18: Partition Functions and Ideal Gases
- Page ID
- 11814
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In chemistry, we are typically concerned with a collection of molecules. However, if the molecules are reasonably far apart as in the case of a dilute gas, we can approximately treat the system as an ideal gas system and ignore the intermolecular forces. The present chapter deals with systems in which intermolecular interactions are ignored. In ensemble theory, we are concerned with the ensemble probability density, i.e., the fraction of members of the ensemble possessing certain characteristics such as a total energy E, volume V, number of particles N or a given chemical potential μ and so on. The molecular partition function enables us to calculate the probability of finding a collection of molecules with a given energy in a system. The equivalence of the ensemble approach and a molecular approach may be easily realized if we treat part of the molecular system to be in equilibrium with the rest of it and consider the probability distribution of molecules in this subsystem (which is actually quite large compared to systems containing a small number of molecules of the order of tens or hundreds).
- 18.1: Translational Partition Functions of Monotonic Gases
- This page explores the translational partition function of a monatomic gas particle within a cubic box. It derives the function by summing energy levels based on quantum numbers, using integrals for closely spaced levels. The resulting function relates to volume and the de Broglie wavelength, indicating numerous available states for gas molecules. An example calculation for an iodine molecule at 300K demonstrates the concept, revealing a substantial number of thermally accessible quantum states.
- 18.2: Most Atoms are in the Ground Electronic State
- This page explains the electronic partition function, which measures accessible electronic states at temperature \(T\) based on electronic energies and their degeneracies. At typical temperatures, only the lowest electronic state, such as in \(\ce{H_2}\) at 300 K, significantly contributes, as higher states are energetically inaccessible. Consequently, the partition function approaches 1, confirming that only the ground state is thermally accessible at this temperature.
- 18.3: The Energy of a Diatomic/Polyatomic Molecule Can Be Approximated as a Sum of Separate Terms
- This page explains the differences in degrees of freedom between monatomic and polyatomic gases, highlighting that polyatomic gases, including diatomic ones, have additional rotational and vibrational energy storage capabilities. It discusses the overall partition function for polyatomic gases, which includes contributions from translational, vibrational, rotational, and electronic states, with the electronic partition function typically equating to one at room temperature.
- 18.4: Most Molecules are in the Ground Vibrational State
- This page discusses vibrational energy levels of diatomic molecules and the derivation of the vibrational partition function, linking it to bond stiffness and vibrational temperature. It includes a sample calculation for \(I_2\) at 300 K, showing the accessibility of vibrational states compared to rotational and translational states.
- 18.5: Most Molecules are Rotationally Excited at Ordinary Temperatures
- This page explains the rotational quantum number \(J\) and its impact on molecular rotations and energy levels, expressed as \(E(J)= \tilde{B} J(J+1)\) with degeneracy \(g(J)=2J+1\). It discusses the partition function \(q_\text{rot}(T)\) and highlights that at room temperature, diatomic gases have fewer energy levels, allowing integral approximations, while light gases need discrete summation at low temperatures.
- 18.6: Rotational Partition Functions of Diatomic Gases Contain a Symmetry Number
- This page covers the rotational energy levels of diatomic molecules, highlighting equations related to rotational constants and moment of inertia, and introduces rotational temperature where thermal energy equals energy level spacing. It discusses the rotational partition function, including the role of degeneracy and symmetry in homonuclear molecules.
- 18.7: Vibrational Partition Functions of Polyatomic Molecules Include the Partition Function for Each Normal Coordinate
- This page explores the partition function of polyatomic molecules by approximating their energy through individual degrees of freedom. It explains translational, rotational, vibrational, and electronic partition functions, with a focus on how normal modes describe vibrational motion. The vibrational energy and entropy relate to the potential energy surface's shape. For \(\ce{NO2}\) at 300 K, calculations indicate limited vibrational states, yielding a vibrational activity product of 1.
- 18.8: Rotational Partition Functions of Polyatomic Molecules Depend on the Sphar of the Molecule
- This page discusses the degrees of freedom in polyatomic molecules, focusing on rotational symmetry and energy calculations for nonlinear and linear molecules. It details the derivation of the rotational partition function for an asymmetric top, exemplified by nitrogen dioxide (\(\ce{NO2}\)), and evaluates thermodynamic properties while comparing classical and improved approaches.
- 18.9: Molar Heat Capacities
- This page describes a learning exercise aimed at evaluating the Dulong-Petit Law through the analysis of specific heat capacity data for monoatomic solids from elements in groups 1-14. Participants will gather data, analyze heat capacities and molar masses, create visual plots, assess deviations from the law, calculate the universal gas constant, and verify theoretical limits concerning heat capacity. Basic knowledge of statistical thermodynamics is required.
- 18.10: Ortho and Para Hydrogen
- This page explores hydrogen's ortho and para forms, highlighting their impact on rotational states and thermodynamics, and detailing the temperature-dependent ortho-para ratio and its behavior during cooling. It notes the slow interconversion rates without catalysts and examines oxygen molecules' contribution to the rotational partition function, emphasizing statistical thermodynamics principles.
- 18.11: The Equipartition Principle
- This page discusses the equipartition theorem, which states that each quadratic degree of freedom contributes \(½k_BT\) to average energy, affecting heat capacity. In polyatomic gases, various motions contribute at high temperatures, while lower temperatures diminish vibrational contributions, leading to a failure of the theorem—resolved through quantum mechanics.
- 18.E: Partition Functions and Ideal Gases (Exercises)
- This page features homework exercises from Chapter 18 of "Physical Chemistry: A Molecular Approach" by McQuarrie and Simon. Key topics include spin degeneracy of nuclei, thermodynamic functions from partition functions, calculations for diatomic and polyatomic molecules, equilibrium constants for chemical reactions, and the impacts of anharmonicity and centrifugal distortion on molecular energy. It also addresses the fraction of atoms in excited states at different temperatures.