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 Mayer’s relation which connects the isochoric and isobaric heat capacities is wrong. This paradox is valid also for real gases because, in a wide range of temperatures and pressures, they only minimally deflect from the ideal gas. It is interesting to note that the obtained result explains the enthalpy paradox. Thermodynamic potentials internal energy, U, and enthalpy, U + PV, are qualitatively different, but, for the ideal gas, they are identical thermodynamically and differ only in the multiplying factor in that U equals 1.5PV, and H equals 2.5PV. If everything were correct in traditional thermodynamics, then U would not be thermodynamically identical to H even for the ideal gas.
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