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Chapter 2: Equation of State

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Chapter 2: Equation of State Chapter 2: Equation of State Introduction The Local Thermodynamic Equilibrium The Distribution Function Black Body Radiation Fermi-Dirac EoS The Complete Degenerate Gas Application to White Dwarfs Temperature Effects Ideal Gas The Saha Equation “Almost Perfect” EoS Adiabatic Exponents and Other Derivatives Outline Introduction The Local Thermodynamic Equilibrium The Distribution Function Black Body Radiation Fermi-Dirac EoS The Complete Degenerate Gas Application to White Dwarfs Temperature Effects Ideal Gas The Saha Equation “Almost Perfect” EoS Adiabatic Exponents and Other Derivatives The EoS, together with the thermodynamic equation, allows to study how the stellar material properties react to the heat, changing density, etc. Introduction Goal of the Chapter: derive the equation of state (or the mutual dependencies among local thermodynamic quantities such as P; T ; ρ, and Ni ), not only for the classic ideal gas, but also for photons and fermions. Introduction Goal of the Chapter: derive the equation of state (or the mutual dependencies among local thermodynamic quantities such as P; T ; ρ, and Ni ), not only for the classic ideal gas, but also for photons and fermions. The EoS, together with the thermodynamic equation, allows to study how the stellar material properties react to the heat, changing density, etc. Thermodynamics Thermodynamics is defined as the branch of science that deals with the relationship between heat and other forms of energy, such as work. The Laws of Thermodynamics: I First law: Energy can be neither created nor destroyed. This is a version of the law of conservation of energy, adapted for (isolated) thermodynamic systems. I Second law: In an isolated system, natural processes are spontaneous when they lead to an increase in disorder, or entropy, finally reaching an equilibrium. I Third law: The entropy of a system at absolute zero is a constant, determined only by the degeneracy of the ground state. For a given set of macroscopic variables, the entropy measures the degree to which the probability of the system is spread out over different possible microstates, according to the fundamental postulate in statistical mechanics: the occupation of any microstate is assumed to be equally probable. Entropy The most general interpretation of entropy is as a measure of our uncertainty about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved variables; maximizing the entropy maximizes our ignorance about the details of the system. Entropy The most general interpretation of entropy is as a measure of our uncertainty about a system. The equilibrium state of a system maximizes the entropy because we have lost all information about the initial conditions except for the conserved variables; maximizing the entropy maximizes our ignorance about the details of the system. For a given set of macroscopic variables, the entropy measures the degree to which the probability of the system is spread out over different possible microstates, according to the fundamental postulate in statistical mechanics: the occupation of any microstate is assumed to be equally probable. The thermodynamic relation The fundamental thermodynamic relation with chemical reactions: X dQ = TdS = dE + PdV − µi dNi i where µi is the chemical potential of the ith species @E µi = : @Ni S;V Chemical reactions determine the species number density Ni , which may have the units of number per gram of material; i.e., Ni = ni /ρ. Outline Introduction The Local Thermodynamic Equilibrium The Distribution Function Black Body Radiation Fermi-Dirac EoS The Complete Degenerate Gas Application to White Dwarfs Temperature Effects Ideal Gas The Saha Equation “Almost Perfect” EoS Adiabatic Exponents and Other Derivatives The equilibrium of chemical reactions means X µi dNi = 0; i appropriate for a star, since the mean free path of particles is generally short, compared to the variation scales of the −1 thermodynamic quantities; e.g., λph = (κρ) is on order of 1 cm ( except for stellar photospheres). The time scale to establish the LTE is also typically short enough, except for nuclear reactions, compared to the evolutionary time scale of a star with the exception of various explosions. The Local Thermodynamic Equilibrium LTE means that in a sufficiently small volume at any position in a star, the complete thermodynamic (mechanical, thermal, and chemical) equilibrium is very nearly true. The time scale to establish the LTE is also typically short enough, except for nuclear reactions, compared to the evolutionary time scale of a star with the exception of various explosions. The Local Thermodynamic Equilibrium LTE means that in a sufficiently small volume at any position in a star, the complete thermodynamic (mechanical, thermal, and chemical) equilibrium is