Glossary

5.4 Entropy and the Second Law

3 min read•july 23, 2024

is a key concept in thermodynamics, measuring disorder and energy availability in systems. It' crucial for understanding how energy flows and transforms, affecting everything from engine efficiency to the universe's evolution.

The of Thermodynamics states that entropy always increases in isolated systems. This principle explains why certain processes are irreversible and provides a thermodynamic arrow of time, shaping our understanding of the universe's past and future.

Entropy and Its Significance

Physical significance of entropy

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  • Quantifies degree of disorder or randomness in a system
    • Higher entropy indicates greater disorder (gas molecules randomly distributed in a container)
    • Lower entropy indicates greater order (solid crystal structure)
  • Measures unavailability of a system's thermal energy for conversion into mechanical work
    • As entropy increases, less thermal energy available for useful work (heat engine efficiency)
  • Related to number of microscopic configurations () a system can have while maintaining same macroscopic properties
    • System with more possible microstates has higher entropy (flipping coins, more ways to arrange heads and tails)
  • Extensive property depends on amount of substance in the system
    • Doubling amount of substance doubles entropy (ideal gas, entropy proportional to number of moles)

Entropy changes in thermodynamic processes

  • Reversible process, change in entropy (ΔS\Delta S):
    • ΔS=∫dQrevT\Delta S = \int \frac{dQ_{rev}}{T}, dQrevdQ_{rev} is heat exchanged reversibly, TT is absolute temperature
  • Irreversible process, change in entropy greater than integral of dQT\frac{dQ}{T}:
    • ΔS>∫dQT\Delta S > \int \frac{dQ}{T}
  • (constant temperature), change in entropy:
    • ΔS=QT\Delta S = \frac{Q}{T}, QQ is heat exchanged, TT is constant absolute temperature
  • (no heat exchange), change in entropy is zero:
    • ΔS=0\Delta S = 0
  • Ideal gas, change in entropy:
    • ΔS=nRln⁡V2V1\Delta S = nR \ln \frac{V_2}{V_1}, nn is number of moles, RR is universal gas constant, V1V_1 and V2V_2 are initial and final volumes

The Second Law and Irreversibility

Second law and isolated systems

  • States total entropy of an always increases over time
    • Isolated system does not exchange energy or matter with surroundings (insulated container)
  • In isolated system, always lead to entropy increase until system reaches equilibrium
    • At equilibrium, entropy of system is at maximum value (gas expands to fill container)
  • Implies impossible for process to occur in isolated system that decreases entropy

Entropy vs irreversibility

  • Irreversible processes cannot be reversed without leaving change in system or surroundings
    • Heat transfer across finite temperature difference (hot object to cold object)
    • Fluid flow through pipe with friction (pressure drop along pipe)
    • Unrestrained expansion of gas (gas expands into vacuum)
  • Irreversible processes always lead to increase in total entropy of system and surroundings
    • Increase in entropy measures of process
  • Reversible processes do not result in net change in total entropy of system and surroundings
    • Idealized and cannot be achieved in practice due to dissipative factors (friction, heat transfer)

Entropy's implications for time

  • Second Law and entropy increase provide thermodynamic arrow of time
    • Arrow of time is one-way direction or asymmetry of time, distinguishes past from future
  • As universe evolves, total entropy of universe increases, defining direction of arrow of time
    • Universe started in low-entropy state (Big Bang) and evolving towards high-entropy state (heat death)
  • Second Law and entropy increase explain why certain processes are irreversible and why time cannot spontaneously reverse
    • Broken glass cannot spontaneously reassemble, would require decrease in entropy, prohibited by Second Law

Key Terms to Review (19)

