A soap bubble of radius r is blown up to form | vedclass

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(D) soap bubble has two surfaces (inner and outer). Initially, the surface area of the soap bubble is $A_1 = 4 \pi r^2$. Since it has two surfaces, th

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$24$

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(D) soap bubble has two surfaces (inner and outer). Initially, the surface area of the soap bubble is $A_1 = 4 \pi r^2$. Since it has two surfaces, the effective initial area is $S_1 = 2 	imes (4 \pi r^2) = 8 \pi r^2$. Under isothermal conditions, the radius becomes $2r$. The new surface area is $A_2 = 4 \pi (2r)^2 = 16 \pi r^2$. The effective final area is $S_2 = 2 	imes (16 \pi r^2) = 32 \pi r^2$. The increase in surface area is $\Delta S = S_2 - S_1 = 32 \pi r^2 - 8 \pi r^2 = 24 \pi r^2$. The energy spent $(W)$ is given by $W = T 	imes \Delta S$. Therefore, $W = T 	imes 24 \pi r^2 = 24 \pi T r^2$.

$A$ soap bubble of radius $r$ is blown up to form a bubble of radius $2r$ under isothermal conditions. If $T$ is the surface tension of the soap solution, what is the energy spent in blowing the bubble (in $\pi T r^2$)?

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