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Ch5 van der Waals

Ch5 van der Waals - Chapter 5 Phase Transition I van der...

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Chapter 5 Phase Transition I: van der Waals Gas 5.1 General remarks Phase transitions occur at discontinuities of the equilibrium thermodynamic functions with varying certain physical parameters such as temperature, density, magnetic field, etc. For a finite system, thermodynamic functions such as grand potential are analytical for all physi- cal parameters, though it could have singularities when the parameters are taken as imaginary numbers (by continuation to the complex plane). In the thermodynamic limit, the singularities approach to the real axis of the physical parameters, and the discontinuity and hence the phase transition takes place. 5.1.1 Order of phase transition The energy, free energy, and grand potential of a system are always continuous with varying the temperature, volume, magnetic field, etc. But their di ff erentials may not. Phase transitions can be classified according to which order of the di ff erentials is discontinuous. To be specific, we consider the grand potential . The discussions apply to energy U , free energies F or G similarly. We consider a macroscopic system with volume V → ∞ or number of particles N → ∞ . So we would rather consider the potential per unit volume, i,e., the pressure P = / V , (5.1) which is a function of the density ρ (or specific volume v V / N = ρ 1 ) and the temperature T . A specific physical system is described by its state of equation P = P ( v , T ) . (5.2) For example, the state of equation of an ideal gas is P = k B T v . (5.3) For an ideal gas, P is analytical for temperature and volume as real numbers, so there is no phase transition. When the n th order di ff erential of P regarding T or V is discontinuous and all 63
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64 CHAPTER 5. PHASE TRANSITION I: VAN DER WAALS GAS the lower order di ff erentials are continuous, the phase transition is called an n th order one. For example, the n th order di ff erential of the isothermal curve [ n P ( v , T ) v n ] T (5.4) is discontinuous for the n th order phase transition. Examples of first-order phase transitions are melting of ice and condensation of vapor. 5.1.2 Finite systems: No phase transition With the sole exception of Bose-Einstein condensation, all phase transitions are due to inter- action between particles. 1 Let us consider an interacting gas with potential of the following form u ( r ) = + , r < d > u 0 , d r < D 0 , r D . (5.5) Such a model simulates atoms of strong repulsion at short distances and short-ranged weak attraction at long distance. The partition function of a grand canonical ensemble is Ξ = N = 0 z N Z N , (5.6) where z exp( βµ ) and Z N = Tr e β H ( N ) , (5.7) is the partition function of a canonical ensemble. The interaction potential in Eq. (5.5) means that two particles cannot be closer than d and hence the number of particles in a finite volume V cannot exceed a certain number N 0 V / d 3 . Thus the partition function is 2 Ξ ( V , T ) = N 0 N = 0 z N Z N ( V , T ) . (5.8) The pressure is P V = k B T V ln Ξ ( V , N ) . (5.9) When Ξ is positive, the pressure is analytical. For a finite system, the partition function is a
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Ch5 van der Waals - Chapter 5 Phase Transition I van der...

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