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

for some constant . Bubble B expands isothermally, so

The initial volumes and pressures of the two bubbles are the same so

When the bubbles reach the surface of the lake, the pressure has reduced to so the volumes of the bubbles are

Since where is the number of degrees of freedom, and

so . 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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