Glossary

measures in systems, from gas molecules to stacked books. It's linked to - possible particle arrangements. As systems move towards equilibrium, they maximize entropy, increasing accessible microstates. This concept is crucial for understanding thermodynamic processes.

Irreversible processes, like heat transfer or , always increase entropy. The second law of thermodynamics states that isolated systems' total entropy always rises. This drives spontaneous processes and the arrow of time, shaping our understanding of energy and change.

Entropy on a Microscopic Scale

Entropy and microstates

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  • Entropy measures the or in a system
    • Higher entropy signifies more disorder or randomness (gas molecules spread out in a room)
    • Lower entropy signifies more order or predictability (neatly stacked books on a shelf)
  • Microstates depict the possible arrangements of particles in a system
    • Each microstate corresponds to a specific configuration of particle positions and energies (molecules in a gas occupying different positions and having different velocities)
    • The number of microstates increases with the number of particles and the system's energy (more molecules in a gas lead to more possible arrangements)
  • Entropy is related to the number of accessible microstates (Ω\Omega) through the : S=kBln⁡ΩS = k_B \ln \Omega
    • kBk_B represents the Boltzmann constant (1.38×10−231.38 \times 10^{-23} J/K)
    • A larger number of accessible microstates leads to higher entropy (more ways to arrange particles in a system)
  • As a system evolves towards equilibrium, it tends to move towards the state with the highest number of accessible microstates, maximizing entropy (a gas expands to fill its container, increasing the number of accessible microstates)
  • The accounts for the number of ways particles can be arranged in space

Entropy changes in irreversible processes

  • Irreversible processes cannot be reversed without leaving a trace in the environment
    • Heat transfer from hot to cold objects (coffee cooling down to room )
    • Gas expansion into a vacuum (air rushing out of a punctured tire)
    • (cream swirling into coffee)
  • (ΔS\Delta S) for an irreversible process is always positive, indicating an increase in entropy
    • For a heat transfer process: ΔS=QT\Delta S = \frac{Q}{T}, where QQ is the heat transferred and TT is the absolute temperature (heat flowing from a hot pan to a cold countertop)
    • For an isothermal expansion of an ideal gas: ΔS=nRln⁡V2V1\Delta S = nR \ln \frac{V_2}{V_1}, where nn is the number of moles, RR is the gas constant, and V1V_1 and V2V_2 are the initial and final volumes (a balloon expanding as it is inflated)
  • The second law of thermodynamics states that the total entropy of an isolated system always increases during an irreversible process
    • This means that the entropy of the universe is constantly increasing (the total disorder in the universe grows over time)
    • Entropy increase drives spontaneous processes and the arrow of time (a shattered glass spontaneously reassembles)
  • The relates entropy changes to the spontaneity of processes in systems at constant temperature and

Absolute zero and entropy

  • is the lowest possible temperature, corresponding to 0 kelvin or -273.15°C
    • At , a system has the minimum possible energy and the lowest possible entropy (a perfect crystal at absolute zero)
  • As temperature approaches absolute zero, the number of accessible microstates decreases
    • This occurs because particles have less thermal energy to occupy higher energy states (electrons in a metal settling into the lowest available energy levels)
    • In the limit of absolute zero, the system would theoretically occupy only the ground state or lowest energy microstate (a Bose-Einstein condensate)
  • The states that the entropy of a perfect crystalline substance approaches zero as the temperature approaches absolute zero
    • In practice, reaching absolute zero is impossible due to the finite steps in the cooling process (limitations of current refrigeration technology)
    • However, the concept of absolute zero provides a lower limit for entropy and a reference point for understanding the behavior of matter at extremely low temperatures (superconductivity and superfluidity emerging near absolute zero)

Statistical mechanics and entropy

  • provides a framework for understanding macroscopic properties in terms of microscopic behavior
  • The assumes that, over long time scales, a system will explore all possible microstates with equal probability
  • represents all possible states of a system, with each point corresponding to a unique microstate
  • connects entropy to the amount of information needed to describe a system's state

