9.2 Partition functions and their applications

Partition functions are the secret sauce of statistical thermodynamics. They bridge the gap between tiny molecular movements and big-picture thermodynamic properties. By understanding partition functions, you can predict how systems behave on a larger scale.

These mathematical tools help us calculate important stuff like free energy, internal energy, and heat capacity. Mastering partition functions is key to unlocking the mysteries of how molecules interact and influence the world around us.

Partition Functions

Definition and Molecular Partition Function

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• Partition function $q$ is a fundamental concept in statistical thermodynamics that describes the statistical properties of a system in thermodynamic equilibrium
• Connects microscopic properties of a system (energy levels) to macroscopic thermodynamic properties (temperature, pressure, volume)
• Molecular partition function $q_{mol}$ is the product of the translational, rotational, vibrational, and electronic partition functions for a single molecule
  • $q_{mol} = q_{trans} \cdot q_{rot} \cdot q_{vib} \cdot q_{elec}$
• Each partition function corresponds to a specific type of molecular motion or energy

Translational and Rotational Partition Functions

• Translational partition function $q_{trans}$ describes the contribution of translational motion to the molecular partition function
  • Depends on the mass of the molecule $m$ and the temperature $T$
  • For a particle in a 3D box: $q_{trans} = \left(\frac{2\pi mkT}{h^2}\right)^{3/2}V$, where $k$ is the Boltzmann constant, $h$ is Planck's constant, and $V$ is the volume
• Rotational partition function $q_{rot}$ describes the contribution of rotational motion to the molecular partition function
  • Depends on the moment of inertia $I$ and the temperature $T$
  • For a linear molecule: $q_{rot} = \frac{8\pi^2IkT}{\sigma h^2}$, where $\sigma$ is the symmetry number
  • For a nonlinear molecule: $q_{rot} = \frac{\sqrt{\pi}}{\sigma}\left(\frac{8\pi^2kT}{h^2}\right)^{3/2}\sqrt{I_AI_BI_C}$, where $I_A$, $I_B$, and $I_C$ are the principal moments of inertia

Vibrational and Electronic Partition Functions

• Vibrational partition function $q_{vib}$ describes the contribution of vibrational motion to the molecular partition function
  • Depends on the vibrational frequency $\nu$ and the temperature $T$
  • For a harmonic oscillator: $q_{vib} = \frac{1}{1-e^{-h\nu/kT}}$
  • For a molecule with multiple vibrational modes: $q_{vib} = \prod_i \frac{1}{1-e^{-h\nu_i/kT}}$
• Electronic partition function $q_{elec}$ describes the contribution of electronic energy levels to the molecular partition function
  • Depends on the electronic energy levels $\epsilon_i$ and the temperature $T$
  • $q_{elec} = \sum_i g_i e^{-\epsilon_i/kT}$, where $g_i$ is the degeneracy of the $i$-th electronic state

Thermodynamic Properties

Relationship between Partition Functions and Thermodynamic Properties

• Partition functions provide a link between microscopic properties and macroscopic thermodynamic properties
• Thermodynamic properties can be derived from the partition function using statistical mechanics
• Key thermodynamic properties include free energy, internal energy, and heat capacity

Free Energy and Internal Energy

• Free energy $A$ is a measure of the useful work that can be extracted from a system
  • Related to the partition function by: $A = -kT \ln Q$, where $Q$ is the canonical partition function
• Internal energy $U$ is the total energy of a system, including both kinetic and potential energy
  • Can be calculated from the partition function using: $U = kT^2 \left(\frac{\partial \ln Q}{\partial T}\right)_{N,V}$
• Both free energy and internal energy provide insights into the stability and behavior of a system

Heat Capacity

• Heat capacity $C$ is a measure of the amount of heat required to change the temperature of a system by a given amount
  • Can be calculated from the partition function using: $C = \left(\frac{\partial U}{\partial T}\right)_{N,V} = kT \left(\frac{\partial^2 \ln Q}{\partial T^2}\right)_{N,V}$
• Heat capacity provides information about the thermal properties of a system
  • Helps understand how a system responds to changes in temperature
  • Useful for predicting phase transitions and other thermal phenomena