Question

A spherical soap bubble with a surface-tension of 0.005 lbf/ft is expanded from a diameter of 0.5 in to 3.0 in. How much work, in Btu, is required to expand this bubble?

Short answer

Answer: The work required to expand the soap bubble from a diameter of 0.5 inches to 3 inches is approximately 1.2272 x 10^(-6) Btu.

Step-by-step solution

Convert dimensions to consistent units

First, we will convert the given dimensions to consistent units. The diameter's units should be in feet, as the surface tension's units are lbf/ft. We can do this by using the following conversion factor: 1 ft = 12 inches Initial diameter (in feet) = 0.5 inches * (1 ft / 12 inches) = 0.0416667 ft Final diameter (in feet) = 3.0 inches * (1 ft / 12 inches) = 0.25 ft

Compute the initial and final surface areas

Next, we will calculate the initial and final surface areas of the spherical soap bubble, using the formula for the surface area of a sphere: A = 4 * π * R^2, where R is the radius. Initial radius = Initial diameter / 2 = 0.0416667 ft / 2 = 0.0208333 ft Final radius = Final diameter / 2 = 0.25 ft / 2 = 0.125 ft Initial surface area (A1) = 4 * π * (0.0208333 ft)^2 = 0.00544637 ft^2 Final surface area (A2) = 4 * π * (0.125 ft)^2 = 0.196349 ft^2

Calculate the change in surface energy

Now we can compute the change in surface energy that occurs when the bubble expands from its initial to final size. Surface energy change (ΔE) is equal to the surface tension (γ) times the change in surface area (ΔA): ΔE = γ * ΔA ΔA = A2 - A1 = 0.196349 ft^2 - 0.00544637 ft^2 = 0.190903 ft^2 Surface tension (γ) = 0.005 lbf/ft ΔE = γ * ΔA = 0.005 lbf/ft * 0.190903 ft^2 = 0.000954516 lbf*ft

Convert work from lbf*ft to Btu

To convert the work from lbf*ft to Btu (British thermal units), we will use the conversion factor: 1 Btu = 778.169 lbf*ft Work done (in Btu) = 0.000954516 lbf*ft * (1 Btu / 778.169 lbf*ft) = 1.2272 x 10^(-6) Btu The work required to expand the soap bubble from a diameter of 0.5 inches to 3 inches is approximately 1.2272 x 10^(-6) Btu.