# Thermodynamics

# Reversible adiabatic oscillations

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

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An adiabatic and horizontal cylinder separated into two compartments by an adiabatic piston of mass , moving without friction, contains, at the initial state, one mole of ideal gas on each side.

At time , the operator removes the piston from its equilibrium position of small compared with the length ( ).

## Question

Study the small oscillations of the system.

### Hint

### Solution

It is assumed that the transformations are isentropic. So for the gas at the left :

Similarly, for the gas at the right :

The second law of Newton for the piston gives :

## Question

Justify the assumptions of adiabatic and reversibility.

### Hint

### Solution

For assuming adiabatic, one can evaluate the long time associated with heat exchange by thermal diffusion.

For the reversibility hypothesis, we must neglect the friction due to the viscosity of the air and the piston.

This assumption is questionable because the friction inevitably leads to the stop of the piston.