Boltzmann Statistics Notes

Boltzmann Statistics

Introduction

• This chapter provides a detailed approach to compute thermodynamic quantities using microscopic models and first principles.

• Expands upon earlier models such as the two-state paramagnet, Einstein solid, and monatomic ideal gas, incorporating more complex scenarios.

• Introduces essential theoretical tools designed for handling intricate models where direct combinatorics becomes challenging.

The Boltzmann Factor

• The Boltzmann factor is introduced to determine the probability of a system existing in a specific microstate when thermodynamically balanced with a reservoir at a defined temperature.

• A single atom with microstates that correspond to its energy levels exemplifies this system.

  • Energy levels are dictated by the solutions to the Schrodinger equation for the atom, solved given the atom's potential.

• Degeneracy: Degeneracy occurs when an energy level is associated with multiple independent states.

  • Mathematically, if multiple states $s1, s2, …, sn$ all have energy $$E(s1) = E(s2) = … = E(sn) then this energy level is said to be degenerate.

  • Degeneracy is often associated with symmetry, where different quantum numbers can yield the same energy.

• The fundamental postulate in statistical mechanics states that every accessible microstate in an isolated system has equal probability.

  • Accessible refers to states that are permissable given the macroscopic constraints on the system like total energy, number of particles, and volume.

• The combination of the atom and the reservoir creates an isolated system.

• 
  $: Represents the reservoir's multiplicity when the atom is in state$ $s_1$.

• The likelihood of the atom being in a particular state is directly proportional to the count of microstates available to the reservoir.

• $\frac{P(s2)}{P(s1)}=\frac{\left|(s2)\right.}{\Omega(s1)}$

• Expressing multiplicity via entropy using $S = k \ln \Omega$:

$\frac{P(s<em>2)}{P(s</em>1)} = \frac{e^{S<em>R(s</em>2)/k}}{e^{S<em>R(s</em>1)/k}} = e^{[S<em>R(s</em>2) - S<em>R(s</em>1)]/k}$

• The thermodynamic identity is defined as:

$dS<em>R = \frac{1}{T} dU</em>R + P dV<em>R - \mu dN</em>R$

• The energy, volume, and particle number changes in the reservoir are linked to the atom's energy changes.

• $S<em>R(s</em>2) - S<em>R(s</em>1) = \frac{1}{T}[U<em>R(s</em>2) - U<em>R(s</em>1)] = - \frac{1}{T}[E(s<em>2) - E(s</em>1)]$

• The probability ratio is:

$\frac{P(s<em>2)}{P(s</em>1)} = e^{-[E(s<em>2) - E(s</em>1)]/kT} = \frac{e^{-E(s<em>2)/kT}}{e^{-E(s</em>1)/kT}}$

• Boltzmann factor:

$e^{-E(s)/kT}$

• The probability of each state:
  $P(s) = \frac{1}{Z} e^{-E(s)/kT}$

  • $Z$ represents the partition function, a normalizing constant to ensure the total probability equals 1.

  • Equation 6.8, also known as the Boltzmann or canonical distribution, is fundamental in statistical mechanics.

• Setting $E_0 = 0$ for the ground state, states having energies significantly less than $kT$ have probabilities close to $1/Z$, whereas those with energies far greater than $kT$ have negligible probabilities.

• A constant shift in all energies does not impact probabilities. All Boltzmann factors get an added factor of $e^{-E_0/kT}$, which also multiplies $Z$, leading to cancellation.

The Partition Function

• The cumulative probability of the atom existing in any state must be 1:

$1 = \sum<em>s P(s) = \sum</em>s \frac{1}{Z} e^{-E(s)/kT} = \frac{1}{Z} \sum_s e^{-E(s)/kT}$

• Solving for $Z$ yields:

$Z = \sum_s e^{-E(s)/kT} = \text{sum of all Boltzmann factors}$

• The partition function effectively calculates the number of accessible states for the atom, weighting each by its probability.

