Entropy

Entropy

Entropy (symbol $S$) is a fundamental thermodynamic property that measures the degree of disorder or randomness in a system. It quantifies the number of microscopic configurations that correspond to a thermodynamic system's macroscopic state.

Definition

Entropy is related to the number of possible microstates ($\mathrm{\Omega }$) by the Boltzmann equation:

$S=k_B\ln \mathrm{\Omega }$

where:

• $S$ = entropy
• $k_B$ = Boltzmann constant ($1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{\mathrm{-}\mathrm{1}}$)
• $\mathrm{\Omega }$ = number of accessible microstates of the system

Thermodynamic Definition

In classical thermodynamics, the change in entropy for a reversible process is defined as:

$dS=\frac{\delta Q_{\text{rev}}}{T}$

where:

• $dS$ = infinitesimal change in entropy
• $\delta Q_{\text{rev}}$ = infinitesimal heat absorbed reversibly
• $T$ = absolute temperature

For a finite reversible process between states 1 and 2:

$\mathrm{\Delta }S=S_2-S_1=\underset{1}{\overset{2}{\int }}\frac{\delta Q_{\text{rev}}}{T}$

Second Law of Thermodynamics

The Second Law states that the total entropy of an isolated system never decreases:

$\mathrm{\Delta }{S}_{\text{total}}\ge 0$

where equality holds for reversible processes and inequality for irreversible processes.

This law implies that natural processes tend to move towards increased entropy or disorder.

Entropy Change in Chemical Reactions

The entropy change of a system during a chemical reaction is:

$\mathrm{\Delta }S=\sum S_{\text{products}}-\sum S_{\text{reactants}}$

This change helps determine the spontaneity of reactions when combined with enthalpy changes in the Gibbs free energy:

$\mathrm{\Delta }G=\mathrm{\Delta }H-T\mathrm{\Delta }S$

Statistical Interpretation

Entropy can also be viewed as a measure of uncertainty or information content in the system's microscopic state, connecting thermodynamics with information theory.

Applications

• Predicting spontaneity and direction of chemical reactions
• Explaining phase transitions and mixing phenomena
• Understanding biological processes and energy transfer
• Engineering systems such as engines and refrigerators

References

• Atkins, P., & de Paula, J. (2010). Physical Chemistry. Oxford University Press.
• Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics. Wiley.