   Next: 2.4 Transport processes Up: 2. Elements of Kinetic Previous: 2.2 The Maxwell-Boltzmann distribution

# 2.3 Thermodynamics of dilute gases

The pressure of a gas in a container is produced by the incessant drumming of the gas molecules upon the walls. At each such wall collision - say, against the right wall of a cubic box - the respective momentum of the molecule is reversed. The momentum transferred to the wall is thus . The force acting on the unit area of the wall is then just the time average of the momentum transfer: (2.35)

where is the number of wall impacts within the time .

To obtain a theoretical prediction for the value of the pressure we argue as follows:

The number of particles having -velocity and impinging on the right wall per unit time is obviously proportional to . The momentum transfer from such a particle to the wall is . Thus we find (2.36)

Inserting the Boltzmann density for and performing the integrations we have (2.37) Applet Hspheres: Start Where do we stand now? Just by statistical reasoning we have actually arrived at a prediction for the pressure - a thermodynamic quantity!

Having thus traversed the gap to macroscopic physics we take a few steps further. From thermodynamics we know that the pressure of a dilute gas at density and temperature is . Comparing this to the above formula we find for the mean energy of a molecule (2.38)

and the parameter (which was introduced in connection with the equilibrium density in velocity space) turns out to be just the inverse of .

The further explication of the dilute gas thermodynamics is easy. Taking the formula for the internal energy,  , and using the First Law (2.39)

we immediately find that amount of energy that is needed to raise the temperature of a gas by degree - a.k.a. the heat capacity : (2.40)   Next: 2.4 Transport processes Up: 2. Elements of Kinetic Previous: 2.2 The Maxwell-Boltzmann distribution
Franz Vesely
2005-01-25