A soap bubble of diameter a is produced using the soap solution of surface tension T. Find the energy required to double the radius of the bubble without change of temperature.

Explanation

For smaller bubble `P_(1) = (P_(0) + (4T)/(r )) and V_(1) = (4)/(3) pi r^(3)`
For larger bubble `P_(2) = P_(0) + (4T)/(R ) - (sigma^(2))/(2in_(0)) and V_(2) = (4)/(3) pi R^(3) " where " sigma= (q)/(4pi R^(2))`
for air in the bubble, `P_(1) V_(1) = P_(2)V_(2)`
`(P_(0) + (4T)/(r )) r^(3) = [(P_(0) + (4T)/(R ))- (q^(2))/(16pi^(2) R^(4) xx 2 in_(0))]R^(3)`
`P_(0) [R^(3) -r^(3)] + 4T [R^(2) - r^(2)] - (q^(2))/(32pi^(2) in_(0)R)=0`
But R= 2r `P_(0) [7r^(3)] + 4T [3r^(2)] - (q^(2))/(32pi^(2) in_(0)(2r))=0`
`(q^(2))/(64pi^(2) in_(0)r) = 7P_(0)r^(3) + 12Tr^(2)`
`q^(2)= 64pi^(2) in_(0)r^(3) [7P_(0)r + 12T]`
`q= 8pi r [in_(0)r (7P_(0)r + 12T)]^(½)`