The work done in increasing the size of a soap bubble is equal to the increase in its surface energy, which is the product of surface tension and the change in surface area.

1. Identify Surface Area: A soap bubble has two free surfaces (inner and outer) in contact with air. Therefore, the effective surface area is $2 \times 4\pi r^2 = 8\pi r^2$.

2. Calculate Initial and Final Areas: Initial Radius ($r_1$) = $r$ Initial Surface Area ($A_1$) = $2 \times 4\pi r^2 = 8\pi r^2$ Final Radius ($r_2$) = $2r$ (radius is doubled) Final Surface Area ($A_2$) = $2 \times 4\pi (2r)^2 = 2 \times 4\pi (4r^2) = 32\pi r^2$

3. Calculate Change in Area ($\Delta A$): $\Delta A = A_2 - A_1 = 32\pi r^2 - 8\pi r^2 = 24\pi r^2$

4. Calculate Work Done ($W$): $W = T \times \Delta A = T \times 24\pi r^2 = 24\pi r^2 T$