Under isothermal conditions, two soap bubbles of radii ' \(r_1\) ' and ' \(r_2\) ' combine to forms a single soap bubble of radius ' \(R\) '. The surface tension of soap solution is ( \(\mathrm{P}=\) outside pressure)

Pressure inside the first bubble \(=\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_1}\)
Pressure inside the second bubble \(=\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_2}\)
Using the formula \(\mathrm{PV}=\mathrm{nR} \theta \quad(\theta=\) absolute temp \()\)
\(\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_1}\right) \cdot \frac{4 \pi}{3} \mathrm{r}_1^3=\mathrm{n}_1 \mathrm{R} \theta \quad\) ( \(\mathrm{R}\) is molar gas constant)
\(\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_2}\right) \cdot \frac{4 \pi}{3} \mathrm{r}_2^3=\mathrm{n}_2 \mathrm{R} \theta\)
and \(\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{R}}\right) \cdot \frac{4 \pi}{3} \mathrm{R}^3=\left(\mathrm{n}_1+\mathrm{n}_2\right) \mathrm{R} \theta\)
\(\therefore\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{R}}\right) \cdot \frac{4 \pi}{3} \mathrm{R}^3=\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_1}\right) \cdot \frac{4 \pi}{3} \mathrm{r}_1^3+\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_2}\right) \cdot \frac{4 \pi \mathrm{r}_2^3}{3}\)
\(\therefore\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{R}}\right) \mathrm{R}^3=\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_1}\right) \mathrm{r}_1^3+\left(\mathrm{P}+\frac{4 \mathrm{~T}}{\mathrm{r}_2}\right) \mathrm{r}_2^3\)
on solving: \(\mathrm{T}=\frac{\mathrm{P}\left(\mathrm{R}^3-\mathrm{r}_1^3-\mathrm{r}_2^3\right)}{4\left(\mathrm{r}_1^2+\mathrm{r}_2^2-\mathrm{R}^2\right)}\)