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. \( 4 \lambda_1=2 \lambda_2=2 \lambda_3=\lambda_4\)

As an electron makes a transition from an excited state to the ground state of a hydrogen-like atom/ion:

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. \(\nu_L/25\)