Two soap bubble of radii `r_(1)` and `r_(2)` combime to form a single bubble of radius r under isothermal conditions . If the external pressure is P, prove that surface tension of soap solution is given by `S=(P(r^(3)-r_(1)^(3)-r_(2)^(3)))/(4(r_(1)^(2)+r_(2)^(2)-r^(2)))`.

Options

A. `(P_(0)(R^(3)+R_(1)^(3)+R_(2)^(3)))/(4(R^(2)+R_(1)^(2)+R_(2)^(2)))`
B. `(P_(0)(R_(1)^(3)+R_(2)^(3)-R^(3)))/(4(R^(2)-R_(1)^(2)-R_(2)^(2)))`
C. `P_(0)(R_(1)^(3)+R_(2)^(3)-R^(3))`
D. `4P_(0)(R_(1)^(3)+R_(2)^(3)-R^(3))`

Explanation

According to isothermal process, `PV=P_(1)V_(1)+P_(2)V_(2)`
`(P_(0)+(4T)/(R))(4)/(3)piR^(3)=(P_(0)+(4T)/(R_(1)))(4)/(3)piR_(1)^(3)+(P_(0)+(4T)/(R_(2)))(4)/(3)piR_(2)^(3)`
`P_(0)(R^(3)-R_(1)^(3)-R_(2)^(3))=4T(R_(1)^(2)+R_(2)^(2)-R^(2))`