Entropy
Entropy is a thermodynamic state function that quantifies the degree of disorder, randomness, or the number of microstates available to a system at a given macrostate. Macroscopically, it is defined via the Clausius inequality as the ratio of reversible heat exchange to absolute temperature; microscopically, Boltzmann's formula connects it to the number of microscopic configurations. Entropy always increases in irreversible processes in isolated systems, driving systems toward equilibrium and explaining the thermodynamic arrow of time.
Key Formula
S = k_B × ln(Ω)
LaTeX: S = k_B \ln \Omega
| Symbol | Meaning | Unit |
|---|---|---|
| S | Entropy of the system | J/K |
| k_B | Boltzmann constant | J/K |
| Ω | Number of accessible microstates (multiplicity) | dimensionless |
Worked Example
Problem
Calculate the entropy change when 1 mole of an ideal gas expands isothermally and reversibly from volume V₁ = 10 L to V₂ = 20 L at temperature T = 300 K.
Solution
Step 1: For an isothermal reversible expansion, ΔU = 0, so Q = W. Step 2: Work done by gas: W = nRT ln(V₂/V₁) = 1 × 8.314 × 300 × ln(20/10). Step 3: W = 2494.2 × ln(2) = 2494.2 × 0.6931 = 1729 J. Step 4: ΔS = Q_rev / T = 1729 / 300 = 5.76 J/K.
Answer
ΔS = 5.76 J/K (entropy increases as gas expands into larger volume)
Entropy Changes in Common Thermodynamic Processes
| Process | System | ΔS (System) | ΔS (Surroundings) | ΔS (Universe) |
|---|---|---|---|---|
| Isothermal reversible expansion | Ideal gas | > 0 | < 0 | = 0 |
| Free (irreversible) expansion | Ideal gas | > 0 | = 0 | > 0 |
| Freezing of water at 0 °C | Water | < 0 | > 0 | = 0 |
| Ice melting at 0 °C | Ice | > 0 | < 0 | = 0 |
| Combustion of fuel | Fuel + O₂ | > 0 | > 0 | >> 0 |
| Adiabatic reversible process | Gas | = 0 | = 0 | = 0 |
Interactive Tools
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Related Terms
Second Law of Thermodynamics
The Second Law of Thermodynamics states that in any spontaneous process, the total entropy of an isolated system can only increase or remain constant, never decrease. This gives thermodynamics a preferred direction of time, explaining why heat flows from hot to cold, why mechanical energy converts irreversibly to heat, and why perpetual motion machines of the second kind are impossible. It is the thermodynamic basis for the arrow of time and sets fundamental efficiency limits on all heat engines.
Boltzmann Constant
The Boltzmann constant (k_B) is a fundamental physical constant that relates the average kinetic energy of particles in a gas to the absolute temperature of the gas, acting as the bridge between macroscopic thermodynamic quantities and microscopic statistical mechanics. It appears in Boltzmann's entropy formula S = k_B ln Ω, the ideal gas law in per-particle form, and the Maxwell-Boltzmann energy distribution, making it one of the most universal constants in physics. Since the 2019 SI redefinition, the Boltzmann constant has an exact defined value of 1.380649 × 10⁻²³ J/K.
Carnot Cycle
The Carnot Cycle is an idealized, reversible thermodynamic cycle consisting of two isothermal and two adiabatic processes, first described by Sadi Carnot in 1824 as the most efficient possible heat engine operating between two fixed temperatures. No real engine can exceed the efficiency of a Carnot engine operating between the same hot and cold reservoirs, making it the theoretical upper bound for heat engine performance. The Carnot efficiency depends only on the absolute temperatures of the reservoirs and sets the fundamental limit imposed by the Second Law of Thermodynamics.
Coined by Rudolf Clausius in 1865 from Greek "en" (in) + "trope" (transformation or turning), meaning "a transformation content." Clausius deliberately paralleled the word with "energy" to highlight the two fundamental thermodynamic quantities.