Key Terms to Review (32)

Absolute temperature scale: An absolute temperature scale is a thermodynamic temperature scale that uses absolute zero as its null point. The two most common absolute temperature scales are Kelvin and Rankine.
Absolute zero: Absolute zero is the lowest possible temperature where all molecular motion ceases. It is 0 Kelvin, or -273.15 degrees Celsius.
Absolute Zero: Absolute zero is the lowest possible temperature, at which the motion of atoms and molecules comes to a complete stop. It is the point at which a system reaches its minimum energy state, and is the coldest possible temperature that can be achieved in the physical universe.
Boltzmann Equation: The Boltzmann equation is a fundamental equation in statistical mechanics that describes the statistical distribution of particles in a system in thermodynamic equilibrium. It relates the microscopic properties of individual particles to the macroscopic properties of the system as a whole.
Coefficient of volume expansion: The coefficient of volume expansion is a material-specific constant that quantifies the fractional change in volume per degree change in temperature. It is typically denoted by $\beta$ and measured in $\text{K}^{-1}$.
Configurational Entropy: Configurational entropy is a measure of the number of possible arrangements or configurations of a system at the microscopic level. It quantifies the disorder or randomness inherent in the spatial arrangement of the system's components, reflecting the system's propensity to explore different microstates.
Disorder: Disorder in thermodynamics refers to the randomness or chaos within a system. It is closely associated with entropy, which quantifies this randomness.
Disorder: Disorder refers to the level of randomness or chaos in a system, often associated with the arrangement of particles or states within that system. In the context of entropy, disorder is a crucial concept as it describes how energy is distributed among the particles in a substance and how this distribution affects the system's overall energy state.
Entropy: Entropy is a measure of the disorder or randomness in a system. It quantifies the number of possible microscopic configurations that correspond to a thermodynamic system's macroscopic state.
Entropy Change: Entropy change is a measure of the disorder or randomness that occurs in a system during a process or transformation. It is a fundamental concept in thermodynamics that describes the tendency of a system to move towards a more disordered state over time.
Ergodic Hypothesis: The ergodic hypothesis is a fundamental concept in statistical mechanics that describes the relationship between the time-averaged behavior of a system and its ensemble-averaged behavior. It suggests that over a sufficiently long period of time, the time average of a system's properties will converge to the ensemble average, provided the system is ergodic.
Gas Expansion: Gas expansion refers to the increase in the volume of a gas when it is subjected to a decrease in pressure or an increase in temperature. This process is governed by the fundamental laws of thermodynamics and is a crucial concept in understanding the behavior of gases on a microscopic scale.
Gibbs Free Energy: Gibbs free energy is a thermodynamic quantity that combines the concepts of energy, entropy, and temperature to determine the spontaneity and feasibility of a chemical process. It is a measure of the useful work that can be extracted from a system at constant temperature and pressure.
Information Theory: Information theory is the mathematical study of the quantification, storage, and communication of information. It provides a framework for understanding and analyzing the transmission, processing, and storage of data, as well as the fundamental limits of such processes.
Isentropic: An isentropic process is a thermodynamic process in which entropy remains constant. This implies the process is both adiabatic and reversible.
Ludwig Boltzmann: Ludwig Boltzmann was an Austrian physicist known for his foundational contributions to statistical mechanics and thermodynamics, particularly in understanding the behavior of particles in gases. His work established a connection between the microscopic properties of matter and macroscopic observations, illustrating how molecular speeds distribute among particles in a gas and how these distributions relate to entropy on a microscopic scale.
Macrostates: Macrostates refer to the macroscopic properties of a system that describe its overall condition, such as temperature, pressure, and volume. These properties are defined on a large scale and result from the collective behavior of countless microscopic components, like atoms and molecules, which may exist in many different configurations or arrangements, known as microstates. Understanding macrostates is essential for grasping concepts like entropy, as it highlights how different arrangements at the microscopic level can lead to the same observable macroscopic behavior.
