As shown schematically in the figure, two vessels contain water solutions (at temperature T) of potassium permanganate ({KMnO}₄) of different concentrations n₁ and n₂ (n₁ > n₂) molecules per unit volume with Δ n = (n₁ - n₂) [JEE Adv 2020 P1]

The force causing the molecules to move across the tube is $\Delta n k_B T S$
Force balance implies $n_1 \beta v l = \Delta n k_B T$
Total number of molecules going across the tube per sec is $\left( \frac{\Delta n}{l} \right) \left( \frac{k_B T}{\beta} \right) s$
Rate of molecules getting transferred through the tube does not change with time

Solution:

\therefore P_1 = n_1 k_B T, P_2 = n_2 k_B T

\therefore \Delta F = (P_1 - P_2) S

= (n_1 - n_2) k_B T S

\Delta F = \Delta n k_B T S

...(i)

\therefore

Balancing of force,

\beta v \times (

\times n_1 ) = \Delta n k_B T S

\Rightarrow n_1 \beta v l = \Delta n k_B T

...(ii)

\therefore v = \frac{\Delta n k_B T}{\beta n_1 l}

\therefore

Number of molecules per second

\frac{\Delta N}{\Delta t} = \frac{(v d t) \times S \times n_1}{d t}

= v S n_1

= \frac{\Delta n k_B T}{\beta n_1 l} \times S n_1

\frac{\Delta N}{\Delta t} = \left( \frac{\Delta n}{l} \right) \left( \frac{k_B T}{\beta} \right) S

...(iii)

∵ Δn decreases with time, so rate of transfer decreases with time.

\therefore

Answer

(A, B, C)

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