c
(c) de-Broglie wavelength \(\lambda = \frac{h}{{m{v_{rms}}}}\), rms velocity of a gas particle at the given temperature \((T)\) is given as \(\frac{1}{2}mv_{rms}^2 = \frac{3}{2}kT\)

\( \Rightarrow {v_{rms}} = \sqrt {\frac{{3\,kT}}{m}} \)

\( \Rightarrow m{v_{rms}} = \sqrt {3\,mk\;T} \)

 \(\lambda = \frac{h}{{m{v_{rms}}}} = \frac{h}{{\sqrt {3\,mkT} }}\)

\( \Rightarrow \frac{{{\lambda _H}}}{{{\lambda _{He}}}} = \sqrt {\frac{{{m_{He}}{T_{He}}}}{{{m_H}{T_H}}}} = \sqrt {\frac{{4\,(273 + 127)}}{{2\,(273 + 27)}}} = \sqrt {\frac{8}{3}} \)