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(D) soap bubble has two surfaces (inner and outer). The surface area of a spherical bubble of radius $r$ is $A = 4\pi r^2$. Since there are two surfac

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$24$

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(D) soap bubble has two surfaces (inner and outer). The surface area of a spherical bubble of radius $r$ is $A = 4\pi r^2$. Since there are two surfaces, the total surface area is $A_{total} = 2 	imes 4\pi r^2 = 8\pi r^2$. The initial surface area is $A_1 = 8\pi r^2$. When the radius is doubled to $R = 2r$, the final surface area is $A_2 = 8\pi (2r)^2 = 8\pi (4r^2) = 32\pi r^2$. The change in surface area is $\Delta A = A_2 - A_1 = 32\pi r^2 - 8\pi r^2 = 24\pi r^2$. The work done (energy required) is $W = T 	imes \Delta A = T 	imes 24\pi r^2 = 24\pi r^2 T$.

The radius of a soap bubble is $r$, and the surface tension of the soap solution is $T$. Without increasing the temperature, how much energy is required to double its radius (in $\pi r^2 T$)?

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