Exact Calculation of the Internal Energy of the Ideal Gas in Statistical Mechanics

Previously, in the calculation of the internal energy of the ideal gas in statistical mechanics, it has been supposed that the volume is a constant, which does not depend on any arguments. However, the volume depends on pressure and temperature and its partial derivative is not equal to zero. In this paper, the dependence of the volume on pressure and temperature is taken into account, and the internal energy is calculated exactly. It differs from the traditional internal energy by the product of the pressure and volume. This explains three paradoxes in thermodynamics. It follows that the isochoric heat capacity equals the isobaric one. It is shown that the derivation of the Mayer’s relation which connects the isochoric and isobaric heat capacities, is wrong.