Hydrogen atoms in their ground state are bombarded with electrons. The energies of the \(n=1\) and \(n=3\) levels of hydrogen are \(-13.6\ \mathrm{eV}\) and \(-1.51\ \mathrm{eV}\). The minimum speed of an electron that can raise an atom to the \(n=3\) level by collision is about (Take \(m=9.1\times10^{-31}\ \mathrm{kg}\) and \(1\ \mathrm{eV}=1.6\times10^{-19}\ \mathrm{J}\).)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(2.06\times10^{6}\ \mathrm{m\,s^{-1}}\)
A striking electron can give its kinetic energy to the atom. To lift the atom from \( n=1 \) to \( n=3 \), it must bring at least the energy gap \( E_3-E_1 \).
Energy needed:
\( \begin{aligned} \Delta E &= E_3-E_1 \\ &= -1.51-(-13.6) \\ &= 12.09\ \mathrm{eV} \end{aligned} \)
\( \begin{aligned} \Delta E &= 12.09\times1.6\times10^{-19}\ \mathrm{J} \\ &= 1.93\times10^{-18}\ \mathrm{J} \end{aligned} \)
The kinetic energy \( \frac{1}{2}mv^{2} \) of the electron must be at least \( \Delta E \).
\( \begin{aligned} v &= \sqrt{\frac{2\Delta E}{m}} \\ &= \sqrt{\frac{2\times1.93\times10^{-18}}{9.1\times10^{-31}}} \\ &= 2.06\times10^{6}\ \mathrm{m\,s^{-1}} \end{aligned} \)
From the above, \( v=2.06\times10^{6}\ \mathrm{m\,s^{-1}} \). Hence, slower electrons cannot raise the atom to the \( n=3 \) level.
NCERT (2026-27):
Energy levels – “At room temperature, most of the hydrogen atoms are in ground state. When a hydrogen atom receives energy by processes such as electron collisions, the atom may acquire sufficient energy to raise the electron to higher energy states. … Similarly, E3 = –1.51 eV and E3 – E1 = 12.09 eV, or to excite the hydrogen atom from its ground state (n = 1) to second excited state (n = 3), 12.09 eV energy is required, and so on.”