Two soap bubbles coalesce to form a single bubble. If \(V\) is the subsequent change in volume of contained air and \(S\) change in total surface area, \(T\) is the surface tension and \(P\) atmospheric pressure, then which of the following relation is correct?

\(\therefore {P_1} = P + \frac{{4T}}{{{R_1}}}.\,\,\,{P_2} = P + \frac{{4T}}{{{R_2}}}\,\,and\,\,{P_3} = P + \frac{{4T}}{{{R_3}}}\)

\(Also\,{P_1}{V_1} + {P_2}{V_2} = {P_3}{V_3}\)

\(\therefore \left( {P + \frac{{4T}}{{{R_1}}}} \right)\frac{{4\pi }}{3}R_1^3 + \left( {P + \frac{{4T}}{{{R_2}}}} \right)\frac{{4\pi }}{3}R_2^3\)

\( = \left( {p + \frac{{4T}}{{{R_3}}}} \right)\frac{{4\pi }}{3}R_3^3\)

\(P\left( {\frac{{4\pi }}{3}R_1^3 + \frac{{4\pi }}{3}R_2^3 - \frac{{4\pi }}{3}R_3^2} \right)\)

\( + \frac{{4T}}{3}\left( {4\pi R_1^2 + 4\pi R_2^2 - 4\pi R_3^2} \right) = 0\)

\(P\left( {{V_1} + {V_2} - {V_3}} \right) + \frac{{4T}}{3}\left( {{S_1} - {S_2} - {S_3}} \right) = 0\)

\(PV + \frac{{4T}}{3}S = 0\,\,\,\,\,\,\, \Rightarrow \,\,\,\,3PV + 4ST = 0\)