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.

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