Find the work to be performed in order to blow a soap bubble of radius `R` if the outside air pressure is equal to `p_0` and the surface tension of the soap water solution is equal to `alpha`.

Explanation

Work must be done to stretch the soap film and compress the air inside. The former is simply `2 alpha xx 4 pi R^2 = 8 pi R^2 alpha`, there being two sides of the film. To get the latter we note that the compression is isothermal and work done is
`-int_(V_1 = V_0)^(V_f = V) pdV` where `V_0 p_0 = (p_0 + (4 alpha)/(R)).V, V = (4 pi)/(3)R^3`
or `V_0 = (pV)/(p_0), p =p_0 + (4 alpha)/(R)`
and minus sign is needed because we are calculating work done on the system. Thus since `pV` remains constants, the work done is
`pV 1n (V_0)/(V) = pV 1n (p)/(p_0)`
So `A' = 8 pi R^2 alpha + p V 1n (p)/(p_0)`.