An excited $He^+$ ion emits two photons in succession, with wavelengths $108.5 \, nm$ and $30.4 \, nm$, in making a transition to the ground state. The quantum number $n$, corresponding to its initial excited state is $n = ........$ (for a photon of wavelength $\lambda$, energy $E = \frac{1240 \, eV}{\lambda \, (in \, nm)}$).

When an electron is excited from its $4^{\text{th}}$ orbit to its $5^{\text{th}}$ stationary orbit,the change in the angular momentum of the electron is approximately (Planck's constant $= h = 6.63 \times 10^{-34} \text{ J s}$)

$A$ hydrogen atom,a deuteron atom,a $He^+$ ion,and a $Li^{++}$ ion each contain a single electron orbiting their nucleus. The wavelengths of the electromagnetic radiation emitted during the transition of the electron from the $n = 2$ orbit to the $n = 1$ orbit are found to be $\lambda_1, \lambda_2, \lambda_3$,and $\lambda_4$ respectively. Then:

The wavelength of the energy emitted when an electron transitions from the fourth orbit to the second orbit in a hydrogen atom is $20.397 \, cm$. The wavelength of the energy for the same transition in $He^+$ is .......... $cm$.

The emission series of hydrogen atom is given by $\frac{1}{\lambda}=R\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)$ where,$R$ is the Rydberg constant. For a transition from $n_{2}$ to $n_{1}$,the relative change $\Delta \lambda / \lambda$ in the emission wavelength,if hydrogen is replaced by deuterium (assume that,the mass of proton and neutron are the same and approximately $2000$ times larger than that of electrons) is ........... $\%$

The frequency of revolution of an electron in the $n^{th}$ Bohr orbit is $v_n$. The graph between $\log n$ and $\log (v_n / v_1)$ may be