The Partition Function and the Average Molecular Energy

The partition function, \( q \), is fundamental to statistical thermodynamics. It describes how energy is distributed across molecular energy levels and enables the calculation of average molecular properties, such as energy.

Average Molecular Energy

The total energy of the system is written as:

\[E_{\text{tot}} = N_1 E_1 + N_2 E_2 + \cdots = \sum_i N_i E_i\]

where \( N_i \) is the number of particles at energy level \( E_i \). Using the Boltzmann distribution to express the population at each energy level:

\[N_i = N \frac{e^{-\beta E_i}}{q},\]

with \( \beta = \frac{1}{k_B T} \), we rewrite \( E_{\text{tot}} \) as:

\[E_{\text{tot}} = \sum_i E_i \cdot N \frac{e^{-\beta E_i}}{q} = N \cdot \frac{\sum_i E_i e^{-\beta E_i}}{q}.\]

Recognizing the average energy per molecule as:

\[\langle \epsilon \rangle = \frac{E_{\text{tot}}}{N} = \frac{\sum_i E_i e^{-\beta E_i}}{q},\]

The expression $\sum_i E_i e^{-\beta E_i}$ is related to the derivative of the partition function $q$ with respect to the Boltzmann factor $\beta$:

\[\left(\frac{\partial q}{\partial \beta}\right)_{N,V} = -\sum_i E_i e^{-\beta E_i}\]

Therefore, the average molecular energy relates directly to the derivative of the partition function:

\[\langle \epsilon \rangle = -\frac{N}{q} \left(\frac{\partial \ q}{\partial \beta} \right)_{N,V} = -\left(\frac{\partial \ln q}{\partial \beta} \right)_{N,V}.\]

Relation to Temperature

Since \( \beta = \frac{1}{k_B T} \), we find:

\[\frac{d \beta}{dT} = -\frac{1}{k_B T^2}.\]

Using this relationship, the temperature dependence of \( \langle \epsilon \rangle \) is:

\[\langle \epsilon \rangle = k_B T^2 \left( \frac{\partial \ln q}{\partial T} \right)_{V}.\]

This expression highlights the direct dependence of the average energy on temperature through the partition function.

 

The Canonical Partition Function and Molar Energy

Previously, we introduced the molecular partition function, \( q \), which characterizes the distribution of energy states for a single particle. To describe an entire system of particles, we extend this concept to the canonical partition function, \( Q_N \), which accounts for the collective energy states of the system. This transition is necessary to connect the microscopic properties of individual particles to macroscopic thermodynamic quantities like molar energy.

System Energy States

A canonical ensemble is a statistical ensemble of identical systems at constant temperature \( T \), constant volume \( V \), and a constant number of particles \( N \). Each system in the ensemble can exchange energy with a heat reservoir, but the number of particles and the volume remain fixed for each system. This ensemble is used in statistical mechanics to describe systems where the total energy fluctuates, but the average energy is constant due to the fixed temperature. For simplicity, we often assume particles are independent. However, this assumption is restrictive because particle interactions can affect the system's energy states and configuration.

Canonical Distribution

The canonical partition function for a system of \( N \) particles, denoted \( Q_N \), is defined as:

\[ Q_N = \sum_i \exp(-\beta E_i), \]

where:

  • \( \beta = \frac{1}{k_B T} \) is the inverse thermal energy,
  • \( E_i \) represents the energy of the \( i \)-th state of the system,
  • The summation runs over all possible energy states of the system.

The formulation of \( Q_N \) depends on whether the particles are distinguishable or indistinguishable:

  • Distinguishable Particles: If the particles can be individually identified, the partition function is:
    \[ Q_N = q^N, \]
    where \( q \) is the molecular (single-particle) partition function.
  • Indistinguishable Particles: If the particles are identical and indistinguishable, permutations of particles do not represent distinct states. To avoid overcounting, we divide by \( N! \):
    \[ Q_N = \frac{q^N}{N!}. \]

Distinguishable Particles

For distinguishable particles, each particle has a unique identity, and we can differentiate between them. An example is molecules in a crystal lattice, where each molecule occupies a fixed position that does not change. The energy levels for the system are quantized as combinations of individual particle energy states, such as:

\[ E_{\text{tot}} = E_{A,0} + E_{B,0}, \quad E_{A,1} + E_{B,0}, \quad \text{and so on.} \]

