# Thermodynamics

# Rüchardt's experience

Take 15 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.

If you have more questions, feel free to ask them on the forum.

Rüchardt's method for determining the ratio of thermal capacity of an ideal gas at constant pressure and volume, studying (see figure) the movement of a ball in a glass tube.

The metal ball, a diameter close to that of the tube behaves as a sealed piston.

Neglecting friction.

When we loose ball in the tube of cross-section , we observe oscillations around an equilibrium position.

The method consists in measuring the period of oscillation of the ball in the tube or, equivalently, the period of oscillation of the pressure of the air contained inside the bottle.

For this, pressure is recorded using a pressure sensor for about .

The air is treated as an ideal gas.

It is noted the position of the ball center at time (the origin is chosen at the equilibrium position of the ball) ; and denote the temperature and pressure of the air inside the bottle.

Data :

(mass of the ball), and (pressure and temperature of atmospheric air), (inner cross section of the tube), (Earth gravity field) and (total volume, for ).

**A Java simulation of this experiment (JJ Rousseau)** : click **HERE**

## Question

Applying second law of Newton to the ball, establish the differential equation of motion of the ball.

What is the pressure balance ?

### Hint

### Solution

The second law of Newton applied to the ball in the laboratory frame assumed Galilean is written, in projection on (Ox) :

So, with :

## Question

From a thermodynamic point of view, the compressions and expansions of gas within the bottle are regarded as practically reversible and adiabatic.

The gap of pressure and of volume are low, is approximated by and by .

Deduce as a function of .

### Hint

### Solution

It is assumed that the ball oscillates quickly enough so that air within the bottle does not have time to receive heat transfer from the outside (but still slowly enough that this transformation can be considered reversible !).

Ultimately, the transformation is considered reversible adiabatic (is isentropic).

Therefore, the Laplace law is applicable and is written :

Or, in differential form :

Is here :

Whence :

## Question

Determine the differential equation satisfied by .

What is the period of oscillations of pressure ?

We measure .

Determine the value of .

Comment the obtained value.

What can be the encountered experimental problems ?

### Hint

### Solution

The differential equation of the ball movement then becomes :

Whence :

It is the characteristic differential equation of a harmonic oscillator of angular frequency :

And period :

Coefficient is therefore expressed as a function of the measured period of the pressure oscillations in the bottle :

Numerical Application :

We find .

The experimental value obtained is consistent with the expected theoretical value (either ) for air which is composed of diatomic molecules (oxygen and nitrous).

The experimental method proposed in this exercise was originally developed by the physicist Rüchardt in .

This method has limits because it is based on three assumptions (gas likened to a perfect gas, negligible friction between the ball and the tube of the bottle and reversible adiabatic transformations gas) that are not necessarily verified.