The single perfect gas

Let /I and/e be the chemical potentials of a pure gas at the pressures p and respectively, and the same temperaJture T. The gas will be said to be perfect if 

Since (3 2) is an identity it is permissible to equate the partial differential coefficients of the left-hand and right-hand sides.f Thus from the equation we obtain the pressure and temperature coefficients of the chemical potential of a perfect gas 

Either of these two relations leads immediately to the normal experimental criterion of the perfect gas. Now for any single component fluid we have rom (2 1116) and (2 1126) 

The following elementary theorems concerning the perfect gas are easily verified  

It may be noted that the definition of a perfect gas, as given above, does not imply that the heat capacities are independent of temperature. This occurs only in the special case of the monfi tomic gas. However, over small ranges of temperature, especially in regions where the rotational degrees of freedom are fully excited but where the vibrational modes are hardly excited at all, it is often permissible to take the heat capacities as being approximately constant. Under such conditions an important equation may be obtained as follows. 

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