The canonical partition function for distinguishable particles is expressed as:

\[ Q_N = \sum_N \exp\left[-\beta \left( E_{A,i} + E_{B,j} + \cdots \right) \right], \]

which simplifies to:

\[ Q_N = \left( \sum_i \exp(-\beta E_{A,i}) \right) \cdot \left( \sum_j \exp(-\beta E_{B,j}) \right) \cdot \cdots = q_A \cdot q_B \cdot \cdots. \]

If the partition functions for all particles are identical (e.g., \( q_A = q_B = q \)), the total partition function becomes a product of the the molecular (single-particle) partition function \( q \):

\[ Q_N = \prod_N q_N = q^N. \]

Indistinguishable Particles

For indistinguishable particles, such as molecules in a gas or solution, it is not possible to differentiate between particles based on their identity. Therefore, permutations of particles do not represent distinct sates but have the same energy configuration and are thus indistinguishable and must not be overcounted. For instance, consider three particles in three different states:

  • \( E_{\text{tot}} = E_1 + E_2 + E_3 \),
  • \( E_{\text{tot}} = E_2 + E_1 + E_3 \),
  • \( E_{\text{tot}} = E_3 + E_2 + E_1 \).

These configurations are identical. To correct for overcounting like:

\[ Q_N = \left( \sum_N \exp(-\beta E_{N}) \right) = q^N. \]

we divide the partition function by \( N! \), the number of permutations of \( N \) particles:

\[ Q_N = \frac{q^N}{N!}. \]

Microscopic and Macroscopic Connection

The canonical ensemble can be represented as the product of molecular partition functions. This representation connects the microscopic description of a system (e.g., quantum energy levels) to the macroscopic thermodynamic properties (e.g., internal energy, entropy), bridging statistical thermodynamics with classical thermodynamics.

Molar Energy

The average molar energy, \( \langle E \rangle \), is related to the derivative of the canonical partition function \( Q_N \) with respect to \( \beta \), at constant volume:

\[ \langle E \rangle = -\left( \frac{\partial \ln Q_N}{\partial \beta} \right)_{V}. \]

Substituting the definition of \( Q_N \):

\[ \ln Q_N = \ln \left( \sum_i \exp(-\beta E_i) \right). \]

Taking the derivative of \( \ln Q_N \):

\[ \left( \frac{\partial \ln Q_N}{\partial \beta} \right)_{V} = \frac{1}{Q_N} \cdot \left( \frac{\partial Q_N}{\partial \beta} \right)_{V}, \]

where:

\[ \frac{\partial Q_N}{\partial \beta} = -\sum_i E_i \exp(-\beta E_i). \]

Substituting back, we find:

\[ \langle E \rangle = \frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}. \]

This confirms that the molar energy is the weighted average of the system's energy levels, where the weights are the Boltzmann probabilities \( \exp(-\beta E_i) \).

 

The Partition Function and Molecular Energy Levels

Molecular Degrees of Freedom

The total energy of a molecule is the sum of its contributions from different degrees of freedom:

\[ E_{\text{tot}} = E_{\text{tr}} + E_{\text{rot}} + E_{\text{vib}} + E_{\text{el}}, \]

where:

  • \( E_{\text{tr}} \): Translational energy,
  • \( E_{\text{rot}} \): Rotational energy,
  • \( E_{\text{vib}} \): Vibrational energy,
  • \( E_{\text{el}} \): Electronic energy.

Decomposing the Partition Function

The molecular partition function, \( q_{\text{tot}} \), can be expressed as a product of contributions from each degree of freedom. It is written as:

\[ q_{\text{tot}} = \sum g_{\text{total}} \cdot \exp(-\beta E_{\text{tot}}), \]

where:

  • \( g_{\text{total}} \): Total degeneracy of the energy states,
  • \( E_{\text{tot}} \): Total molecular energy.

