5: Boltzmann
- Page ID
- 426430
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 5.1: The Boltzmann Factor is used to Approximate the Fraction of Particles in a Large System
- The proportionality constant \(k\) (or \(k_B\)) is named after Ludwig Boltzmann. It plays a central role in all statistical thermodynamics. The Boltzmann factor is used to approximate the fraction of particles in a large system. The Boltzmann factor is given by: \(e^{-\beta E_i}\).
- 5.2: The Boltzmann Distribution represents a Thermally Equilibrated Distribution
- The Boltzmann distribution represents a thermally equilibrated most probable distribution over all energy levels. There is always a higher population in a state of lower energy than in one of higher energy.
- 5.3: The Average Ensemble Energy is Equal to the Observed Energy of a System
- The probability of finding a molecule with energy \(E_i\) is equal to the fraction of the molecules with energy \(E_i\). The average energy is obtaining by multiplying \(E_i\) with its probability and summing over all \(i\): \[\langle E \rangle=\sum_i{E_iP_i} \nonumber\]. Using the Boltzmann distribution for \(P_i\) allows us to show that the average ensemble energy is equal to the observed energy of the system.
- 5.4: Heat Capacity at Constant Volume is the Change in Internal Energy with Temperature
- The heat capacity at constant volume, denoted \(C_V\), is defined to be the change in thermodynamic energy with respect to temperature.
- 5.5: Partition Functions can be Decomposed into Partition Functions of Each Degree of Freedom
- Just as the partition function of a system of \(N\) particles can be decomposed into the product of partition functions for each molecule, a molecular partition function can be decomposed into the product of the partition functions for each degree of freedom.
- 5.6: Translational Partition Functions of Monotonic Gases
- The energy levels of translation are very closely spaced, so a large number of translational states are accessible and available for occupation by the molecules of a gas. This result is very similar to the result of the classical kinetic gas theory.
- 5.7: Most Atoms are in the Ground Electronic State
- The energy difference between the ground electronic state of a system and its first excited state are typically much larger the thermal energy, \(kT\). This means that most atoms are in their ground electronic state, unless the temperature of the system is very high.
- 5.8: The Energy of a Diatomic/Polyatomic Molecule Can Be Approximated as a Sum of Separate Terms
- A reasonable partition function of a diatomic/polyatomic molecule is the product of the partition function for the translational, vibrational, rotational, and electronic degrees of freedom. The total energy of the molecule then becomes the sum of the translational, vibrational, rotational, and electronic energies.
- 5.9: Most Molecules are in the Ground Vibrational State
- At room temperature, most molecules are in the ground vibrational state. This is because the vibrational energies of molecules are larger than the average thermal energy available.
- 5.10: Most Molecules are Rotationally Excited at Ordinary Temperatures
- At room temperature, many rotational states will be populated. This is due to the smaller rotational energies compared to vibrational or electronic energies.
- 5.11: Rotational Partition Functions of Diatomic Gases Contain a Symmetry Number
- Homonuclear diatomic molecules have a high degree of symmetry and rotating the molecule by 180° brings the molecule into a configuration which is indistinguishable from the original configuration. This leads to an overcounting of the accessible states. To correct for these symmetry factors, we divide the partition function by \(σ\), which is called the symmetry number.