Second Law of Thermodynamics: Spontaneity, Reversibility, and Entropy

Historical Context of Perpetual Motion and Thermodynamics

• The Concept of "No Free Lunch": In economics and thermodynamics, the principle exists that every desirable outcome requires the expenditure of effort, money, or energy. Perpetual motion machines—devices that produce energy while requiring none—attempt to circumvent this reality.

• History of Perpetual Motion Machines:

  • Vilan de Honecot (13th Century): Proposed the first recorded perpetual motion machine.

  • Leonardo da Vinci: Created numerous drawings for machines intended to produce energy at no cost.

  • Robert Fludd (1600s): A prominent scientist who correctly identified the Sun as the center of the universe and described blood as a carrier for life-important gases, but also believed lightning was an act of God. He proposed well-known early perpetual motion designs.

  • M. C. Escher's "Waterfall" (1961): An artistic representation of impossible cyclical motion reminiscent of Fludd’s designs.

• The Case of Charles Redheffer (1812):

  • Redheffer claimed to have a machine requiring no energy source in Philadelphia. He sought city funding for a larger version but restricted access to the device.

  • Isaiah Lukens: A local engineer who built a replica based on Redheffer’s principles to expose the fraud.

  • Robert Fulton (New York City): After Redheffer fled to New York, the inventor Robert Fulton noticed the machine operated in a "wobbly manner." Fulton challenged Redheffer, offering compensation if his fraud accusation was false.

  • The Exposure: Fulton removed wall boards to find a thin cord of catgut leading to another room. He discovered an elderly man eating a crust of bread while turning a crank with his other hand. Redheffer disappeared following this discovery.

• Thermodynamic Limitations of Perpetual Motion:

  • Most machines fail because they violate the First Law of Thermodynamics.

  • Falling Water Example: While falling water can turn a crank to do work, there is insufficient energy available to lift the water back to the starting reservoir to repeat the cycle, especially considering energy losses due to friction.

The Second Law of Thermodynamics and the Ammonia Engine

• John Gamgee and the Ammonia Engine (1880s):

  • Gamgee attempted to persuade the US Navy to use an ammonia-powered engine for ship propulsion. President Garfield even inspected the device.

  • Mechanism: The engine used the heat of ocean water to evaporate liquid ammonia. The expanding vapor drove a piston (similar to gasoline combustion). During expansion, the ammonia vapor cooled, condensed back into liquid, and replenished the reservoir.

  • Violation: While it seemed plausible, this engine violated the Second Law of Thermodynamics, which relates to the criteria for spontaneity and energy conversion limits.

Criteria for Spontaneity and State Functions

• Spontaneity: To understand why certain processes occur naturally, one must analyze how the state of a system changes.

• State Functions: Properties that define a state and are independent of the path taken to reach it. These include:

  • Temperature ($T$)

  • Internal Energy ($E$)

  • Enthalpy ($H$)

• Path Functions: Properties whose values depend on the specific path taken between states. These include:

  • Heat transfer ($q$)

  • Work ($w$)

• Key Insight: Understanding the differences between paths (reversible vs. irreversible) is critical to understanding spontaneity.

Reversible and Irreversible Processes

• Sadi Carnot (1824):

  • A 20-year-old French engineer who analyzed the efficiency of steam engines.

  • He determined it is impossible to convert the energy content of a fuel completely into work, as a significant amount of heat is always lost to the surroundings.

  • His work laid the foundation for the discipline of thermodynamics.

• Reversible Process:

  • A specific change of state where the system and surroundings can be restored to their original states by exactly reversing the change.

  • No net change occurs to either the system or its surroundings upon restoration.

  • Produces the maximum amount of work that can be done by a system on its surroundings.

• Irreversible Process:

  • A process that cannot be reversed to restore both the system and surroundings to their original states.

  • The amount of work extracted from a process depends on the path taken (reversible vs. irreversible).

Heat Transfer and Reversibility

• Spontaneous Heat Flow: Heat flows from a hotter object to a colder object. This is inherently irreversible because heat cannot flow in the opposite direction spontaneously.

• Conditions for Reversible Heat Transfer:

  • Heat can flow reversibly only if the temperature difference ($\Delta T$) between the system and surroundings is infinitesimally small.

  • Scenario A: If the system is at $T + \Delta T$, an infinitesimal amount of heat flows from the system to the surroundings.

  • Scenario B: If the system is lowered to $T - \Delta T$, an infinitesimal amount of heat flows from the surroundings to the system.

• Theoretical Nature: For a process to be truly reversible, the heat must be infinitesimally small and transferred infinitely slowly. No observable real-world process is truly reversible.

Entropy and Equilibrium

• Determining Entropy Change ($\Delta S$):

  • To measure entropy change experimentally, the heat transfer must be measured for a reversible process ($q_{rev}$).

  • Entropy change is calculated using: $\Delta S = \frac{q_{rev}}{T}$.

• Phase Changes and Equilibrium:

  • The melting of ice and freezing of water at $0\,^{\circ}\text{C}$ is an example of a process often treated as reversible at equilibrium.

  • Adding heat in small increments converts ice to water; removing heat in small increments converts water back to ice without altering the surroundings.

• Spontaneity and Reversibility:

  • Spontaneous processes are not reversible; they occur in one direction.

  • At every step of a reversible pathway, the system remains at equilibrium.

  • Spontaneous processes follow irreversible pathways and involve non-equilibrium conditions.

Isothermal Expansion of an Ideal Gas

• Isothermal Process: A process occurring at a constant temperature.

• Irreversible Expansion into a Vacuum:

  • When a partition is removed, gas spontaneously fills the evacuated space.

  • Work ($w$): Because the gas expands against zero external pressure ($P_{ext} = 0$), the expansion work is zero ($w = 0$).

  • Compression: To restore the gas to its original volume, the surroundings must perform work on the system (w > 0).

  • Conclusion: Since the return path requires different values of work and heat than the initial expansion, the process is irreversible.

• Hypothetical Reversible Isothermal Expansion:

  • This would require the external pressure ($P_{ext}$) to exactly balance the internal gas pressure ($P_{gas}$) at all times.

  • If $P_{ext}$ is reduced infinitely slowly, the piston moves outward while the gas pressure readjusts to maintain equilibrium.

  • This cycle (expansion and compression) would result in no net change to the surroundings.

• Real-World Application: All real processes are irreversible. The reverse of any spontaneous process is non-spontaneous and requires work from the surroundings. Spontaneous processes are inherently irreversible.