In a sample of $\mathrm{H}$-atoms, all the atoms are in $3^{\mathrm{rd}}$ excited state. They de-excite to the ground state in one or more transitions but no photons is observed in Paschen series. The ratio of longest and shortest wavelengths of emitted photons is $\mathrm{x}:1$. The value of '$\mathrm{x}$' is.

An electron beam can undergo diffraction by crystals. Through what potential should a beam of electrons be accelerated so that its wavelength becomes equal to $1.54 \mathrm{\AA }$? Values of ${\mathrm{m}}_e=9.1\times {10}^{-31 }\mathrm{kg}$ and $\mathrm{h}=6.62\times {10}^{-34} \mathrm{J}.\mathrm{s}$.

Give answer after multiplying with 10 and rounding off to the nearest integer value.

An electron in ${\mathrm{Li}}^{2+}$ ion makes a transition from higher state ${\mathrm{n}}_2$ to lower state ${\mathrm{n}}_1=6.$ The emitted photon is used to ionize an electron in $\mathrm{H}$-atom from $2^{\text{nd }}$ excited state. The electron on leaving the $\mathrm{H}$-atom has a de-Broglie wavelength $\mathrm{\lambda }=12.016Å.$ Find the value of ${\mathrm{n}}_2.$

Note: Use $(12.016{)}^2=\frac{150\times 144}{13.6\times 11},{\mathrm{\lambda }}_{Å}=\sqrt{\frac{150}{{\mathrm{KE}}_{\mathrm{eV}}}}.$

If the energy difference between the ground state of an atom and its excited state is $4.4\times {10}^{-19} \mathrm{J}$, The wave length of the photon (in meters) required to produce this transition is $\frac{\mathrm{x}}{2}\times {10}^{-7}\mathrm{m}$. The value of $\mathrm{x}$ is

Round off your answer up to the nearest integer.