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Third Law of Thermodynamics: Absolute Entropy Guide

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The third law of thermodynamics states that the entropy of a perfect crystalline solid at absolute zero (0 K) is exactly zero. At 0 K a perfect crystal has only one possible arrangement of its atoms, so from S = k ln W it has zero disorder and zero entropy.

This law matters because it gives entropy an absolute starting point. The first and second laws only deal with changes in entropy, but the third law lets chemists tabulate absolute (third-law) entropies for every substance and compute the entropy change of any reaction from them.

Key Concepts

  • Absolute zero = 0 K = −273.15 °C, the temperature at which all thermal motion would cease.
  • Unattainability statement: absolute zero cannot be reached in a finite number of steps, though we can get arbitrarily close.
  • Standard molar entropy S°: absolute entropy of one mole at 1 bar and 298 K, in J/mol·K. Unlike standard enthalpies of formation, S° of elements is not zero.
  • Residual (zero-point) entropy: imperfect crystals such as solid CO or N2O keep some disorder even at 0 K, so their entropy does not quite reach zero.
  • Heat capacities fall toward zero as T → 0 (the Debye T³ law), which is what makes the entropy integral below converge.

Formulas

  • S(T) = ∫(0→T) (Cp/T) dT, plus ΔH/transition ÷ T for each phase change: absolute entropy is built up by reversibly warming the perfect crystal from 0 K.
  • ΔS°(reaction) = Σ νS°(products) − Σ νS°(reactants), with ν the stoichiometric coefficients.
  • Assumptions: a perfect crystal at 0 K, reversible heating, and heat-capacity data extrapolated to 0 K with the Debye model.

Worked Example

Find ΔS° for 2H2(g) + O2(g) → 2H2O(l) using standard tabulated entropies: S°(H2) ≈ 130.7, S°(O2) ≈ 205.1, S°(H2O, l) ≈ 69.9 J/mol·K.

  1. Reactants: 2 × 130.7 + 205.1 = 261.4 + 205.1 = 466.5 J/mol·K.
  2. Products: 2 × 69.9 = 139.8 J/mol·K.
  3. ΔS° = 139.8 − 466.5 = −326.7 J/mol·K.

The entropy drops sharply because three moles of disordered gas become two moles of compact liquid — a large loss of microstates, exactly what the third-law scale quantifies.

Common Misconceptions

  • "The entropy of elements is zero." That is true for standard enthalpies of formation, not entropy. Every substance, element or compound, has S° > 0 at 298 K.
  • "The third law says we can reach 0 K." It says the opposite in practice: 0 K is a limit that can be approached but never attained in finite steps.
  • "Every solid has zero entropy at 0 K." Only perfect crystals do. Glasses and orientationally disordered solids like CO retain residual entropy.

Real-World Applications

  • Predicting spontaneity: ΔG° = ΔH° − TΔS° needs ΔS°, which comes from third-law tables.
  • Cryogenics: designing dilution refrigerators and adiabatic demagnetisation cooling relies on entropy data near 0 K.
  • Materials science: ordering transitions in alloys and superconductors are analysed through their entropy changes.

The third law turns entropy from a relative bookkeeping quantity into an absolute property of matter: start every substance at zero entropy at 0 K, warm it up, add up Cp/T, and you know exactly how disordered it is at any temperature.

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