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's Entropy Formula: Boltzmann's entropy formula is a foundational equation in statistical mechanics that relates the entropy of a system to the number of microstates accessible to it. The formula is expressed as $$S = k imes ext{ln}(W)$$, where $S$ is the entropy, $k$ is Boltzmann's constant, and $W$ is the number of possible microstates. This relationship highlights how disorder at the microscopic level correlates with macroscopic thermodynamic properties, connecting the concept of entropy with probability and the Second Law of Thermodynamics.
Clausius Inequality: The Clausius inequality is a fundamental principle in thermodynamics that relates to the change in entropy of a system, stating that the change in entropy for any process is greater than or equal to the heat transferred divided by the temperature at which the transfer occurs. This inequality highlights the directionality of spontaneous processes and indicates that real processes cannot occur with 100% efficiency. It establishes a connection between irreversible processes and entropy, serving as a cornerstone for understanding the Second Law of Thermodynamics.
Closed System: A closed system is a type of thermodynamic system that can exchange energy, but not matter, with its surroundings. This means that while energy in the form of heat or work can enter or leave the system, the total mass remains constant as no substances can cross its boundaries. Understanding closed systems is essential for analyzing energy conservation and various thermodynamic processes.
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.
Equilibrium State: An equilibrium state is a condition in which a thermodynamic system's properties remain constant over time because it is in balance with its surroundings. In this state, there are no net changes occurring within the system or between the system and its environment, leading to stability. Understanding equilibrium is crucial as it helps differentiate between various thermodynamic processes, whether they are reversible or irreversible, and plays a significant role in the concepts of entropy and the Second Law of 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.
Irreversibility: Irreversibility refers to the natural tendency of processes to move towards a state of increased disorder, meaning they cannot spontaneously revert to their original state without external work or intervention. This concept is central to understanding the directionality of thermodynamic processes and plays a crucial role in concepts like entropy and the second law of thermodynamics, as well as in analyzing both equilibrium and non-equilibrium states.
Isolated System: An isolated system is a type of thermodynamic system that does not exchange matter or energy with its surroundings. This means that both energy transfer and mass transfer are completely restricted, allowing the system to evolve according to its own internal processes without external interference. In this context, understanding isolated systems helps in grasping the fundamental principles of thermodynamics, the interaction between systems and their environments, the behavior of entropy, and the statistical mechanics related to entropy in microcanonical ensembles.
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.
Ludwig Boltzmann: Ludwig Boltzmann was an Austrian physicist who made significant contributions to the field of statistical mechanics and thermodynamics, particularly known for his formulation of the statistical interpretation of entropy. His work established a crucial link between microscopic particle behavior and macroscopic physical properties, providing a deeper understanding of the second law of thermodynamics and the nature of entropy.
Macrostates: Macrostates are the overall, observable states of a system defined by macroscopic properties such as temperature, pressure, and volume. They provide a broad overview of the system's behavior, but each macrostate can correspond to many different microstates, which are the specific configurations of particles that make up the system. Understanding macrostates is crucial in exploring concepts like entropy, residual entropy, and statistical mechanics, as they bridge the gap between microscopic particle interactions and macroscopic thermodynamic properties.
Microstates: Microstates are the distinct arrangements of particles within a thermodynamic system that correspond to the same macroscopic state, representing a fundamental concept in statistical mechanics. Each microstate reflects a unique configuration of energy and position of particles, contributing to the overall entropy of the system. Understanding microstates is essential for linking microscopic behaviors to macroscopic thermodynamic properties, including energy distribution and probability.
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.
Rudolf Clausius: Rudolf Clausius was a German physicist and mathematician known for his foundational work in thermodynamics, particularly in defining the concept of entropy and formulating the second law of thermodynamics. His contributions helped establish the quantitative understanding of energy transfer and its limitations, shaping the laws governing heat engines and natural processes.
S: In thermodynamics, 's' represents entropy, a measure of the disorder or randomness in a system. It quantifies the amount of energy in a physical system that is unavailable to do work and reflects the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time. As processes occur, entropy tends to increase, highlighting the direction of spontaneous processes and the inherent inefficiency of energy conversions.
Second Law: The Second Law of Thermodynamics states that the total entropy of an isolated system can never decrease over time, and is often associated with the concept of irreversibility in natural processes. This principle highlights how energy transformations are not 100% efficient and that systems naturally evolve toward a state of greater disorder or entropy. It connects closely with the concept of entropy, as it provides a framework for understanding how energy disperses and how heat transfer occurs between systems.
Spontaneous Processes: Spontaneous processes are natural occurrences that happen without the need for external energy input, often leading to a decrease in free energy and an increase in entropy within a system. These processes are characterized by their tendency to move towards equilibrium, indicating a shift from ordered states to more disordered ones. Understanding spontaneous processes is crucial, as they are closely linked to the concepts of entropy and the second law of thermodynamics, revealing how energy transformations occur in the universe.
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