For an ideal gas having a fixed number N of particles, the internal energy depends only on the temperature and not at all on the volume: E = (ν/2)NkT. Furthermore, the pressure, volume, and temperature are related by the ideal gas law, pV = NkT. The first law for a system of a fixed number of particles reads dE = dQ−pdV. Using this information and the definition of molar heat capacities, show that (a) CV = (ν/2)R, (b) Cp = CV + R, where R = NAk is the molar gas constant.
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