Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to :-

Dual Nature of Radiation PYQ — JEE Main 2021 · 31 Aug · Shift 2 · Q50 with Solution

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Correct answer: C

For a particle,

\lambda = \frac{h}{p}

The uncertainty in position is proportional to the de Broglie wavelength:

\Delta x \propto \lambda \propto \frac{1}{p}

So, for electron and proton gases at the same temperature, we compare their typical momenta.

For an ideal gas at temperature $T$ , the average kinetic energy is

\frac{p^2}{2m} \sim kT

Thus,

p \sim \sqrt{2mkT} \propto \sqrt{m}

when $T$ is the same for both gases.

Since

\lambda \propto \frac{1}{p} \propto \frac{1}{\sqrt{m}}

we get

\Delta x \propto \frac{1}{\sqrt{m}}

Therefore,

\frac{\Delta x_e}{\Delta x_p} = \frac{1/\sqrt{m_e}}{1/\sqrt{m_p}} = \sqrt{\frac{m_p}{m_e}}

\sqrt{\frac{m_p}{m_e}}

corresponds to Option C.

Comparison with stored answer: Stored correct answer is C, which matches our derived answer.