Substituting \( E_{\text{tot}} = E_{\text{tr}} + E_{\text{rot}} + E_{\text{vib}} + E_{\text{el}} \), the partition function becomes:

\[ q_{\text{tot}} = \sum g_{\text{tr}} \cdot g_{\text{rot}} \cdot g_{\text{vib}} \cdot g_{\text{el}} \cdot \exp\left[-\beta (E_{\text{tr}} + E_{\text{rot}} + E_{\text{vib}} + E_{\text{el}})\right]. \]

Since the terms are separable, \( q_{\text{tot}} \) can be decomposed into individual contributions:

\[ q_{\text{tot}} = q_{\text{tr}} \cdot q_{\text{rot}} \cdot q_{\text{vib}} \cdot q_{\text{el}}. \]

Translational Partition Function

The translational partition function can be derived from the quantum mechanical particle-in-a-box model. For one dimension, it is expressed as:

\[ q_{\text{tr,1D}} = \sum_{n} \exp\left(-\beta \cdot \frac{n^2 h^2}{8mL_x^2}\right), \]

where:

  • \( n \): Quantum number,
  • \( h \): Planck's constant,
  • \( m \): Mass of the particle,
  • \( L_x \): Length of the box in one dimension.

To simplify the summation, we approximate it as an integral for large \( n \). The result becomes:

\[ q_{\text{tr,1D}} = \int_0^\infty \exp\left(-\beta \cdot \frac{n^2 h^2}{8ma_x^2}\right) dn \]

with $A = \frac{\beta h^2}{8ma_x^2}$ we solve the integral:

\[ \int_0^\infty \exp\left(-An^2\right) dn = \frac{1}{2} \sqrt\frac{\pi}{A}\]

We then define the expression for the thermal wavelength, which is a statistical version of the de Broglie wavelength that determines whether quantum effects play a significant role in the system of interest. For example, if the thermal wavelength is smaller than the interparticle spacing the system behaves classically, while if it is comparable or larger, then the quantum effects dominate (e.g. Bose-Einstein condensate).

\[ \Lambda = \sqrt{\frac{h^2 \beta}{2\pi m}}, \]

Therefore, substituting the thermal de Broglie wavelength in the expression for the one-dimensional translational partition function we get:

\[ q_{\text{tr,1D}} = \frac{a_x}{\Lambda}. \]

Substituting \( \Lambda \) explicitly, we also write:

\[ q_{\text{tr,1D}} = L_x \cdot \sqrt{\frac{2 \pi m k_B T}{h^2}}. \]

This result shows that the partition function scales with the length of the system, the particle mass, and the temperature.

Extending this result to three dimensions, we find:

\[ q_{\text{tr}} = \left(\frac{V}{\Lambda^3}\right), \]

where \( V \) is the volume of the system, expressed as \( V = a_x \cdot a_y \cdot a_z \).

This result shows that the translational partition function depends on the volume of the system and the thermal de Broglie wavelength, which decreases as the particle mass or temperature increases.

 

Rotational Partition Function

The rotational partition function describes the contribution of rotational motion to the total molecular partition function. For a rigid rotator, rotational energy levels are quantized and given by:

\[ E_J = hcB J(J+1), \]

where:

  • \( J = 0, 1, 2, \dots \): Rotational quantum number,
  • \( h \): Planck’s constant,
  • \( c \): Speed of light,
  • \( B \): Rotational constant, which depends on the molecule's moment of inertia.

The rotational constant \( B \) is related to the moment of inertia \( I \) of the molecule by:

\[ B = \frac{h}{8\pi^2 c I}. \]

The moment of inertia \( I \) for a diatomic molecule is given by:

\[ I = \mu r^2, \]

where \( \mu = \frac{m_1 m_2}{m_1 + m_2} \) is the reduced mass of the two atoms, and \( r \) is the bond length between the two atoms.

