Glossary

15.3 Applications to ideal gases and solids

3 min read•july 23, 2024

Ideal gases and partition functions are key concepts in thermodynamics. They help us understand how energy is distributed among particles in a system. The describes the probability of particles occupying different energy states, while partition functions connect microscopic properties to macroscopic thermodynamic quantities.

Solids and density of states are crucial for understanding the behavior of materials. The density of states represents the distribution of energy levels in a solid, while heat capacity models like Einstein and Debye help predict how solids absorb thermal energy. These concepts are essential for explaining phenomena in solid-state physics and materials science.

Ideal Gases and Partition Functions

Boltzmann distribution for ideal gases

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  • Describes probability of a system being in a particular energy state ii given by Pi=e−Ei/kTZP_i = \frac{e^{-E_i/kT}}{Z}
    • EiE_i energy of state ii
    • kk Boltzmann constant (relates energy at the particle level with temperature)
    • TT absolute temperature (in Kelvin)
    • ZZ normalizes the distribution ensures probabilities sum to 1
  • Examples
    • Probability of gas molecules occupying high vs low energy states at different temperatures
    • Distribution of molecular speeds in a gas (Maxwell-Boltzmann distribution)

Partition functions in thermodynamics

  • Sum of all Boltzmann factors e−Ei/kTe^{-E_i/kT} for each possible energy state ii
    • Z=∑ie−Ei/kTZ = \sum_{i} e^{-E_i/kT}
    • Connects microscopic properties (energy states) to macroscopic thermodynamic quantities
  • For ideal gases can be separated into translational, rotational, and vibrational components
    • Z=ZtransZrotZvibZ = Z_{trans} Z_{rot} Z_{vib}
    • Each component calculated based on specific energy levels of gas molecules
      • Translational: kinetic energy of linear motion
      • Rotational: kinetic energy of rotation about an axis
      • Vibrational: potential and kinetic energy of atomic vibrations
  • Examples
    • Monatomic ideal gases (He, Ne, Ar) only have translational component
    • Diatomic gases (N2, O2) have translational and rotational components
    • Polyatomic gases (CH4, CO2) have all three components
  • Partition function ZZ relates to thermodynamic properties
    • Helmholtz free energy: F=−kTln⁡ZF = -kT \ln Z
    • : U=kT2(∂ln⁡Z∂T)V,NU = kT^2 \left(\frac{\partial \ln Z}{\partial T}\right)_{V,N}
    • : S=kln⁡Z+kT(∂ln⁡Z∂T)V,NS = k \ln Z + kT \left(\frac{\partial \ln Z}{\partial T}\right)_{V,N}
    • Heat capacity at constant volume: CV=(∂U∂T)V,N=kT(∂2ln⁡Z∂T2)V,NC_V = \left(\frac{\partial U}{\partial T}\right)_{V,N} = kT \left(\frac{\partial^2 \ln Z}{\partial T^2}\right)_{V,N}
  • Examples
    • Calculating molar heat capacity of monatomic, diatomic, and polyatomic gases
    • Predicting changes in entropy and internal energy with temperature

Solids and Density of States

Density of states in solids

  • Number of energy states per unit energy interval g(E)g(E)
    • Represents distribution of energy states in a solid
    • Depends on lattice structure, atomic mass, and other properties of the solid
  • Partition function for a solid calculated using density of states
    • Z=∫g(E)e−E/kTdEZ = \int g(E) e^{-E/kT} dE
    • Integration performed over all possible energy states
  • Examples
    • Density of states for electrons in metals (conduction band) and semiconductors (valence and conduction bands)
    • Phonon density of states for lattice vibrations in solids

