Thermodynamics

At the macroscopic scale : a perfect gas is a (real !) gas studied at low pressures (less than a few bars, is the normal atmospheric pressure).

Gas is characterized by measurable macroscopic parameters on our scale : volume , pressure , temperature (in Kelvin) or ( ) and the number of moles .

These parameters are called "state variables".

These variables are defined as "thermodynamic equilibrium" of gas.

Relationship between (Kelvin scale) and (Celsius scale) :

In the IS (the International System), the unit of pressure is the Pascal.

Also used the bar :

The equation of state of an ideal gas is :

Where is the number of moles, the mass of gas and its molecular weight.

Consider a monatomic gas (Ar, He, H, ...) at temperature , pressure and occupying a volume .

We note :

• : number of moles

•  : total number of particles in the volume

• : density (number of particles per unit volume),

Assumptions of the model :

• The particles have a constant movement (thermal agitation).

• The particles do not interact with each other but only with the wall during impacts therewith. They are punctual.

• The velocities of the particles are distributed in a statistical law called "Maxwell -Boltzmann's distribution law".

  The distribution of particle velocities is homogeneous (independent of the point where we place) and isotropic (independent of the chosen direction).

• The particle density is the same at all points of the container (homogeneous distribution of the particles).

In the following, we choose a simple (simplistic) model of molecular speed distribution law :

• The particles (mass ) all have the same speed (in norm), denoted .

• The particles can move in three dimensions (Ox, Oy and Oz) and therefore six directions.

• The kinetic pressure results of incessant impacts of particles against the wall.

• At particle scale, these shocks are elastic (conservations of momentum and kinetic energy).

The momentum gained by the wall in an impact is (conservation of momentum) :

During the time interval , the force exerted by the wall is :

Where is the amount of movement received by the wall in the impact of particles falling on the wall during .

If is the number of particles that shock the wall during the time interval , then :

The kinetic pressure is deduced :

Calculation of (see figure below) :

The particles have a "six chance" to hit the wall :

We deduce the kinetic pressure P :

So :

A more realistic calculation, taking into account the Maxwell-Boltzmann distribution led to a similar result, if we replace by the quadratic (RMS) speed  :

Knowing that :

It comes :

This equation is to compare to the macroscopic equation of state of a perfect gaz :

Yet :

Whence :

Thus deduced in the expression of the quadratic (RMS = root mean square) velocity in function of the absolute temperature :

Where is the molar mass.

We note :

the Boltzmann constant ( ).

Finally :

Some numerical applications :

At , for helium, and for Argon, .

The average kinetic energy of a particle, denoted , is expressed in terms of temperature  :

Temperature is a measure of the thermal motion of the particles.

It is convenient sometimes to express a temperature in , or (which are energy units) : the correspondence between in and in is obtained from the previous relationship.

Thus, if is , whereas, in :

Since we neglect the forces of mutual interaction between particles, the potential energy of mutual interaction (defined in mechanics, in the case of two interacting bodies for instance) is zero.

In the laboratory frame (in which the container containing the gas is assumed immobile), the mechanical energy of the system of particles is called "internal energy" and noted .

It is :

It represents the energy contained within the container, due to the thermal agitation (whence the name of internal energy).

With :

It comes :

Thus :