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?

$\begin{array}{l}\text{ Let moles of air in firs t bubble of radius }{\mathbf{R}}_1{\text{ be n}}_1\text{ and that in second bubble of radius }{\mathbf{R}}_2\text{ be }{\mathbf{n}}_2.\\ \text{ Since, number of moles of gas is conserved. }\\ {\mathbf{n}}_1+{\mathbf{n}}_2={\mathbf{n}}^{\text{'}}\\ {\text{ where n' is the number of moles of gas in the final single bubble of radius R}}^{\text{'}}\text{ . }\\ \Rightarrow \frac{{\mathbf{P}}_1{\mathbf{V}}_1}{\mathbf{R}T}+\frac{{\mathbf{P}}_2{\mathbf{V}}_2}{\mathbf{R}T}=\frac{{\mathbf{P}}^{\text{'}}{\mathbf{V}}^{\text{'}}}{\mathbf{R}T}\\ \Rightarrow {\mathbf{P}}_1{\mathbf{V}}_1+{\mathbf{P}}_2{\mathbf{V}}_2={\mathbf{P}}^{\text{'}}{\mathbf{V}}^{\text{'}}\\ Now,\\ {\mathbf{P}}^{\text{'}}=\mathbf{P}+\frac{4T}{{\mathbf{R}}^{\text{'}}}\\ and\\ {\mathbf{V}}^{\text{'}}=\frac{4\pi }{3}{\mathbf{R}}^{\text{'}3}\\ \therefore (\mathbf{P}+\frac{4 \text{T}}{{\mathbf{R}}_1})\left(\frac{4\pi }{3}{\mathbf{R}}_1^3\right)+(\mathbf{P}+\frac{4 \text{T}}{{\mathbf{R}}_2})\left(\frac{4\pi }{3}{\mathbf{R}}_2^3\right)=(\mathbf{P}+\frac{4\mathbf{T}}{{\mathbf{R}}^{\text{'}}})\left(\frac{4\pi }{3}{\mathbf{R}}^{\text{'}3}\right)\\ \Rightarrow \mathbf{P}(\frac{4\pi }{3}{\mathbf{R}}_1^3+\frac{4\pi }{3}{\mathbf{R}}_2^3-\frac{4\pi }{3}{\mathbf{R}}^{\text{'}3})+\frac{4T}{3}(4\pi {\mathbf{R}}_1^2+4\pi {\mathbf{R}}_2^2-4\pi {\mathbf{R}}^2)=0\\ \Rightarrow \mathbf{PV}+\frac{4T}{3}\mathbf{S}=\mathbf{0}\\ \Rightarrow \mathbf{3}\mathbf{PV}+4\mathbf{ST}=\mathbf{0}\end{array}$