$A$ particular hydrogen-like ion emits radiation of frequency $3 \times 10^{15} \,Hz$ when it makes a transition from $n=2$ to $n=1$. The frequency of radiation emitted in a transition from $n=3$ to $n=1$ is $\frac{x}{9} \times 10^{15} \,Hz$, where $X = \text{ . . . . . . }$.

The figure below is the plot of potential energy versus internuclear distance $(d)$ of $H_2$ molecule in the electronic ground state. What is the value of the net potential energy $E_0$ (as indicated in the figure) in $kJ \ mol^{-1}$, for $d=d_0$ at which the electron-electron repulsion and the nucleus-nucleus repulsion energies are absent? As reference, the potential energy of $H$ atom is taken as zero when its electron and the nucleus are infinitely far apart.
Use Avogadro constant as $6.023 \times 10^{23} \ mol^{-1}$.

The ratio of the radii of the second and third orbits of an $H$-atom is . . . . . . .

What is the number of de Broglie wavelengths associated with an electron revolving in the $n^{th}$ allowed orbit of a Bohr atom?

$A$ moving Hydrogen atom makes a head-on collision with a stationary Hydrogen atom. Before the collision,both atoms are in the ground state,and after the collision,they move together. The minimum kinetic energy of the moving Hydrogen atom,such that one of the atoms reaches the excitation state,is (in $eV$)

In $Li^{++}$ ion,an electron in the first Bohr orbit is excited to a higher energy level by a radiation of wavelength $\lambda$. When the ion de-excites to the ground state in all possible ways (including intermediate emissions),a total of six spectral lines are observed. What is the value of $\lambda$ in $nm$? (Given: $h = 6.63 \times 10^{-34} \, J \cdot s, c = 3 \times 10^8 \, m/s, 1 \, eV = 1.6 \times 10^{-19} \, J$)