To calculate the work done in increasing the volume of a soap bubble, we need to consider the surface tension of the soap solution and the change in radius of the bubble. In this case, we are given the initial radius of the bubble, and we need to find the work done when the volume is increased to 27 times the initial volume.


The formula for surface tension is given by:

T = (P * r) / 2

Where T is the surface tension, P is the pressure inside the bubble, and r is the radius of the bubble.


The volume of a sphere is given by the formula:

V = (4/3) * π * r^3

If we increase the volume by a factor of n, the new volume will be:

V' = n * V

So, in this case, the new volume is:

V' = 27 * V


We can calculate the change in radius using the formula for volume:

V = (4/3) * π * r^3

Solving for r, we get:

r = (3V / 4π)^(1/3)

So, the initial radius is:

r = (3V / 4π)^(1/3)

And the final radius, when the volume is increased to 27 times, is:

r' = (3V' / 4π)^(1/3)

Substituting the value of V' from earlier, we get:

r' = (3 * 27V / 4π)^(1/3)

Simplifying further, we get:

r' = 3^(1/3) * r


The work done in increasing the volume of the soap bubble can be calculated using the formula:

W = T * ΔA

Where W is the work done, T is the surface tension, and ΔA is the change in surface area.

The initial surface area of the bubble is given by:

A = 4π * r^2

And the final surface area, when the volume is increased to 27 times, is given by:

A' = 4π * r'^2

Substituting the value of r' from earlier, we get:

A' = 4π * (3^(1/3) * r)^2

Simplifying further, we get:

A' = 36π * r^2

The change in surface area is then:

ΔA = A' - A = 36π * r^2 - 4π * r^2 = 32π * r^2

Finally, substituting the values of T and ΔA into the formula for work done, we get:

W = T * ΔA = (P * r / 2) * (32π * r^2)

Simplifying further, we get:

W = 16π * P *