As shown schematically in the figure, two vessels contain water solutions (at$$ temperature 𝑇$$) of potassium permanganate (KMnO4) of different concentrations 𝑛$$1$$ and 𝑛$$2$$ $$($$𝑛$$1$$ 𝑛$$2)$$ molecules per unit volume with Δ$$n=n1$$−$$n2$$≪$$n1$$ . . When they are connected by a tube of small length ℓ and $$cross-sectional$$ area S, $$KMnO4$$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed v of the molecules is limited by the viscous force −βv on each molecule, where β is a constant. Neglecting all terms of the order $$($$Δ$$n)2$$, which of the following $$is/are$$ correct? $$($$𝑘𝐵 is the Boltzmann $$constant)-$$

Question

As shown schematically in the figure, two vessels contain water solutions (at$$ temperature 𝑇$$) of potassium permanganate (KMnO4) of different concentrations 𝑛$$1$$ and 𝑛$$2$$ $$($$𝑛$$1$$ > 𝑛$$2)$$ molecules per unit volume with Δ$$n=n1$$−$$n2$$≪$$n1$$ . . When they are connected by a tube of small length ℓ and $$cross-sectional$$ area S, $$KMnO4$$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed v of the molecules is limited by the viscous force −βv on each molecule, where β is a constant. Neglecting all terms of the order $$($$Δ$$n)2$$, which of the following $$is/are$$ correct? $$($$𝑘𝐵 is the Boltzmann $$constant)-$$

Infinity Learn · Accepted Answer

$$PV=NkBT$$⇒$$P=NVkBT=nkBTn1$$>>$$n1$$−$$n2=$$Δn $$F=P1-P2S=n1$$−$$n2kBTS=$$ΔnkBTSOption (A) is correct.Viscous Force $$=$$ Total number of molecules ×β$$v=n1lS$$βvIf viscous force is balanced by pressure force, then Δ$$nkBTS=$$ℓ$$n1S$$βv⇒$$n1$$βvℓ$$=$$ΔnkBT    Option (B) is correct. Total number of molecules per $$sec$$$$=n1vdtSdt=n1vS=$$ΔnkBTvSβvℓ$$=$$ΔnℓkBTβSOption (C) is correct.As ∆n will decrease with time therefore rate of molecules getting transfer decreases with time.Option (D) is incorrect.

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