Statistical Mechanics - Physics at Oregon State University

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78 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES. In our discussion of the ideal gas we have assumed that the orbitals are characterized by three quantum numbers nx, ny, and nz. These three quantum numbers describe the motion of the center of mass of the particle, but did not include any internal degrees of freedom. Therefore, the previous formulas only are valid for a gas of mono-atomic molecules. A gas of poly-atomic molecules is easily treated if we assume that the energy associated with the internal degrees of freedom does not depend on (1) the presence of other particles and (2) the motion of the center of mass. Hence the rotation of a diatomic molecule is not hindered by neighboring molecules or changed by its own motion. In that case we can write, using int for the collective internal degrees of freedom: ɛ(nx, ny, nz, int) = 2 π 2 (n 2M L 2 x + n 2 y + n 2 z) + ɛint (4.43) Then internal degrees of freedom represent the rotational quantum numbers (of which there are at most three) and the vibrational quantum numbers. If we have a molecule with N atoms, there are 3N internal degrees of freedom. Three are used for the center of mass, r for the rotations (r is 2 or 3) and hence 3(N − 1) − r for the vibrational state. Changes in the partition function. In the classical regime µ ≪ −kBT and hence λ = e µ k B T ≪ 1. In the partition function for a given orbital terms with λ 2 and higher powers can be neglected for bosons and in all cases we find Zo(T, µ, V ) = 1 + λ where the internal partition function is defined by int Zint(T ) = e − ɛo+ɛint ɛo kB T − k = 1 + λZinte B T (4.44) int e − ɛ int k B T (4.45) The average number of particles in orbital o, independent of its internal state, is therefore given by < no >= int λe − ɛ int k B T e − ɛo k B T 1 + λ int e− ɛ int k B T e − ɛo k B T ɛo − k ≈ λZinte B T (4.46) where we used the fact that the denominator is approximately equal to one because λ is very small. Hence in the Boltzmann distribution function we have to replace λ by λZint(T ) and we find N = λZint(T )nQ(T )V (4.47) n µ = kBT log( ) − log(Zint) nQ(T ) (4.48)
4.4. DEGENERATE GAS. 79 n F = NkBT log( ) − 1 + Fint(T, N) (4.49) nQ(T ) Fint(T, N) = −NkBT log Zint(T ) (4.50) Experimental access to the internal degrees of freedom. For the entropy and internal energy we also find formulas that show that these quantities are the sum of the entropy/energy for the motion of the center of mass plus a term pertaining to the internal degrees of freedom. Since we have assumed that the energy of the internal motion does not depend on other particles, Fint does not depend on volume and the ideal gas law pV = NkT remains valid! The heat capacities do change, however, since the internal energy is different. Therefore, the value of the ratio γ is different. As we will see later, the value of this ratio is a direct measure of the number of active degrees of freedom. Hence by measuring the pressure-volume relation in adiabatic expansion gives us direct information about the internal structure of the molecules! Do we have them all? The entropy is the sum of the entropy associated with the center of mass motion and the entropy of the internal state of the molecules. Hence the entropy is now larger than the entropy of mono-atomic molecules. We need more information to describe the state of large molecules, and hence we have more ”un-knowledge”, which translates into a larger entropy. How do we know if we have all internal degrees of freedom accounted for? In principle, we do not know, because there could always be hidden degrees of freedom. For example, do we have to separate electrons and nucleï? Neutrons and protons? Quarks? Hence the real value of the entropy could even be larger as what we calculated here. We can also ask the question from an experimental point of view. In that case, the answer is simple, cool the system down to a low temperature. But what is low? Sometimes we see phase transitions at very low temperatures, and we are never sure if we went low enough. For example, the nuclear spins could order below a certain temperature, and the values of such temperatures are small indeed. 4.4 Degenerate gas. The entropy of an ideal or Boltzmann gas is given by the Sackur-Tetrode formula. The temperature enters this equation through the quantum concentration. In the limit T → 0, the entropy is approximately 3 2NkB log(T ) and approaches −∞, which is obviously incorrect. The Sackur-Tetrode formula is not
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Statistical Mechanics Henri J.F. Ja
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IV CONTENTS 4.3 Gas of poly-atomic
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VI CONTENTS
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VIII INTRODUCTION Introduction. The
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X INTRODUCTION In chapter nine we d
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2 CHAPTER 1. FOUNDATION OF STATISTI
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10 CHAPTER 1. FOUNDATION OF STATIST
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128 CHAPTER 6. DENSITY MATRIX FORMA
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140 CHAPTER 7. CLASSICAL STATISTICA
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158 CHAPTER 8. MEAN FIELD THEORY: C
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192 CHAPTER 9. GENERAL METHODS: CRI
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230 APPENDIX A. SOLUTIONS TO SELECT
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