Question 406: Properties of Matter

Question

As shown schematically in the figure, two vessels contain water solutions (at temperature T) of
potassium permanganate (KMnO
4
) of different concentrations n
1
and n
2
(n
1
> n
2
) molecules per
unit volume with $$\Delta $$n = (n
1
− n
2
) << n
1
. When they are connected by a tube of small length l and
cross-sectional area S, KMnO
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)

Accepted Answer

$${n_1} >  > ({n_1} - {n_2}) = \Delta n$$
$$ \because $$ $${p_1} = {{{n_1}RT} \over {{N_A}}}$$ and $${p_2} = {{{n_2}RT} \over {{N_A}}}$$
$$F = ({n_1} - {n_2}){k_B}TS = \Delta n{k_B}TS$$
$$V = {{\Delta n{k_B}TS} \over \beta }$$
$$\Delta n{k_B}TS = {\ln _1}S\beta v$$
$$ \Rightarrow {n_1}\beta vl = \Delta n{k_B}T$$
Total number of molecules/second
$$ = {{({n_1}vdt)S} \over {dt}}$$
$$ = {n_1}vS = {{\Delta n{k_B}TvS} \over {\beta vl}} = \left( {{{\Delta n} \over l}} \right)\left( {{{{k_B}T} \over \beta }} \right)S$$
As $$\Delta $$n will decrease with time, therefore rate of molecules getting transfer decreases with time.

Answer

the force causing the molecules to move across the tube is $$\Delta n{k_b}TS$$

Answer

force balance implies $${n_1}\beta vl = \Delta n{k_B}T$$

Answer

total number of molecules going across the
tube per sec is $$\left( {{{\Delta n} \over l}} \right)\left( {{{{k_B}T} \over \beta }} \right)S$$

Answer

rate of molecules getting transferred through the tube does not change with time

Properties of Matter

Choose the correct answer: