Question

As shown schematically in the figure, two vessels contain water solutions (at temperature $T$ ) of potassium permanganate $\left(KMn{O}_4\right)$ of different concentrations $n_1$ and $n_2\left(n_1>n_2\right)$ molecules per unit volume with $\Delta n=\left(n_1-n_2\right)<<n_1$. When they are connected by a tube of small length $\ell$ and cross-sectional area $S,KMn{O}_4$ 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 $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta n{)}^2$, which of the following is/are correct? ( $k_B$ is the Boltzmann constant)

• A. the force causing the molecules to move across the tube is $\Delta n{k}_{B}TS$
• B. force balance implies $n_l\beta vl=\Delta n{k}_{B}T$
• C. total number of molecules going across the tube per $\sec$ is $\left(\frac{\Delta n}{l}\right)\left(\frac{k_{B}t}{\beta }\right)S$
• D. rate of molecules getting transferred through the tube does not change with time

Answer 1

Answer: (A), (B), (C)

$PV=NKT$
$P_1=n_1{K}_{B}T$
$P_2=n_2{K}_{B}T$
$n_1,n_2\to$ no. of molecules per unit volume
Force acting on the molecules
$F=\Delta PS$
$=\left(P_1-P_2\right)S$
$F=\left(n_1-n_2\right)K_{B}TS=\Delta n{k}_{B}TS\dots$..(i)
So Option (A) is correct
Total no. of molecules in the tube $=\left(n_{1}S\ell \right)$
Force acting on each molecules
$F=-\beta v$
So total force acting on ( $n_{1}S\ell$ ) molecules $=-$
$(\beta v)(n_{1}s\ell$ )...(ii)
From (i) and (ii)
$\Delta n{K}_{B}TS=(\beta v)\left(n_{1}S\ell \right)$
So, $\Delta n{K}_{B}T=\left(\beta v{n}_1\ell \right)\dots$ (iii)
So, Option (B) is correct.
No. of molecules going across the tube per second
$=\left({svn}_1\right)$
$=\left(s{n}_1\right)\left(\frac{\Delta n{K}_{\beta }T}{\beta n_1\ell }\right)=\left(\frac{S\Delta n{K}_{\beta }T}{\beta \ell }\right)$
So, Option (C) is correct.
With respect to time $\Delta$ changes hence rate of molecules getting transferred through the tube changes with time.
So, Option (D) is incorrect.