Consider two separate ideal gases of electrons and protons having...

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:

Options

A $\sqrt{\dfrac{{m}_{p}}{{m}_{e}}}$
B $\dfrac{{m}_{p}}{{m}_{e}}$
C $\left(\dfrac{{m}_{p}}{{m}_{e}}\right)\dfrac{3}{2}$
D $\sqrt{\dfrac{{m}_{e}}{{m}_{p}}}$

Correct answer

A. $\sqrt{\dfrac{{m}_{p}}{{m}_{e}}}$

Step-by-step solution

K . E . = \dfrac{3}{2}kT

⇒ \dfrac{{p}^{2}}{2m} = \dfrac{3}{2}kT

⇒ p = \sqrt{3kTm}

From Heisenberg uncertainty principle

∴ \Deltax\Deltap = \dfrac{h}{4\pi}

\Deltax = \dfrac{h}{4\pi\Deltap} = \dfrac{h}{4\pi\sqrt{3kTm}}

∴ \Deltax ∝ \dfrac{1}{\sqrt{m}}

∴ \dfrac{\Delta{x}_{e}}{\Delta{x}_{p}} = \sqrt{\dfrac{{m}_{p}}{{m}_{e}}}

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