Overview
Definition
Third Law of Thermodynamics states: entropy of a perfect crystal approaches zero as temperature approaches absolute zero (0 K).
Historical Context
Formulated independently by Walther Nernst (1906-1912). Also called Nernst Heat Theorem. Motivated by low-temperature thermodynamics.
Scope
Applies to thermodynamic systems in equilibrium. Crucial for defining absolute entropy scales and understanding cryogenic behavior.
Formulation of the Third Law
Nernst Heat Theorem
Entropy change ΔS for any isothermal process approaches zero as T → 0.
Planck's Statement
Entropy of a perfect crystalline substance is zero at absolute zero temperature.
Mathematical Expression
limT→0 S = 0 for perfect crystals in thermodynamic equilibrium.
Entropy at Absolute Zero
Conceptual Basis
Entropy quantifies disorder or number of accessible microstates. At 0 K, perfect crystal has unique ground state, hence zero disorder.
Thermodynamic Consequence
Absolute entropy scale defined: S(0 K) = 0. Enables determination of absolute entropies from calorimetric data.
Implications for Spontaneity
Helps in evaluating Gibbs free energy at low temperatures. Stability predictions require absolute entropy values.
Residual Entropy
Definition
Nonzero entropy at 0 K due to degeneracy of ground state or disorder frozen into crystal lattice.
Examples
Carbon monoxide crystals, ice with proton disorder, glasses exhibiting frozen-in configurational states.
Calculation
Residual entropy Sres = R ln W, where W = number of equivalent microstates at 0 K.
Heat Capacity Behavior Near 0 K
Vanishing Heat Capacity
Heat capacities (Cp, Cv) approach zero as T → 0, consistent with Third Law.
Debye Model
Predicts C ∝ T3 behavior for solids at low temperatures.
Experimental Observations
Measured heat capacities confirm theoretical predictions, supporting Third Law validity.
| Temperature (K) | Heat Capacity (J/mol·K) |
|---|---|
| 0.5 | 0.002 |
| 1.0 | 0.016 |
| 2.0 | 0.128 |
Thermodynamic Implications
Zero Entropy Reference
Allows absolute entropy determination; avoids ambiguity in entropy scale.
Equilibrium and Spontaneity
Enables accurate Gibbs free energy calculations at low temperatures; influences phase stability.
Impossibility of Reaching 0 K
Third Law implies absolute zero unattainable in finite steps; entropy changes vanish asymptotically.
Statistical Mechanics Perspective
Microstates and Entropy
S = kB ln W, where W = number of accessible microstates.
Ground State Uniqueness
For perfect crystals, W = 1 at 0 K; hence S = 0.
Degenerate Ground States
Multiple ground states yield residual entropy; violates idealized Third Law assumptions.
S = k_B ln WW = 1 → S = 0 at 0 KW > 1 → S > 0 (residual entropy)Applications
Cryogenics
Design of low-temperature systems; entropy control critical for refrigeration cycles.
Thermodynamic Data
Standard absolute entropies tabulated based on Third Law; essential for chemical thermodynamics.
Material Science
Characterization of defects, disorder, and phase transitions at low temperatures.
Limitations and Exceptions
Non-crystalline Solids
Amorphous solids and glasses exhibit residual entropy due to frozen disorder.
Magnetic Systems
Spin glasses and frustrated magnets can have nonzero entropy at 0 K.
Quantum Effects
Zero-point energy and quantum degeneracy can complicate strict application.
Experimental Verifications
Calorimetric Measurements
Heat capacity integrations show entropy approaches zero for pure crystals.
Low-Temperature Spectroscopy
Confirms ground state uniqueness and energy gaps consistent with Third Law.
Residual Entropy Detection
Measured via entropy differences; confirms exceptions and disorder effects.
Formulas and Calculations
Entropy Calculation from Heat Capacity
S(T) = ∫(0 to T) (C_p/T) dT + S(0)For perfect crystals, S(0) = 0.
Residual Entropy Formula
S_residual = R ln WR: universal gas constant; W: number of equivalent microstates.
Debye Heat Capacity Approximation
C_v = 12π^4/5 * R * (T/Θ_D)^3Θ_D: Debye temperature; valid near 0 K.
| Parameter | Symbol | Units |
|---|---|---|
| Entropy | S | J·mol-1·K-1 |
| Heat Capacity | C_p, C_v | J·mol-1·K-1 |
| Gas Constant | R | 8.314 J·mol-1·K-1 |
| Debye Temperature | Θ_D | K |
References
- Atkins, P.W., Physical Chemistry, 10th ed., Oxford University Press, 2014, pp. 85-90.
- Nernst, W., "The New Heat Theorem," Zeitschrift für Physikalische Chemie, 60, 1906, pp. 214-225.
- Planck, M., Treatise on Thermodynamics, Dover Publications, 1945, pp. 130-135.
- Callen, H.B., Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, 1985, pp. 290-295.
- Fultz, B., Howe, J.M., Transmission Electron Microscopy and Diffractometry of Materials, 4th ed., Springer, 2013, pp. 121-125.