Heat capacity models for solids

    • Assumes all atoms vibrate at the same frequency ωE\omega_E
    • Einstein temperature: ΘE=ℏωE/k\Theta_E = \hbar \omega_E / k
    • Heat capacity: CV=3Nk(ΘET)2eΘE/T(eΘE/T−1)2C_V = 3Nk \left(\frac{\Theta_E}{T}\right)^2 \frac{e^{\Theta_E/T}}{(e^{\Theta_E/T} - 1)^2}
      • NN number of atoms in the solid
    • Predicts exponential decrease in heat capacity at low temperatures
    • Considers range of vibrational frequencies up to maximum ωD\omega_D
    • Debye temperature: ΘD=ℏωD/k\Theta_D = \hbar \omega_D / k
    • Heat capacity: CV=9Nk(TΘD)3∫0ΘD/Tx4ex(ex−1)2dxC_V = 9Nk \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x - 1)^2} dx
      • Integral evaluated numerically
    • Predicts CV∝T3C_V \propto T^3 at low temperatures
  • Examples
    • Comparing heat capacity predictions of Einstein and Debye models for different solids (diamond, copper, silicon)
    • Explaining deviations from Dulong-Petit law (CV=3NkC_V = 3Nk) at low temperatures

Key Terms to Review (22)

