground state in all possible ways (including intermediate emissions), a total of six spectral lines are observed. What is the value of \(\lambda\)? (Given: \(h=6.63 \times 10^{-34} ~\text{Js}\), \(c=3 \times 10^8 ~\text{ms}^{-1}\))
1. \(12.3~\text{nm}\)
2. \(10.8~\text{nm}\)
3. \(9.4~\text{nm}\)
4. \(11.4~\text{nm}\)

The electron in a hydrogen atom first jumps from the third excited state to the second excited state and subsequently to the first excited state. The ratio of the respective wavelengths, \(\lambda_1/\lambda_2\) of the photons emitted in this process is:
1. \(22/5\)
2. \(7/5\)
3. \(9/7\)
4. \(20/7\)

In a hydrogen atom the electron makes a transition from \((n+1)^{th}\) level to the \(n^{th}\) level. If \(n>>l\), the frequency of radiation emitted is proportional to:
1. \(\frac{1}{n^4}\)
2. \(\frac{1}{n^3}\)
3. \(\frac{1}{n^2}\)
4. \(\frac{1}{n}\)