15.3 Applications to ideal gases and solids

Ideal Gases and Partition Functions

The partition function is the central tool connecting the microscopic world of energy states to the macroscopic thermodynamic quantities you can actually measure. For ideal gases, it lets you derive internal energy, entropy, heat capacity, and more from first principles. For solids, similar ideas apply, but the energy states come from lattice vibrations rather than molecular motion.

Boltzmann Distribution for Ideal Gases

The Boltzmann distribution gives the probability of finding a system in a particular energy state $i$:

$P_i = \frac{e^{-E_i / kT}}{Z}$

where:

• $E_i$ is the energy of state $i$
• $k$ is the Boltzmann constant, which relates energy at the particle level to temperature
• $T$ is the absolute temperature in Kelvin
• $Z$ is the partition function, which normalizes the distribution so that all probabilities sum to 1

The exponential factor $e^{-E_i / kT}$ is called the Boltzmann factor. It tells you that higher-energy states are exponentially less likely to be occupied, and that raising the temperature makes those high-energy states more accessible.

Two classic applications:

• Population of energy levels: At room temperature, most gas molecules sit in low-energy states. As $T$ increases, the population spreads out across higher-energy states.
• Maxwell-Boltzmann speed distribution: The familiar bell-shaped curve of molecular speeds in a gas is a direct consequence of the Boltzmann distribution applied to translational kinetic energy.

Partition Functions in Thermodynamics

The partition function sums all Boltzmann factors across every possible energy state:

$Z = \sum_{i} e^{-E_i / kT}$

Think of $Z$ as a measure of how many states are thermally accessible at temperature $T$. A large $Z$ means many states are significantly populated; a small $Z$ means the system is confined to just a few low-energy states.

Factorization for ideal gases. Because the different modes of molecular motion are independent, the total partition function factors into separate contributions:

$Z = Z_{\text{trans}} \, Z_{\text{rot}} \, Z_{\text{vib}}$

• Translational ($Z_{\text{trans}}$): kinetic energy of the molecule's center-of-mass motion through space. Every gas has this.
• Rotational ($Z_{\text{rot}}$): kinetic energy of rotation about molecular axes. Requires at least two atoms.
• Vibrational ($Z_{\text{vib}}$): energy stored in stretching and bending of chemical bonds. Becomes significant at higher temperatures.

Which components matter depends on molecular structure:

Note that for diatomic gases at moderate temperatures, vibrational modes are often "frozen out" because the vibrational energy spacing is large compared to $kT$. They only contribute significantly at high temperatures.

From Partition Function to Thermodynamic Quantities

Once you have $Z$, you can extract every equilibrium thermodynamic property through derivatives of $\ln Z$. These relations are worth memorizing:

• Helmholtz free energy: $F = -kT \ln Z$
• Internal energy: $U = kT^2 \left(\frac{\partial \ln Z}{\partial T}\right)_{V,N}$
• Entropy: $S = k \ln Z + kT \left(\frac{\partial \ln Z}{\partial T}\right)_{V,N}$
• Heat capacity at constant volume: $C_V = \left(\frac{\partial U}{\partial T}\right)_{V,N}$

The pattern here is that $\ln Z$ plays the role of a generating function. Successive temperature derivatives give you progressively higher-order thermal properties.

Practical example: For a monatomic ideal gas with only translational degrees of freedom, the equipartition theorem gives $U = \frac{3}{2}NkT$ and therefore $C_V = \frac{3}{2}Nk$. A diatomic gas at moderate temperatures adds two rotational degrees of freedom, giving $C_V = \frac{5}{2}Nk$. These results follow directly from evaluating the partition function components.

Solids and Density of States

Density of States in Solids

In a solid, energy levels are so closely spaced that they form a near-continuum. Instead of summing over discrete states, you describe the system using the density of states $g(E)$, which gives the number of energy states per unit energy interval.

The partition function then becomes an integral rather than a sum:

$Z = \int g(E) \, e^{-E/kT} \, dE$

The form of $g(E)$ depends on what kind of excitation you're considering:

• Electrons in metals: $g(E)$ near the Fermi level determines electrical and thermal properties of the conduction band.
• Electrons in semiconductors: separate $g(E)$ functions for the valence and conduction bands, with the band gap in between.
• Phonons (quantized lattice vibrations): the phonon density of states governs how a solid absorbs thermal energy, and it's the foundation for the heat capacity models below.

Heat Capacity Models for Solids

At high temperatures, most solids obey the Dulong-Petit law: $C_V = 3Nk$ (or about 25 J/mol·K per mole of atoms). This comes from the equipartition theorem applied to 3 dimensions of vibration, each with kinetic and potential energy. But experiments show that $C_V$ drops well below this value at low temperatures. The Einstein and Debye models explain why.

Einstein Model

Einstein's approach is the simpler one. It assumes every atom in the solid vibrates at the same single frequency $\omega_E$.

• Einstein temperature: $\Theta_E = \frac{\hbar \omega_E}{k}$
• Heat capacity:

$C_V = 3Nk \left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E / T}}{(e^{\Theta_E / T} - 1)^2}$

At high $T$ ($T \gg \Theta_E$), this reduces to the Dulong-Petit value $3Nk$. At low $T$, the heat capacity drops off exponentially. This qualitatively captures the experimental trend, but the exponential decay is too fast compared to what's actually observed in most solids.

Debye Model

Debye improved on Einstein by recognizing that a solid supports a range of vibrational frequencies, from zero up to a maximum cutoff frequency $\omega_D$.

• Debye temperature: $\Theta_D = \frac{\hbar \omega_D}{k}$
• Heat capacity:

$C_V = 9Nk \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D / T} \frac{x^4 e^x}{(e^x - 1)^2} \, dx$

The integral must be evaluated numerically in general, but the limiting behavior is what matters most:

• High $T$ limit ($T \gg \Theta_D$): recovers $C_V = 3Nk$ (Dulong-Petit)
• Low $T$ limit ($T \ll \Theta_D$): predicts $C_V \propto T^3$

The $T^3$ dependence at low temperatures matches experimental data far better than Einstein's exponential decay. This is the Debye model's main advantage.

Typical Debye temperatures give you a sense of when quantum effects become important: diamond has $\Theta_D \approx 2230 \, \text{K}$ (so quantum effects persist to high temperatures, and $C_V$ is well below Dulong-Petit even at room temperature), while copper has $\Theta_D \approx 343 \, \text{K}$ and silicon has $\Theta_D \approx 645 \, \text{K}$. A solid only reaches the classical Dulong-Petit limit when $T$ is well above its Debye temperature.