1: Principles
- Page ID
- 556040
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 1.1: Overview
- This page motivates the need for a statistical treatment of energy distribution amongst collections of particles on a macroscopic scale.
- 1.2: Microscopic Statistics
- This page explains the distribution of thermal energy in twelve quantum harmonic oscillators, simulating gas molecules. It presents two scenarios for allocating 2 quanta of vibrational energy: one with a single molecule excited (12 configurations) and another with two excited molecules (66 configurations). Key definitions of macrostate, configuration, microstate, and weight are discussed to enhance understanding of thermodynamic principles in the system.
- 1.3: Configuration Weights
- This page gives a general formula for calculating the weight of thermodynamic configurations, addressing the challenges in enumerating configurations as systems grow larger. It emphasizes that the most probable configurations dictate the behavior of macroscopic systems, especially as particle numbers and energy distribution increase. Additionally, it raises the question of how to identify the most probable way of distributing thermal energy.
- 1.4: The Boltzmann Distribution
- Boltzmann's method enables the determination of the most probable configuration using logarithms, Stirling's approximation, and Lagrange's multipliers, arriving at a formula for the Boltzmann distribution.
- 1.5: The Boltzmann Constant
- This page explores the meaning of the parameter beta that arises during derivation of the Boltzmann Distribution. A thought experiment illustrates that increasing beta results in fewer particles at higher energy states, signifying reduced thermal energy. It introduces the Boltzmann constant which connects temperature to particle distribution in thermal equilibrium, bridging microscopic quantum behaviour with macroscopic thermodynamics.
- 1.6: Dealing with Degeneracy
- This page explains the concept of degenerate solutions to the Schrödinger equation, focusing on quantum states with identical energy, such as degenerate 2p atomic orbitals. It emphasizes a method for simplifying calculations by computing each Boltzmann factor once and multiplying by the number of degenerate states.
- 1.7: The Partition Function
- This page defines the partition function as the denominator of the Boltzmann distribution expression, corresponding to the sum over all Boltzmann factors. An example illustration of how the partition function changes with temperature and energy level spacings is also provided.
- 1.8: Internal Energy
- This page explains the calculation of thermal energy using Boltzmann populations. It describes alternative methods ranging from a simple energy sum to more efficient energy-weighted sums and differentiation of the partition function. It also notes that the thermal energy increases with temperature, affected by energy level spacings, demonstrating linear growth at high temperatures and a faster-than-linear decrease near absolute zero.
- 1.9: Entropy
- This page discusses the relationship between macroscopic entropy and microscopic molecular behavior in statistical thermodynamics. It explains that higher entropy indicates more ways to distribute thermal energy. At absolute zero, entropy is zero, while at infinite temperature, entropy tends towards a constant value as thermal energy becomes equally distributed amongst all available energy levels.