Question

Two soap bubbles coalesce to form a single bubble. if $V$ is the subsequent change in volume of contained air and $S$ the change in total surface area, $T$is the surface tension and $P$ atmospheric pressure, which of the following relations is correct ?
(A) $4PV + 3ST = 0$
(B) $3PV + 4ST = 0$
(C) $2PV + 3ST = 0$
(D) $3PV + 2ST = 0$

Answer 1

Hint: Excess pressure formed due to shape of bubble. Use an isothermal equation under which temperature is constant${\text{PV = constant}}$ and conservation of energy.

Complete step by step answer:
Let ${r_1}$ is the radius of first bubble before coalesce.
${r_2}$ is the radius of second bubble before coalesce.
${r_3}$ is the radius of sphere after coalesce.
${P_1}$ is excess pressure inside the first bubble.
${P_1} = P + \dfrac{{4T}}{{{r_1}}}$ … (i)
Here $P$is atmospheric pressure,$T$ is surface tension.
Similarly
${P_2}$is excess pressure inside the second bubble
${P_2} = P + \dfrac{{4T}}{{{r_2}}}$ … (ii)
and ${P_3}$is excess pressure inside the bubble which formed after coalesce of first and second bubble is
${P_3} = P + \dfrac{{4T}}{{{r_3}}}$ … (iii)
and ${V_1}$is volume of first before coalesce
${V_1} = \dfrac{4}{3}\pi r_1^3$ … (iv)
${V_2}$is volume of second bubble before coalesce
${V_2} = \dfrac{4}{3}\pi r_2^3$ … (v)
${V_3}$is volume of bubble formed after coalesce
${V_3} = \dfrac{4}{3}\pi r_3^3$ … (vi)
According to energy conservation.
${P_1}{V_1} + {P_2}{V_2} = {P_3}{V_3}$ … (vii)
Use above equation in equation (vii)
$\left( {P + \dfrac{{4T}}{{{r_1}}}} \right)\dfrac{4}{3}\pi r_1^3 + \left( {P + \dfrac{{4T}}{{{r_2}}}} \right)\dfrac{4}{3}\pi r_2^3 = \left( {P + \dfrac{{4T}}{{{r_3}}}} \right)\dfrac{4}{3}\pi r_3^3$
$P\left[ {\dfrac{4}{3}\pi r_1^3 + \dfrac{4}{3}\pi r_2^3 - \dfrac{4}{3}\pi r_3^3} \right] + \dfrac{{4T}}{3}\left[ {4\pi r_1^2 + 4\pi r_2^2 - 4\pi r_3^2} \right] = 0$ … (viii)
In equation (viii)
$V = $ change in volume of bubble
$ = \left[ {initial\,\,volume} \right] - \left[ {find\,\,volume} \right]$
$V = \left[ {\dfrac{4}{3}\pi r_1^3 + \dfrac{4}{3}\pi r_2^3} \right] - \left[ {\dfrac{4}{3}\pi r_3^3} \right]$ … (ix)
$S = $change in surface area of bubble
$ = \,\left[ {initial\,\,surface\,\,area} \right] - \left[ {find\,\,surface\,\,area} \right]$
$ = \left[ {4\pi r_1^2 + 4\pi r_2^2} \right] - \left[ {4\pi r_3^2} \right]$ … (x)
Hence equation (viii) reduces to
$PV + \dfrac{{4T\,S}}{3} = 0$
$3PV + 4T\,S = 0$

So, the correct answer is “Option B”.

Note:
 Let \[p\]and \[{p_a}\]be the pressure inside the bubble and outside the bubble respectively. The bubble can exist only if\[p > {p_a}\]. The difference in pressure \[\left( {p - {p_a}} \right)\]is known as excess pressure inside the bubble.