8: Boltzmann's and Saha's Equations
- Page ID
- 6698
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 8.1: Introduction
- This page explains the relationship between spectrum line strength and atom quantity in energy levels during transitions, highlighting the roles of emission and absorption lines. It emphasizes the importance of Boltzmann's equation for atomic distribution and Saha's equation for evaluating elemental abundance by factoring in ionization stages.
- 8.2: Stirling's Approximation. Lagrangian Multipliers.
- This page covers two key mathematical concepts for Boltzmann's equation: Stirling's Approximation and Lagrangian Multipliers. Stirling's Approximation aids in approximating logarithms of large factorials, especially useful in statistical mechanics. Lagrangian Multipliers are introduced for finding maximum values of functions under constraints, leading to conditions necessary for achieving extrema in multi-dimensional spaces with multiple constraints.
- 8.3: Some Thermodynamics and Statistical Mechanics
- This page outlines prerequisites for understanding Boltzmann's equation, emphasizing Stirling's approximation, Lagrangian multipliers, and fundamental concepts from thermodynamics and statistical mechanics. It presents the key thermodynamic relation \(dU = TdS - PdV\) and links entropy to thermodynamic probability via \(S = k \ln X\). The author notes that while a deep understanding of the derivation is beneficial, focusing on the application of Boltzmann's equation is more important.
- 8.4: Boltzmann's Equation
- This page covers the dynamics of energy levels in hot, dense gases, focusing on the balance of collisional excitation and radiative de-excitation. It introduces Boltzmann's Equation for population distributions, deriving the most probable distribution using Lagrangian multipliers. The discussion includes degenerate energy levels and the necessity of incorporating statistical weights in calculations.
- 8.5: Some Comments on Partition Functions
- This page covers two main topics: the divergence of partition functions in atomic hydrogen, contrasting metals and nonmetals' energy levels, and the total energy in simple diatomic molecules. It addresses the issue of infinite Rydberg levels and practical adjustments in partially ionized gases for finite computations.
- 8.6: Saha's Equation
- This page covers the Saha equation, detailing the equilibrium of neutral atoms, ions, and electrons in gases, influenced by temperature and pressure. It emphasizes the role of partition functions in understanding ionization and recombination, especially in astrophysics for stellar classification.
- 8.7: The Negative Hydrogen Ion
- This page explores the negative hydrogen ion (\(H^−\)), which has one proton and two electrons. It explains the capture of the second electron due to an induced dipole moment, creating a unique energy structure with one bound level below its ionization limit of 0.7 eV. The page also discusses the abundance of neutral hydrogen in the solar atmosphere and the role of \(H^−\) in contributing to continuous opacity via bound-free and free-free transitions.
- 8.8: Autoionization and Dielectronic Recombination
- This page explores ionization energy and the dynamics of excited atomic states. It details how atoms above the ionization limit can engage in autoionization, allowing electrons to escape without extra energy, and how excited ions can undergo dielectronic recombination by capturing free electrons. Additionally, it highlights the uncertainties in energy levels caused by the short lifetimes of these excited states, leading to broad spectral lines in elements such as copper, zinc, and cadmium.
- 8.9: Molecular Equilibrium
- This page covers the dissociation of diatomic molecules, relating it to Saha's equation for ionization, and examines the equilibrium of the reaction AB ↔ A + B with an associated equation. It further presents a problem involving methyl cyanate (CH3CNO) at specific conditions, resulting in 14 equations that include the ideal gas law and stoichiometric relations, requiring proficiency in solving multiple nonlinear equations.
- 8.10: Thermodynamic Equilibrium
- This page discusses the intricate nature of temperature in thermodynamics, differentiating it from simpler concepts like entropy and enthalpy. It highlights the Zeroth Law of Thermodynamics, which establishes that bodies in equilibrium have the same temperature. Additionally, it presents various temperature definitions—kinetic, excitation, ionization, vibrational, rotational, effective, and color temperature—emphasizing that these definitions align only in thermodynamic equilibrium.