  • The partition function tells us how probability is distributed among the various energy states for a given temperature.

• At very low temperatures, $Z \approx 1$ since Boltzmann factors for excited states are minimal.

• At elevated temperatures, $Z$ becomes significantly larger, indicating greater accessibility to numerous states.

• Shifting all energies by a constant $E<em>0$ results in the partition function being multiplied by $e^{-E</em>0/kT}$, which is inconsequential when computing probabilities.

Thermal Excitation of Atoms

• Examines a hydrogen atom in the sun's atmosphere at approximately 5800 K.

• The probability ratio for finding the atom in its first excited states ($s<em>2$) versus the ground state ($s</em>1$) is:

$\frac{P(s<em>2)}{P(s</em>1)} = \frac{e^{-E<em>2/kT}}{e^{-E</em>1/kT}} = e^{-(E<em>2 - E</em>1)/kT}$

• With an energy difference of 10.2 eV and $kT$ at $(8.62 \times 10^{-5} \text{ eV/K})(5800 \text{ K}) = 0.50 \text{ eV}$, the probability ratio is approximately $e^{-20.4} = 1.4 \times 10^{-9}$.

• Approximately 1.4 atoms are in any first excited state for every billion in the ground state.

• Considering there are four such excited states with identical energy, the total number of atoms in these states is about 5.6 per billion in the ground state.

• This thermal excitation of atoms leads to phenomena like the Balmer series, observed through missing wavelengths in sunlight, as electrons transition between energy levels.

Average Values

• This section defines how to calculate the average value of any property of the system like its energy, position, or momentum.

• Considering an atom with three states: a ground state at 0 eV, a state at 4 eV, and one at 7 eV. If there are five such atoms, with two in the ground state, two at 4 eV, and one at 7 eV.

• The average energy is calculated as:

$E = \sum<em>s E(s) \frac{N(s)}{N} = \sum</em>s E(s)P(s)$

• The average energy equals the sum of all energies, each weighted by its probability.

• Using probabilities from equation 6.12:

$E = \frac{1}{Z} \sum_s E(s) e^{-E(s)}$

• The average for any variable is found similarly. If the variable is $X$, with a value $X(s)$ in state $s$, then

$X = \sum<em>s X(s)P(s) = \frac{1}{Z} \sum</em>s X(s) e^{-E(s)}$

• Average values are additive; the total average energy of two objects sums their individual average energies.

• For a collection of independent particles, the total average energy is the average energy of one particle times the number of particles:

$U = NE$

• $U$ represents the entire system's average energy.

Useful Formula for Average Energy

• An alternate representation of average energy is:

$E = - \frac{1}{Z} \frac{\partial Z}{\partial \beta} = - \frac{\partial}{\partial \beta} \ln Z$

Standard Deviation

• Quantifies energy fluctuations around the mean:

$\Delta E = \sqrt{E^2 - (E)^2}$

Fluctuation Formula

• Another way to express energy fluctuation is:

$\Delta E = \sqrt{kT^2 C}$

• $C$ denotes heat capacity.

Paramagnetism

• An ideal two-state paramagnet features elementary dipoles with two states: “up” at energy $-\mu B$, and “down” at $+\mu B$.

• The partition function for a single dipole is:

$Z = \sum_s e^{-E(s)} = e^{+ \mu B} + e^{-\mu B} = 2 \cosh(\beta \mu B)$

• The probability of the dipole being in the “up” state:

$P_{\uparrow} = \frac{e^{+ \mu B}}{Z} = \frac{e^{+ \mu B}}{2 \cosh(\beta \mu B)}$

• Conversely, the probability of it being in the “down” state:

$P_{\downarrow} = \frac{e^{-\mu B}}{Z} = \frac{e^{-\mu B}}{2 \cosh(\beta \mu B)}$

Average Energy and Total Energy

• The average energy of the dipole is:

