Entropy measures disorder in systems, while the Second Law of Thermodynamics states that total entropy always increases. These concepts explain why certain processes occur spontaneously, like heat flowing from hot to cold objects.

The Second Law has wide-ranging applications, from predicting energy transformations to determining process feasibility. It helps us understand why some changes happen naturally, while others require external energy input to overcome entropy increases.

Entropy and the Second Law of Thermodynamics

Entropy and Second Law

• Entropy ($S$) measures disorder or randomness in a system
  • Higher entropy indicates greater disorder or randomness (gas molecules spread out in a room)
  • Lower entropy indicates greater order or predictability (neatly stacked books)
• Second Law of Thermodynamics states total entropy of an isolated system always increases over time
  • In any spontaneous process, entropy of the universe increases (ice melting, salt dissolving in water)
  • Entropy can remain constant in reversible processes, but never decreases
• Entropy determines direction of spontaneous processes
  • Processes that increase entropy are more likely to occur spontaneously (heat flowing from hot to cold object)
  • Processes that decrease entropy are not spontaneous and require external energy input (separating mixed salt and pepper)
• The Second Law provides a fundamental explanation for the arrow of time, indicating why certain processes are irreversible

Applications of Second Law

• Energy transformations always involve an increase in total entropy of system and surroundings
  • Heat transfer from hot object to cold object increases entropy (coffee cooling to room temperature)
  • Mixing of two substances increases entropy due to increased randomness (cream mixing with coffee)
• Spontaneous processes occur naturally without external intervention
  • Examples include heat flow from hot to cold, gas expansion, and diffusion (perfume spreading through a room)
  • Non-spontaneous processes require external energy input to overcome entropy increase (separating a mixture of gases)
• Second Law helps predict direction and feasibility of thermodynamic processes
  • Processes that increase entropy are more likely to occur spontaneously (rusting of iron)
  • Processes that decrease entropy are less likely to occur without external energy input (converting waste heat into useful work)

Entropy changes in systems

• Change in entropy ($\Delta S$) for a system can be calculated using formula: $\Delta S = \frac{Q}{T}$
  • $Q$ is heat transferred to or from system (in joules)
  • $T$ is absolute temperature of system (in Kelvin)
• For reversible processes, change in entropy can be calculated by integrating $\frac{dQ}{T}$ over process
  • $\Delta S = \int \frac{dQ}{T}$ (isothermal expansion of an ideal gas)
• For irreversible processes, change in entropy is always greater than $\frac{Q}{T}$
  • Clausius inequality states $\Delta S \geq \frac{Q}{T}$ for irreversible processes (heat transfer through a finite temperature difference)
• Changes in entropy can also be calculated using statistical mechanics
  • Boltzmann's equation relates entropy to number of microstates ($\Omega$): $S = k_B \ln \Omega$
  • $k_B$ is Boltzmann constant ($1.38 \times 10^{-23}$ J/K) (relates microscopic properties to macroscopic thermodynamic quantities)

Thermodynamic Cycles and Efficiency

• Heat engines convert thermal energy into mechanical work
• The Carnot cycle is an ideal thermodynamic cycle that achieves maximum theoretical efficiency
• Free energy concepts, such as Gibbs free energy, help determine the spontaneity and maximum work output of thermodynamic processes