Adiabatic process: An adiabatic process is a thermodynamic process in which no heat is exchanged between the system and its surroundings. This means that any change in the internal energy of the system is entirely due to work done on or by the system, making it a critical concept in understanding various thermodynamic cycles and processes.
Boltzmann distribution: The Boltzmann distribution describes the distribution of particles over various energy states in a system at thermal equilibrium, illustrating how the probability of finding a particle in a particular state depends on the energy of that state. It connects microscopic behavior, such as individual particle states, to macroscopic properties like temperature and pressure, allowing for a deeper understanding of statistical mechanics and thermodynamics.
Charles's Law: Charles's Law states that the volume of a gas is directly proportional to its temperature when the pressure is held constant. This relationship is fundamental in understanding how gases behave under varying temperature conditions and connects to other important concepts such as state variables, thermodynamic equations, and the behavior of ideal gases and solids.
Debye Model: The Debye model is a theoretical framework used to describe the heat capacity of solids at low temperatures by considering the contributions of phonons, or quantized lattice vibrations. This model helps to explain how heat capacity decreases as temperature approaches absolute zero, providing insights into the behavior of solids and their atomic interactions.
Einstein Model: The Einstein Model is a theoretical framework that describes the behavior of solid materials by considering atoms as independent harmonic oscillators. This model helps to explain how the specific heat of solids varies with temperature, predicting that it approaches a constant value at high temperatures, which is relevant in understanding the thermal properties of ideal gases and solids.
Enthalpy: Enthalpy is a thermodynamic property that represents the total heat content of a system, defined as the sum of its internal energy and the product of its pressure and volume. This concept is crucial in understanding how energy is exchanged in processes, especially in the context of thermodynamic systems and the transformations they undergo.
Entropy: Entropy is a measure of the degree of disorder or randomness in a system, reflecting the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state. It connects to various principles of thermodynamics, indicating how energy disperses and the direction of spontaneous processes.
First Law of Thermodynamics: The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another. This principle emphasizes the conservation of energy within a closed system, illustrating how energy transfers and transformations impact thermodynamic processes and systems.
Heat capacity equation: The heat capacity equation relates the amount of heat energy absorbed or released by a substance to the resulting change in its temperature. This equation is essential for understanding how ideal gases and solids respond to thermal energy changes, illustrating the relationship between heat transfer and temperature change, which is crucial for various applications in thermodynamics.
Heat Engines: Heat engines are devices that convert thermal energy into mechanical work by taking in heat from a high-temperature source and releasing some of that heat to a lower temperature sink. This process is governed by the principles of thermodynamics, particularly the first and second laws, which dictate the efficiency and limitations of how heat can be transformed into work.
Ideal Gas Law: The Ideal Gas Law, represented by the equation $$PV=nRT$$, describes the relationship between the pressure (P), volume (V), number of moles (n), gas constant (R), and temperature (T) of an ideal gas. This law combines several earlier gas laws, including Boyle's Law and Charles's Law, into one comprehensive formula, allowing us to predict how a gas will behave under various conditions. The law applies best to gases at high temperatures and low pressures, where real gas behavior closely approximates that of an ideal gas.
Internal Energy: Internal energy is the total energy contained within a thermodynamic system, encompassing kinetic and potential energies of all the molecules in the system. This energy is crucial in determining the state of the system and plays a key role in thermodynamic processes, including heat transfer and work done on or by the system.
Isobaric Process: An isobaric process is a thermodynamic process that occurs at constant pressure. In such a process, any heat transfer into or out of the system results in a change in volume, while the pressure remains unchanged. This constancy of pressure plays a significant role in various energy exchanges and mechanical work done by or on the system.
Isothermal process: An isothermal process is a thermodynamic process in which the temperature of the system remains constant while heat is exchanged with the surroundings. This constant temperature implies that any internal energy changes in the system are fully compensated by heat transfer, making it an essential concept in understanding how systems behave under thermal equilibrium and the laws governing energy conservation.
Joule: A joule is a unit of energy in the International System of Units (SI), representing the amount of work done when a force of one newton displaces an object by one meter. It connects to various important concepts in thermodynamics, including energy transfer, work done by heat engines, and internal energy changes within gases and solids. Understanding the joule is crucial for analyzing how energy is converted and transferred in physical systems, especially when looking at heat engines and energy states in different materials.
Latent heat: Latent heat is the amount of heat energy absorbed or released by a substance during a phase change without a change in temperature. This concept is crucial in understanding how substances transition between solid, liquid, and gas phases, as well as in various thermodynamic processes that involve energy transfer.
Partition function: The partition function is a central concept in statistical mechanics that encapsulates all possible states of a system and their corresponding probabilities. It serves as a crucial link between microscopic properties of particles and macroscopic thermodynamic quantities, allowing for the computation of essential properties like free energy, entropy, and heat capacity.
Phase Diagram: A phase diagram is a graphical representation that shows the relationship between the physical state of a substance and the conditions of temperature and pressure. It helps visualize where different phases, like solid, liquid, and gas, exist and how they transition from one to another under varying conditions. Understanding phase diagrams is crucial for comprehending latent heat, enthalpy changes during phase transitions, and the behaviors of ideal gases and solutions.
Phase Equilibrium: Phase equilibrium refers to a condition in which distinct phases of a substance coexist in a stable manner, with no net change in their respective quantities over time. This balance occurs when the rates of transition between phases, such as solid, liquid, and gas, are equal, leading to an overall stability in the system. Understanding phase equilibrium is essential for analyzing latent heat during phase transitions, chemical potential in thermodynamic systems, the construction of phase diagrams, and the behavior of gases under varying conditions.
Refrigerators: Refrigerators are devices that transfer heat from a lower temperature region to a higher temperature region, utilizing the principles of thermodynamics to keep items cool. They operate on the refrigeration cycle, which involves the compression, condensation, expansion, and evaporation of a refrigerant. This process allows refrigerators to maintain a cold environment inside while releasing heat to the surroundings, showcasing important applications of energy conservation and heat transfer.
Second law of thermodynamics: The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and it dictates the direction of thermodynamic processes. This principle establishes that energy transformations are not 100% efficient, highlighting the inherent tendency for systems to move towards a state of greater disorder or randomness, affecting heat transfer, the performance of engines, and various processes in nature.
Specific Heat Capacity: Specific heat capacity is the amount of heat required to raise the temperature of one unit mass of a substance by one degree Celsius (or one Kelvin). It is a crucial concept in understanding how materials absorb and transfer heat energy, which connects to heat transfer mechanisms, calorimetry, phase changes, and the behavior of ideal gases and solids as well as their internal energy and enthalpy.
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