Degeneracy and the Partition Function. Each energy level \( E_J \) has a degeneracy of \( g_J = 2J + 1 \). The rotational partition function is then expressed as:

\[ q_{\text{rot}} = \sum_J (2J + 1) \cdot \exp\left(-\beta hcB J(J+1)\right), \]

where \( \beta = \frac{1}{k_B T} \) is the inverse thermal energy. For small rotational energy spacings compared to thermal energy (\( k_B T \gg hcB \)), the summation can be approximated as an integral:

\[ q_{\text{rot}} = \int_0^\infty (2J + 1) \cdot \exp\left(-\beta hcB J(J+1)\right) dJ. \]

Solving this integral yields:

\[ q_{\text{rot}} = \frac{1}{\beta hcB}. \]

Substituting \( \beta = \frac{1}{k_B T} \), the rotational partition function becomes:

\[ q_{\text{rot}} = \frac{k_B T}{hcB}. \]

By introducing the rotational temperature \( \Theta_R = \frac{hcB}{k_B} \), which is defined as the temperature where the energy level spacing is comparable to the thermal energy $k_BT$, the rotational partition function can also be written as:

\[ q_{\text{rot}} = \frac{T}{\Theta_R}. \]

Physical Interpretation: The rotational partition function increases linearly with temperature, as more rotational energy levels become accessible at higher temperatures. The term \( \Theta_R \) represents the characteristic temperature scale for rotational motion, with larger \( \Theta_R \) indicating tightly bound systems or heavier molecules (due to smaller \( B \)). This partition function provides insight into how rotational energy contributes to thermodynamic properties such as internal energy and heat capacity.

 

Vibrational Partition Function

The vibrational partition function accounts for the quantized vibrational energy levels of a molecule. For a harmonic oscillator, these energy levels are given by:

\[ E_n = hc \tilde{\nu} \left(n + \frac{1}{2}\right), \]

where:

  • \( n = 0, 1, 2, \dots \): Vibrational quantum number,
  • \( \tilde{\nu} \): Vibrational wavenumber,
  • \( hc \tilde{\nu}/2 \): Zero-point energy, the lowest possible energy of the system.

The vibrational partition function is expressed as:

\[ q_{\text{vib}} = \sum_{n=0}^\infty \exp\left(-\beta E_n\right), \]

Substituting the expression for \( E_n \), we have:

\[ q_{\text{vib}} = \sum_{n=0}^\infty \exp\left(-\beta hc \tilde{\nu} \left(n + \frac{1}{2}\right)\right). \]

Factoring out the term corresponding to the zero-point energy:

\[ q_{\text{vib}} = \exp\left(-\frac{\beta hc \tilde{\nu}}{2}\right) \cdot \sum_{n=0}^\infty \exp\left(-\beta hc \tilde{\nu} n\right). \]

The summation over \( n \) is a geometric series of the form:

\[ \sum_{n=0}^\infty x^n = \frac{1}{1 - x}, \]

where \( x = \exp\left(-\beta hc \tilde{\nu}\right) \). Substituting this into the partition function:

\[ q_{\text{vib}} = \exp\left(-\frac{\beta hc \tilde{\nu}}{2}\right) \cdot \frac{1}{1 - \exp\left(-\beta hc \tilde{\nu}\right)}. \]

Simplifying:

\[ q_{\text{vib}} = \frac{\exp\left(-\frac{\beta hc \tilde{\nu}}{2}\right)}{1 - \exp\left(-\beta hc \tilde{\nu}\right)}. \]

In the high-temperature limit (\( \beta hc \tilde{\nu} \ll 1 \)), this simplifies further, but the exact form above captures the dependence of the vibrational partition function on the molecular vibrational frequency.

For polyatomic molecules, the total vibrational partition function is the product of the contributions from each vibrational mode \( i \), i.e. $3N-5$ and $3N-6$ modes for linear and non-linear molecules, correspondingly:

\[ q_{\text{vib,tot}} = \prod_i q_{\text{vib},i}. \]

 

Electronic Partition Function

The electronic partition function accounts for the electronic energy levels of a molecule. It is expressed as:

\[ q_{\text{el}} = \sum_n g_n \cdot \exp\left(-\beta hc E_n\right), \]

where:

  • \( g_n \): Degeneracy of the \( n \)-th electronic level,
  • \( E_n \): Energy of the \( n \)-th electronic level.

In most cases, the energy difference between electronic states (\( E_n - E_0 \)) is much larger than the thermal energy (\( k_B T \)). As a result, only the ground electronic state contributes significantly at typical temperatures, and:

\[ q_{\text{el}} \approx g_0, \]

where \( g_0 \) is the degeneracy of the ground electronic state.

This simplification highlights that electronic excitation does not significantly affect the partition function at low temperatures.