Isothermal versus adiabatic expansion of an ideal gas bubble

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.38.

As a simple example of isothermal versus adiabatic expansion of an ideal gas, suppose that two identical bubbles form at the bottom of a lake. Bubble A rises quickly so that no heat is exchanged with the surrounding water, while bubble B rises slowly (bumping off the leaves of some lakeweed, for example) so that its temperature remains constant (assuming that the lake’s water temperature is the same everywhere).

Bubble A experiences adiabatic expansion, so it obeys the relation

$\displaystyle PV^{\gamma}=A \ \ \ \ \ (1)$

for some constant ${A}$. Bubble B expands isothermally, so

$\displaystyle PV=NkT=\mbox{constant} \ \ \ \ \ (2)$

The initial volumes ${V_{0}}$ and pressures ${P_{0}}$ of the two bubbles are the same so

 $\displaystyle A$ $\displaystyle =$ $\displaystyle P_{0}V_{0}^{\gamma}\ \ \ \ \ (3)$ $\displaystyle NkT$ $\displaystyle =$ $\displaystyle P_{0}V_{0} \ \ \ \ \ (4)$

When the bubbles reach the surface of the lake, the pressure has reduced to ${P_{1}}$ so the volumes of the bubbles are

 $\displaystyle V_{A}$ $\displaystyle =$ $\displaystyle \left(\frac{P_{0}}{P_{1}}\right)^{1/\gamma}V_{0}\ \ \ \ \ (5)$ $\displaystyle V_{B}$ $\displaystyle =$ $\displaystyle \frac{P_{0}}{P_{1}}V_{0} \ \ \ \ \ (6)$

Since ${\gamma=\left(f+2\right)/f>1}$ where ${f}$ is the number of degrees of freedom, and ${P_{0}>P_{1}}$

$\displaystyle \left(\frac{P_{0}}{P_{1}}\right)^{1/\gamma}<\frac{P_{0}}{P_{1}} \ \ \ \ \ (7)$

so ${V_{B}>V_{A}}$. That is, the the bubble that rises slowly will be larger than the bubble that rises quickly when they reach the surface.