Dual Nature of Matter

JEE Main level, single correct, medium

47 of 200

Question 47 of 200

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:

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Answer: (A) mpme\sqrt{\dfrac{{m}_{p}}{{m}_{e}}}

K.E.=32kTK.E.=\dfrac{3}{2}kT
⇒p22m=32kT\Rightarrow \dfrac{{p}^{2}}{2m}=\dfrac{3}{2}kT
⇒p=3kTm\Rightarrow p=\sqrt{3kTm}
From Heisenberg uncertainty principle
∴ΔxΔp=h4π∴\Delta x\Delta p=\dfrac{h}{4\pi }
Δx=h4πΔp=h4π3kTm\Delta x=\dfrac{h}{4\pi \Delta p}=\dfrac{h}{4\pi \sqrt{3kTm}}
∴Δx∝1m∴\Delta x\propto \dfrac{1}{\sqrt{m}}
∴ΔxeΔxp=mpme∴\dfrac{\Delta {x}_{e}}{\Delta {x}_{p}}=\sqrt{\dfrac{{m}_{p}}{{m}_{e}}}

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