$E = \sum<em>s E(s)P(s) = (-\mu B)P</em>{\uparrow} + (+\mu B)P<em>{\downarrow} = -\mu B(P</em>{\uparrow} - P_{\downarrow}) = -\mu B \frac{e^{\mu B} - e^{-\mu B}}{2 \cosh(\beta \mu B)} = -\mu B \tanh(\beta \mu B)$

• For $N$ such dipoles, the total energy is:

$U = -N \mu B \tanh(\beta \mu B)$

• The average magnetic moment of a dipole along $B$:

$\mu<em>z = \sum</em>s \mu<em>z(s)P(s) = (+\mu)P</em>{\uparrow} + (-\mu)P_{\downarrow} = \mu \tanh(\beta \mu B)$

• Therefore, the sample's total magnetization is:

$M = N \mu_z = N \mu \tanh(\beta \mu B)$

Rotation of Diatomic Molecules

• Considers the rotational motion of an isolated diatomic molecule in low-density gas, where rotational energies are quantized.

• For molecules like CO or HCl, allowed rotational energies are $E(j) = j(j + 1) \epsilon$, with $j$ being 0, 1, 2, etc., and $\epsilon$ inversely proportional to the molecule’s moment of inertia.

• Each level $j$ has $2j + 1$ degenerate states.

• The partition function is a sum over $j$:

$Z<em>{\text{rot}} = \sum</em>{j=0} (2j + 1)e^{-E(j)/kT} = \sum_{j=0} (2j + 1)e^{-j(j+1)\epsilon/kT}$

• The constant $\epsilon$ is generally a small fraction of an electron-volt, setting the energy scale for rotational excitations.

• If $\epsilon = 0.00024$ eV for CO, then $\epsilon/k = 2.8$ K.

• Typically, temperatures of interest far exceed $\epsilon/k$, making $kT/\epsilon$ much greater than 1.

• With many terms contributing significantly, the partition function approximates the area under a smooth curve:

$Z<em>{\text{rot}} \approx \int</em>{0} (2j + 1)e^{-j(j+1)\epsilon/kT} dj = \frac{kT}{\epsilon} \quad (\text{when } kT \gg \epsilon)$

High-Temperature Approximation

• Using formula 6.25, the average rotational energy is:

$E_{\text{rot}} = - \frac{1}{Z} \frac{\partial Z}{\partial \beta} = -(\beta \epsilon) \frac{\partial}{\partial \beta} \frac{1}{\beta \epsilon} = \frac{1}{\beta} = kT \quad (\text{when } kT \gg \epsilon)$

• This aligns with the equipartition theorem, given the two rotational degrees of freedom of a diatomic molecule.

The Equipartition Theorem

• The equipartition theorem is applicable to systems where energy presents in quadratic form, $E(q) = cq^2$, with $c$ being a constant and $q$ a coordinate or momentum variable.

• Z is expressed as:

$Z = \int e^{-\beta cq^2} dq$

Average Energy

• The average energy is:

$E = \frac{1}{2}kT$

The Maxwell Speed Distribution

• This distribution describes the speeds of molecules in a gas, with some moving faster and others slower.

• Probability(v1 . . . v2): $\text{Probability}(v<em>1 \ldots v</em>2) = \int<em>{v</em>1}^{v_2} D(v) dv$, where $D(v)$ is the distribution function.

• The function $D(v)$ is represented as:

$D(v) \propto \text{probability of a molecule having velocity } \vec{v} \times \text{number of vectors } \vec{v} \text{ corresponding to speed } v$

• Each velocity vector corresponds to a molecular state, with probability proportional to the Boltzmann factor $e^{-E(s)/kT}$.

