Tag: the kinetic model of matter

At a depth of h water pressure will be
${P} _{h} = {P} _{a} + \rho gh$                eq(1)
where 
$\rho  =1000 kg/{m}^{3}$
$g  = 9.81 m/{s}^{2} $
$h = \text{depth  of  water  bubble}$
${P} _{a} = {10}^{5}Pa$

initially bubble is below water at h, so
${P} _{1} = {P} _{h}$
finally it rises to surface of lake so
${P} _{2} = {P} _{a}$

we assume that temperature is constant when it is rising from bottom to surface
by ideal gas equation
${P} _{1}{V} _{1} = {P} _{2}{V} _{2}$

volume can be written as $V = \dfrac{4}{3}\pi {r}^{3}$
where r is radius.
${P} _{h} \times \dfrac{4}{3}\pi {r} _{h}^{3} = {P} _{a} \times \dfrac{4}{3}\pi {r} _{s}^{3}$
${r} _{h} = {(\dfrac{{P} _{a}}{{P} _{h}})}^{1/3} r$
${r} _{h} = \dfrac{r}{{(\frac{{P} _{h}}{{P} _{a}})}^{1/3}}$
by eq(1)
${r} _{h} = \dfrac{r}{{(\dfrac{{P} _{a} +  \rho gh}{{P} _{a}})}^{1/3}}$
${r} _{h} = \dfrac{r}{{(1 + 0.0981h)}^{1/3}}$              eq(2)

(a) at depth 30m
h = 30m
put h=30 in eq(2)
${r} _{h} = \dfrac{r}{1.5}$
false

(b) at depth 70m
h = 70m
put h=70 in eq(2)
${r} _{h} = \dfrac{r}{2}$
true

(c) at depth 140m
h = 140m
put h=140 in eq(2)
${r} _{h} = \dfrac{r}{2.4}$
false

(d) at depth 260m
h = 260m
put h=260 in eq(2)
${r} _{h} = \frac{r}{3}$
true

Answer is D.

Let the new pressure inside be $P$ after air introduced.

(a) Temperature must be constant to apply boyle's law
(b)From vanderwaal's equation when non idealities of a gas is undertaken then we can apply boyle's only when temperture is high and pressure is low
(c)Vessel enclosing must be a good conductor so there is no any possibilities of adiabatic process
(d) Temperature must be constant therefore, process must be isothermal
Hence all are correct
Hence option(D)