30 Pressure, Average Translational Energy and Internal Energy
In this section we restrict ourself to a 'non-relativistic' gas, i.e. we suppose that the classical formula may be applied for the kinetic energy of the particles. This limitation is far from being narrow: Given a temperature of 100 Mio K the mean velocity of a Helium atome is less than 0.25 % of the speed of light, while the classical formula for kinetic energy is still perfect even for velocities up to 10 % of c ( have a look at the illustration at the end of E4 ).
Simple statistical considerations show that the pressure an ideal gas exerts on a vessels wall is proportional to the average particle density and can be calculated using the following formula:
The velocity u jof a gas particle ist measured relative to the center of mass of the entire gas. If we consider again (as in section 5) the y-direction of the relative motion, we know exactly how the individual terms of the equation behave. With P' j= P , N' j= N , V' j= V · √ and m' j= m / √ it follows from the above formula
For particle velocities u jor u' , which are small compared to the speed of light, we may use the classical formula for kinetic energy. Hence we get
This equation applies mutatis mutandis for a fast moving observer, since the average kinetic energy transforms like the transverse velocity u :
The average kinetic energy of the particles and V jtransform both by multiplying by the root factor. Therefore the second to last formula is form-invariant. Multiplying that equation by V jwe get
Both sides of this equation are transformed by multiplication by the root term. The mean kinetic energy (always to be related to the center of mass of the gas) is lower to a fast-moving observer, and therefore the heat content of the gas is lower, too. For an ideal gas we have
and following 8 we find again
U' = U · √
U , P · V and H have to be transformed the same way as k ·T and the same way as the mean kinetic energy, that is by multiplication by the root term.