• With translational kinetic energy $\frac{1}{2}mv^2$:

$\text{probability of a molecule having velocity } \vec{v} \propto e^{-mv^2/2kT}$

Number of Vectors

• For speed $v$, the number of vectors $\vec{v}$ is:

$4\pi v^2$

Maxwell Distribution

• This distribution is:

$D(v) = C \cdot 4\pi v^2 e^{-mv^2/2kT}$

• With a constant:

$C = \Big( \frac{m}{2\pi kT} \Big)^{3/2}$

Maxwell Speed Distribution Formula

• The complete distribution is:

$D(v) = \Big( \frac{m}{2\pi kT} \Big)^{3/2} 4\pi v^2 e^{-mv^2/2kT}$

Most Probable Speed

• The most probable speed is at:

$v_{\text{max}} = \sqrt{\frac{2kT}{m}}$

Average Speed

• Average speed is defined by

• $v = \int_{\text{all } v} v D(v) dv$

• $v = \sqrt{\frac{8kT}{\pi m}}$

Partition Functions and Free Energy

• The partition function is denoted as:

$Z(T)$

Free energy Formula

• Free energy is defined as:

$F = -kT \ln Z$

• In terms of the partition function

• $Z = e^{-F/kT}$

Partition Functions for Composite Systems

• Systems of noninteracting, distinguishable particles:

$Z<em>{\text{total}} = Z</em>1 Z_2 \quad (\text{noninteracting, distinguishable particles})$

• Systems of noninteracting, indistinguishable particles:

$Z<em>{\text{total}} = \frac{1}{2} Z</em>1 Z_2 \quad (\text{noninteracting, indistinguishable particles})$

• For N noninteracting, distinguishable systems:

$Z<em>{\text{total}} = Z</em>1 Z<em>2 Z</em>3 \ldots Z_N \quad (\text{noninteracting, distinguishable systems})$

• N noninteracting, indistinguishable particles have:

$Z<em>{\text{total}} = \frac{1}{N!}Z</em>1^N \quad (\text{noninteracting, indistinguishable particles})$

Ideal Gas Revisited

• For an ideal gas, the partition function is:

$Z = \frac{1}{N!} Z_1^N$

• $Z_1$ represents the partition function for a single molecule.

• The Boltzmann factor accounts for both translational kinetic energy ($E<em>{\text{tr}}$) and internal energy ($E</em>{\text{int}}$):

$e^{-E(s)/kT} = e^{-E<em>{\text{tr}}(s)/kT} e^{-E</em>{\text{int}}(s)/kT}$

• Therefore, $Z<em>1 = Z</em>{\text{tr}} Z_{\text{int}}$, with:

$Z<em>{\text{tr}} = \sum</em>{\text{translational states}} e^{-E_{\text{tr}}/kT}$

• and
  $Z<em>{\text{int}} = \sum</em>{\text{internal states}} e^{-E_{\text{int}}/kT}$

Energy Levels for Molecule in 1D box

• The energy levels are:

$E_n = \frac{h^2 n^2}{8mL^2}$

• The partition function in a 1D box is $Z_{\text{1d}} = L/\Lambda$, where $\Lambda = \frac{h}{\sqrt{2 \pi mkT}}$

  • $\Lambda$ is the thermal de Broglie wavelength.

Quantum Volume

• The quantum volume is:

$v_Q = \Lambda^3 = \Big( \frac{h}{\sqrt{2 \pi mkT}} \Big)^3$

Single Particle Parition Function

• The single particle partition function is

• $Z<em>1 = \frac{V}{v</em>Q} \cdot Z_{\text{int}}$

N Molecule Partition Function

• Extended to N molecules, this becomes:

$Z = \frac{1}{N!} \Big( \frac{V Z<em>{\text{int}}}{v</em>Q} \Big)^N$

• And:

$\ln Z = N \Big[ \ln V + \ln Z<em>{\text{int}} - \ln N - \ln v</em>Q + 1 \Big]$

Total Average Energy

• Average energy is

• $U = U_{\text{int}} + \frac{3}{2}NkT$

Thermal properties of an Ideal gas

• The Helmholtz Free Energy is:

$$ F= -NkT \Big[ \ln V - \ln N - \ln vQ + 1 \Big] + F{\text{int