Sackur–Tetrode equation

The Sackur - Tetrode equation is a formula for calculating the entropy S of an ideal gas. It reads:

With:

Otto Sackur and Hugo tetrode presented independently to the equation.

Conclusions

Since the entropy is known of the variables can be temperature, pressure and chemical potential can be derived ( see Microcanonical Ensemble ):

Thus, the inverse temperature is obtained by differentiating according to the energy:

From this one obtains the caloric equation of state:

From this we obtain the thermal equation of state:

With the thermal de Broglie wavelength and the relation for the internal energy can be the Sackur - Tetrode equation be written as:

Derivation

A - atomiges ideal gas was in a sealed box ( constant volume, no energy or particle exchange with the environment, no external fields). It is therefore to describe mikrokanonisch. Here, the required entropy computed over from the partition function.

The microcanonical partition function is:

The gas particles may be individual atoms ( no rotation or vibration, only translational possible), which do not interact with each other. The corresponding Hamiltonian is:

Used in the partition function:

The spatial integrations were run easily. Now it proceeds to dimensional spherical coordinates to simplify the pulse integration. The radius is thus a volume element writes as radius times element surface element.

The integral over the surface ( sphere ) of a 3N -dimensional unit sphere and is:

The delta function can be rewritten as:

Results used in the partition function:

In the limit of large particle numbers one can develop the faculty with the Stirling formula to second order ::

The entropy arises from now:

For large one can neglect the last term. Rearranging yields the Sackur - Tetrode equation:

The case of a harmonic trap potential is discussed as extension to.