very nearly true. The equilibrium of chemical reactions means X µi dNi = 0; i appropriate for a star, since the mean free path of particles is generally short, compared to the variation scales of the −1 thermodynamic quantities; e.g., λph = (κρ) is on order of 1 cm ( except for stellar photospheres). The Local Thermodynamic Equilibrium LTE means that in a sufficiently small volume at any position in a star, the complete thermodynamic (mechanical, thermal, and chemical) equilibrium is very nearly true. The equilibrium of chemical reactions means X µi dNi = 0; i appropriate for a star, since the mean free path of particles is generally short, compared to the variation scales of the −1 thermodynamic quantities; e.g., λph = (κρ) is on order of 1 cm ( except for stellar photospheres). The time scale to establish the LTE is also typically short enough, except for nuclear reactions, compared to the evolutionary time scale of a star with the exception of various explosions. As an example, let’s consider the ionization-recombination reaction: + − 0 H + e $ H + χH where χH = 13:6 eV is the ionization potential from the ground state of H. For simplicity, we have assumed that H has only one bound level and that the gas is pure hydrogen. Let us first consider a classic black body cavity filled with radiation in thermodynamic equilibrium with the wall. The equilibrium of a chemical reaction with photons can be written as X µi dNi + µγ dNγ = 0 i Since photon number is not strictly conserved, in general dNγ 6= 0. While dNi = 0 for the cavity wall (as a whole), we have µγ = 0. So we can neglect the photons here in the discussion of the equilibrium of a chemical reaction. Let us first consider a classic black body cavity filled with radiation in thermodynamic equilibrium with the wall. The equilibrium of a chemical reaction with photons can be written as X µi dNi + µγ dNγ = 0 i Since photon number is not strictly conserved, in general dNγ 6= 0. While dNi = 0 for the cavity wall (as a whole), we have µγ = 0. So we can neglect the photons here in the discussion of the equilibrium of a chemical reaction. As an example, let’s consider the ionization-recombination reaction: + − 0 H + e $ H + χH where χH = 13:6 eV is the ionization potential from the ground state of H. For simplicity, we have assumed that H has only one bound level and that the gas is pure hydrogen. Clearly in such an equilibrium, Ni is closely linked to µi and depends on T and ρ or equivalent thermodynamic quantities, as well as on a catalog of the possible reactions. The ionization-recombination reaction can then be written as 1H+ + 1e− − 1H0 = 0 And in general, a reaction can be written in such a symbolic form: X νi Ci = 0; i where Ci is the names of the participating particle (except for photons). Clearly, dNi / νi . Thus X µi νi = 0 i This is the equation of chemical equilibrium. The ionization-recombination reaction can then be written as 1H+ + 1e− − 1H0 = 0 And in general, a reaction can be written in such a symbolic form: X νi Ci = 0; i where Ci is the names of the participating particle (except for photons). Clearly, dNi / νi . Thus X µi νi = 0 i This is the equation of chemical equilibrium. Clearly in such an equilibrium, Ni is closely linked to µi and depends on T and ρ or equivalent thermodynamic quantities, as well as on a catalog of the possible reactions. Outline Introduction The Local Thermodynamic Equilibrium The Distribution Function Black Body Radiation Fermi-Dirac EoS The Complete Degenerate Gas Application to White Dwarfs Temperature Effects Ideal Gas The Saha Equation “Almost Perfect” EoS Adiabatic Exponents and Other Derivatives and “+” or “−” is for Fermi-Dirac or Bose-Einstein particles. The number of states in the phase space, accounting for the degeneracy of the state gj , is g d 3pd 3r j : h3 The Distribution Function For a particular species of elementary nature in the LTE, the occupation number at a certain quantum state in the coordinate-momentum phase space is 1 F(j; E(p)) = ; exp[−µ + Ej + E(p)]=kT ± 1 where Ej is the internal energy state j referred to some reference energy level, E(p) is the kinetic energy as a function of the momentum p, E(p) = (p2c2 + m2c4)1=2 − mc2, µ is the chemical potential of the species and is to be determined, The number of states in the phase space, accounting for the degeneracy of the state gj , is g d 3pd 3r j : h3 The Distribution Function For a particular species of elementary nature in the LTE, the occupation number at a certain quantum state in the coordinate-momentum phase space is 1 F(j; E(p)) = ; exp[−µ + Ej + E(p)]=kT ± 1 where Ej is the internal energy state j referred to some reference energy level, E(p) is the kinetic energy as a function of the momentum p, E(p) = (p2c2 + m2c4)1=2 − mc2, µ is the chemical potential of the species and is to be determined, and “+” or “−” is for Fermi-Dirac or Bose-Einstein particles.
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