Thermodynamic and Gas Scales of Temperature

Equation (6.1) states that the internal energy is independent of volume at constant temperature, the usual statement of Joule s law. 

Without further assumption about the gas, we can now prove that f(T) = constant X T, so that the pressure of a perfect gas at constant 

Instead of defining a perfect gas as we have done, by Boyle s law and Joule s law, we may prefer to assume that a thermodynamic temperature scale is known, and that the perfect gas satisfies the general gas law PV = const. X T. Then we can at once use the relation (6.2) to calculate the change of internal energy with volume at constant temperature, and find it to be zero. That is, we show directly by thermodynamics that Joule s lawr follows from the gas law, if that is stated in terms of the thermodynamic temperature. 

Thermodynamics is a simple, general, logical science, based on two postulates, the first and second laws of thermodynamics. We have seen in the last chapter how to derive results from these laws, though we have not used them yet in our applications. But we have seen that they are limited. Typical results are like Eq. (5.2) in Chap. II, giving the difference of specific heats of any substance, CP — CV in terms of derivatives which can be found from the equation of state. Thermodynamics can give relations, but it cannot derive the specific heat or equation of state directly. To do that, we must go to the statistical or kinetic methods. Even the second law is simply a postulate, verified because it leads to correct results, but not derived from simpler mechanical principles as far as thermodynamics is concerned. We shall now take up the statistical method, showing how it can lead not only to the equation of state and specific heat, but to an understanding of the second law as well. 

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