Problem 6: Thermodynamics and Oscillation of a Soap Bubble
(a) Molar Specific Heat
The bubble is in a vacuum, so the internal pressure is purely due to surface tension (2 surfaces):
$$ P = \frac{4\sigma}{r} $$Since Volume $V \propto r^3$, we have $r \propto V^{1/3}$. Substituting this into the pressure equation:
$$ P \propto \frac{1}{V^{1/3}} \implies PV^{1/3} = \text{constant} $$This represents a polytropic process $PV^x = \text{const}$ with polytropic index $x = 1/3$.
The molar specific heat for a polytropic process is given by:
For a diatomic gas, $C_V = \frac{5}{2}R$. Substituting $x = 1/3$:
$$ C = \frac{5}{2}R + \frac{R}{1 – 1/3} = 2.5R + \frac{R}{2/3} = 2.5R + 1.5R = 4R $$(b) Angular Frequency of Oscillation
Assumption: The problem states the heat capacity of the film is much greater than the gas. This implies the film acts as a thermal reservoir, maintaining the gas at a constant temperature during oscillations. Thus, the process is Isothermal ($T = \text{const}$, $PV = \text{const}$).
Force Analysis:
Let the radius change by a small amount $x$ ($r’ = r + x$).
Mass of the shell $m = 4\pi r^2 h \rho$.
1. Excess Pressure Force from Gas ($dF_{gas}$):
For isothermal process ($PV = K \implies P \propto r^{-3}$):
At equilibrium $P = \frac{4\sigma}{r}$, so $dP_{gas} = -3 (\frac{4\sigma}{r^2}) x = -\frac{12\sigma}{r^2}x$.
2. Excess Surface Tension Pressure ($dP_{surf}$):
New surface pressure $P_{surf}’ = \frac{4\sigma}{r+x} \approx \frac{4\sigma}{r} (1 – \frac{x}{r}) = \frac{4\sigma}{r} – \frac{4\sigma}{r^2}x$.
Change $dP_{surf} = -\frac{4\sigma}{r^2}x$.
3. Net Restoring Force:
The net pressure pushing outward is $P_{gas}’ – P_{surf}’$. The restoring pressure perturbation is:
Restoring Force $F = \Delta P_{net} \times \text{Area} = (-\frac{8\sigma}{r^2}x) (4\pi r^2) = -32\pi \sigma x$.
Equation of Motion:
$$ ma = -kx \implies (4\pi r^2 h \rho) \omega^2 x = 32\pi \sigma x $$ $$ \omega^2 = \frac{32\pi \sigma}{4\pi r^2 h \rho} = \frac{8\sigma}{\rho h r^2} $$