Heat, work and entropy change of an ideal gas in PV diagram.

See attached file.

Ideal gas with vibrational degrees of freedom.

Consider the model of an ideal gas, where every molecule has an additional vibrational degree of freedom, which can be described by an oscillator potential. Since there is no interaction between the molecules (apart from a very weak interaction which allows a thermal equilibrium to arise) the oscillators can be regarded as decoupled. The total energy of the system is therefore given by the sum of the kinetic energies and the oscillatory energies of the molecules: E = E(kin) + E(osc) =  (E(kin)i + E(osc)i) where i ranges from 1 to N. The entropy of an ensemble of decoupled harmonic oscillators is given by S(osc) = kBN [ln (2E(osc)/h) + 1] a) Use this result t... click for more

Pressure and number of particles in terms of grand potential.

See attached file.

Pressure of ideal Bose and Fermi gas satisfy PV = 2/3 E

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We consider a classical ideal gas of N particles of mass m which are independent of each other and are in a volume V. The particles are in an external potential which has a value epsilon in a part volume v < V and the value 0 in the remaining area V - v. During compression only the volume V changes, i.e. v is constant.

See attached file.