Blundell Thermal Physics · Problem 17.7

Rods, bubbles, and magnets

Physics — Statistical Mechanics · soap-bubble-expansion-work

Problem

A soap bubble of radius

R_1

and surface tension

\gamma

is expanded at constant temperature by forcing in air by driving in a piston containing volume

V_\mathrm{piston}

fully home. Show that the work

\Delta W

needed to increase the bubble's radius to

R_2

\Delta W = p_2 V_2 \ln\frac{p_2}{p_1} + 8\pi\gamma(R_2^2 - R_1^2) + p_0(V_2 - V_1 - V_\mathrm{piston}),

where

p_1

and

p_2

are the initial and final pressures in the bubble,

p_0

is the pressure of the atmosphere and

V_1 = \frac{4}{3}\pi R_1^3

and

V_2 = \frac{4}{3}\pi R_2^3