Maxwell-Boltzmann distribution: The Maxwell-Boltzmann distribution describes the distribution of speeds of particles in a gas. It shows that most particles have speeds around an average value, with fewer particles moving much slower or much faster.
Maxwell-Boltzmann Distribution: The Maxwell-Boltzmann distribution is a statistical model that describes the distribution of molecular speeds in an ideal gas at a given temperature. It is a fundamental concept in the study of the molecular model of an ideal gas, pressure, temperature, and the distribution of molecular speeds, as well as the microscopic understanding of entropy.
Microscopic Scale: The microscopic scale refers to the level of observation and analysis that focuses on the smallest components of a system, such as individual atoms, molecules, and subatomic particles. This scale is crucial for understanding the fundamental processes and properties that govern the behavior of matter and energy at the most fundamental level.
Microstates: Microstates are specific configurations of a system at the microscopic level that correspond to a particular macrostate. Each microstate represents a unique arrangement of particles and their energies, contributing to the overall statistical behavior of the system, which is crucial for understanding concepts like entropy and thermodynamics on a deeper level.
Mixing of fluids: Mixing of fluids refers to the process by which two or more different fluids combine to form a homogeneous mixture. This phenomenon is essential in understanding how energy and matter are transferred, as well as the behavior of particles at a microscopic level, impacting entropy and the overall state of a system.
Phase Space: Phase space is a mathematical representation of the possible states of a system, where each possible state is represented by a unique point in the phase space. It is a powerful tool used to understand the behavior and dynamics of complex systems, particularly in the context of statistical mechanics and thermodynamics.
Pressure: Pressure is the force exerted per unit area on a surface, commonly measured in pascals (Pa). It plays a critical role in understanding how gases behave, how thermal expansion affects materials, and how energy transfers occur in systems. Pressure influences how gases expand or compress, impacts thermodynamic processes, and governs the interactions between molecules at the microscopic level.
Randomness: Randomness refers to the lack of predictability or pattern in a sequence of events or observations. It is the quality of being unpredictable, irregular, and lacking any discernible order or structure. Randomness is a fundamental concept in various fields, including physics, mathematics, and computer science, particularly in the context of entropy on a microscopic scale.
Reversible process: A reversible process is a thermodynamic process that can be reversed without leaving any net change in either the system or the surroundings. These processes are ideal and occur infinitesimally slowly, allowing the system to remain in equilibrium throughout.
Reversible Process: A reversible process is a thermodynamic process that can be reversed without leaving any trace on the surroundings. In other words, a reversible process can be undone, and the system and the surroundings can be returned to their initial states without the expenditure of any work or the absorption of any heat from the surroundings.
Statistical Mechanics: Statistical mechanics is a branch of physics that uses the principles of probability and statistics to study the behavior of systems composed of a large number of interacting particles. It provides a framework for understanding the macroscopic properties of a system, such as temperature, pressure, and energy, in terms of the microscopic behavior of its individual components.
Temperature: Temperature is a measure of the average kinetic energy of the particles (atoms or molecules) in a substance. It quantifies the degree of hotness or coldness of an object and is a fundamental concept in thermodynamics that is closely related to the transfer of heat energy.
Third law of thermodynamics: The third law of thermodynamics states that the entropy of a perfect crystalline substance approaches zero as the temperature approaches absolute zero. This implies it is impossible to reach absolute zero in a finite number of steps.
Third Law of Thermodynamics: The third law of thermodynamics states that as a system approaches absolute zero, its entropy approaches a constant, usually zero, value. It establishes a lower limit to the decrease in entropy that can be achieved by a system as it approaches the absolute zero of temperature.
Volume: Volume is a fundamental physical quantity that describes the three-dimensional space occupied by an object or a substance. It is a measure of the amount of space enclosed within a defined boundary or container.
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