8.6 Entropy

Entropy is a fundamental concept in thermodynamics that serves as a measure of the amount of disorder, randomness, or uncertainty in a system. It also quantifies the amount of thermal energy in a system that is not available to do useful work. The concept is central to the Second Law of Thermodynamics, which defines the "arrow of time" and explains why natural processes are irreversible.

Definition of Entropy

In classical thermodynamics, the change in entropy ($\Delta S$) of a system undergoing a reversible process is defined as the amount of heat ($\Delta Q_{rev}$) added to or removed from the system, divided by the absolute temperature ($T$) at which the transfer occurs.

Mathematical Formulation:

$\Delta S=\frac{\Delta Q_{rev}}{T}$

• $\Delta S$: Change in entropy (Unit: Joules per Kelvin, J/K)
• $\Delta Q_{rev}$: Heat transferred during a reversible process
• $T$: Absolute temperature in Kelvin (K)

Entropy is a state function, meaning its value depends only on the current state of the system, not on the path taken to reach that state.

Entropy and Disorder

Entropy is often described as a measure of disorder. A system with a high degree of randomness and many possible microscopic arrangements has high entropy.

• Low Entropy (High Order): A crystalline solid, where atoms are fixed in an orderly lattice.
• High Entropy (High Disorder): A gas, where molecules move randomly and chaotically throughout their container.

The natural tendency of systems is to move from states of lower probability (order) to states of higher probability (disorder), which corresponds to an increase in entropy.

Effect of Temperature on Disorder

An increase in temperature increases the average kinetic energy of the molecules, causing them to move more rapidly and randomly. This greater molecular agitation increases the disorder of the system and therefore increases its entropy. Conversely, cooling a substance reduces molecular motion and decreases entropy (e.g., liquid water freezing into ice becomes more ordered).

Entropy and the Second Law of Thermodynamics

The Second Law of Thermodynamics can be stated in terms of entropy:

Statement: The total entropy of an isolated system can never decrease over time; it either stays constant or increases.

This can be expressed mathematically as:

$\Delta S_{total}\ge 0$

• For a reversible process: The total change in entropy of the universe (system + surroundings) is zero ($\Delta S_{total}=0$). This is an idealized process where the system is always in equilibrium.
• For an irreversible process: The total entropy of the universe always increases ($\Delta S_{total}>0$). All real-world, spontaneous processes are irreversible.

This law dictates the direction of natural events. Heat spontaneously flows from hot to cold because this process increases the total entropy of the universe.

Systems Tend Toward Disorder Over Time

All isolated systems naturally evolve toward states of greater disorder. This is because disordered (high-entropy) states are statistically far more probable than ordered (low-entropy) states. For example:

• A drop of ink disperses throughout water — it never spontaneously reconcentrates.
• A broken glass does not spontaneously reassemble.
• Heat flows from hot to cold, never the reverse on its own.

This tendency is captured by the inequality $\Delta S_{total}\ge 0$.

Degradation of Energy

As entropy increases, energy becomes less available to do useful work. This is called the degradation of energy.

• In every natural (irreversible) process, some energy is converted into a less organised form (thermal energy spread over the surroundings) that cannot be recovered as useful work.
• This is why no real heat engine can be 100% efficient — some energy is always degraded.

Applications

• Heat Engines: Entropy limits the efficiency of any heat engine. Because heat must be rejected to a cold reservoir to complete a cycle, some energy is inevitably lost as waste heat. This process of dumping heat increases the entropy of the surroundings, ensuring the total entropy of the universe increases, in compliance with the Second Law.
• Degradation of Energy: As entropy increases, the energy becomes less available to do work. This is often referred to as the degradation of energy.

• Ideal Gas: The change in entropy for an ideal gas undergoing a process from an initial state ($T_1,V_1$) to a final state ($T_2,V_2$) can be calculated with the formula:

$\Delta S=n{C}_v\ln \left(\frac{T_2}{T_1}\right)+nR\ln \left(\frac{V_2}{V_1}\right)$

Possible Questions and Answers

Q: What is the "heat death" of the universe?

A: The "heat death" is a hypothetical end-state of the universe where it has reached maximum entropy. In this state, all energy would be uniformly distributed, and there would be no temperature differences. Consequently, heat could no longer flow, no work could be done, and all thermodynamic processes would cease.

Q: Can the entropy of a system ever decrease?

A: Yes, the entropy of a system can decrease, but only if the entropy of its surroundings increases by an equal or greater amount. For example, when water freezes into ice, the water itself becomes more ordered (entropy decreases), but this process releases heat into the surroundings, increasing the surroundings' disorder. The total entropy of the universe still increases.