Third Law of Thermodynamics

Statement of the Third Law

The third law was formulated by Walther Nernst in 1906 and later refined by Max Planck. Nernst original statement, sometimes called the Nernst heat theorem, says that the entropy change associated with any isothermal process approaches zero as the temperature approaches absolute zero. Planck extended this by asserting that the entropy of a perfect crystal at absolute zero is exactly zero, not merely approaching a constant value.

The physical reasoning is straightforward. At absolute zero, a perfect crystal has exactly one possible microscopic arrangement: every atom sits in its precise lattice position with no thermal motion. Since Boltzmann equation gives S = k ln W, and W = 1 for a single microstate, the entropy is S = k ln 1 = 0. There is no randomness, no disorder, and no uncertainty about the microscopic state of the system.

Real materials at very low temperatures sometimes retain residual entropy if they have multiple nearly degenerate ground states. Ice, for example, retains a small residual entropy at absolute zero because water molecules can arrange their hydrogen bonds in many different configurations with nearly the same energy. Glasses and amorphous solids also retain residual entropy because they freeze into disordered configurations that cannot rearrange at low temperatures.

Absolute Entropy and the Standard Reference

The third law allows scientists to determine absolute entropy values for substances by measuring heat capacities from near absolute zero up to the temperature of interest. The procedure involves integrating C{sub}p{/sub}/T from 0 K to the desired temperature, adding entropy contributions from any phase transitions along the way. This gives the standard molar entropy S at any temperature, tabulated in reference books and used in chemical thermodynamics calculations.

Without the third law, only entropy differences would be meaningful. The first and second laws tell you how entropy changes during a process, but they do not specify where the entropy scale begins. The third law anchors the scale at zero for perfect crystals at 0 K, making absolute entropy values physically meaningful and comparable across different substances.

Standard molar entropies at 298 K reveal interesting patterns. Gases have higher standard entropies than liquids, which have higher standard entropies than solids. Larger molecules have higher entropies than smaller ones. These trends reflect the number of microstates available to each substance, confirming the statistical mechanical interpretation of entropy.

The Unattainability of Absolute Zero

A direct consequence of the third law is that absolute zero can never be reached in a finite number of steps. Each cooling step removes some entropy from the system, but as the temperature approaches zero, the amount of entropy that can be removed per step shrinks. You can get arbitrarily close to absolute zero, but reaching it exactly would require an infinite number of operations.

Modern cryogenic techniques have achieved temperatures within billionths of a degree of absolute zero. Laser cooling, magnetic cooling, and evaporative cooling in atomic traps have pushed temperatures below one nanokelvin. At these temperatures, quantum effects dominate and matter exhibits exotic behavior such as Bose-Einstein condensation, where large numbers of atoms occupy the same quantum state and behave as a single quantum entity.

The impossibility of reaching absolute zero is sometimes stated as an alternative formulation of the third law itself. This formulation is physically equivalent to the entropy statement but emphasizes the practical consequence rather than the mathematical foundation.

Heat Capacity Near Absolute Zero

The third law constrains how heat capacity behaves at very low temperatures. As temperature approaches zero, the heat capacity of every substance must also approach zero. If it did not, the integral of C{sub}p{/sub}/T would diverge, producing an infinite entropy, which contradicts the third law.

The Debye model of solids predicts that heat capacity varies as T³ at low temperatures, which approaches zero as required. This T-cubed law has been confirmed experimentally for many crystalline solids and represents one of the early triumphs of quantum mechanics applied to solid-state physics. In metals, an additional linear term in T arises from the electronic contribution to heat capacity, but this also goes to zero at absolute zero.

These low-temperature heat capacity measurements provide one of the most precise ways to determine absolute entropies. Careful calorimetry from liquid helium temperatures (about 4 K) up to room temperature, combined with theoretical extrapolation below 4 K using the Debye model, yields standard entropy values accurate to within a fraction of a percent.

Applications and Significance

The third law is essential for calculating equilibrium constants of chemical reactions from thermal data alone. By combining standard entropies (made possible by the third law) with standard enthalpies, chemists can compute the standard Gibbs free energy change for any reaction and predict its equilibrium position without ever performing the reaction. This capability is the foundation of computational chemistry and thermodynamic databases.

In materials science, the third law helps explain why certain low-temperature phenomena occur. Superconductivity, superfluidity, and Bose-Einstein condensation all involve systems approaching states of very low entropy. Understanding how entropy behaves near absolute zero is critical for designing cryogenic equipment, quantum computers, and other technologies that operate at extremely low temperatures.

The third law also has philosophical significance. Together with the first and second laws, it completes the axiomatic foundation of classical thermodynamics. The three laws (plus the zeroth law defining temperature) provide a complete, self-consistent framework that governs all thermal phenomena. No experimental violation of any of these laws has ever been observed.