2.3 Third law of thermodynamics

Definition and statement

The third law of thermodynamics tells you what happens to entropy as temperature drops toward absolute zero. It sets hard limits on how cold a system can get and how ordered it can become, linking the macroscopic quantity of entropy directly to the quantum mechanical behavior of particles.

Absolute zero temperature

Absolute zero is defined as $0 \, K$ (equivalently $-273.15°C$). At this temperature, a system occupies its lowest possible energy state, and classical thermal motion ceases entirely.

However, particles don't become perfectly still. Quantum mechanics requires a minimum energy called zero-point energy, so even at $0 \, K$ there's residual motion dictated by the Heisenberg uncertainty principle. Absolute zero serves as the universal reference point for thermodynamic temperature scales and entropy calculations.

A critical consequence of the third law: absolute zero is unattainable in a finite number of steps. You can get asymptotically close, but never reach it.

The perfect crystal at absolute zero

The Planck statement of the third law says: the entropy of a perfect crystal at absolute zero is exactly zero.

A perfect crystal means every atom sits in a single, unique arrangement with no disorder whatsoever. Since there's only one microstate ($W = 1$), Boltzmann's formula gives:

$S = k_B \ln W = k_B \ln 1 = 0$

No real material achieves this perfectly. Defects, isotopic mixing, and quantum effects always introduce some residual disorder. But the perfect crystal concept gives you the theoretical baseline against which real materials are compared.

Entropy at absolute zero

As $T \to 0$, entropy approaches a well-defined constant. For a perfect crystal, that constant is zero. For other systems, it can be nonzero, and understanding why is where statistical mechanics becomes essential.

Zero-point entropy

Zero-point entropy is the entropy a system retains at $T = 0 \, K$. It arises when the ground state is degenerate, meaning multiple distinct microstates share the same lowest energy.

Whether a system has zero-point entropy depends on its structure:

• A perfect crystal with a unique ground state has $S_0 = 0$
• A system with $g$ degenerate ground states has $S_0 = k_B \ln g$
• Amorphous or glassy materials can retain significant zero-point entropy due to frozen-in disorder

Residual entropy

Residual entropy is the specific case where a material retains nonzero entropy at absolute zero because of multiple equivalent ground-state configurations.

Two classic examples:

• Ice ($H_2O$): The hydrogen bonds in ice can be arranged in many nearly equivalent ways while satisfying the "ice rules." Linus Pauling estimated the residual entropy as approximately $S_0 = R \ln(3/2) \approx 3.4 \, \text{J/(mol·K)}$, which matches experiments well.
• Carbon monoxide (CO): CO molecules in a crystal can orient as CO or OC with nearly the same energy, giving roughly $S_0 = R \ln 2 \approx 5.8 \, \text{J/(mol·K)}$.

In both cases, you calculate residual entropy using Boltzmann's formula $S = k_B \ln W$, where $W$ counts the number of accessible microstates at $T = 0$.

Nernst heat theorem

Walther Nernst proposed his heat theorem in 1906, before the third law was formally stated. The theorem says:

This is a weaker statement than Planck's version (it doesn't fix the absolute value of entropy, only says entropy differences vanish), but it was the historical foundation from which the full third law developed.

Implications for heat capacity

The Nernst theorem has a direct, testable consequence: heat capacity must go to zero as $T \to 0$.

Here's why. Entropy is related to heat capacity at constant pressure by:

$S(T) = \int_0^T \frac{C_p(T')}{T'} \, dT'$

For this integral to converge (not blow up to infinity), $C_p$ must vanish as $T \to 0$. Classical models like the Dulong-Petit law predict constant heat capacity, which fails badly at low temperatures. Quantum models fix this:

• The Einstein model treats atoms as independent quantum oscillators and predicts an exponential drop-off in $C_v$ at low $T$
• The Debye model accounts for collective lattice vibrations (phonons) and predicts $C_v \propto T^3$ at low temperatures, which matches experiments much better

Each model has a characteristic temperature (the Einstein temperature $\Theta_E$ or Debye temperature $\Theta_D$) below which quantum effects dominate and heat capacity drops sharply.

Entropy changes near 0 K

The third law implies that all perfect crystals converge to the same entropy (zero) at absolute zero. This is enormously useful because it gives you an absolute entropy scale.

To find the absolute entropy of a substance at temperature $T$:

1. Measure $C_p(T')$ from as close to $0 \, K$ as possible up to $T$
2. Account for any phase transitions by adding $\Delta H / T_{\text{transition}}$ at each transition temperature
3. Integrate: $S(T) = \int_0^T \frac{C_p(T')}{T'} \, dT' + \sum_i \frac{\Delta H_i}{T_i}$

Without the third law fixing $S(0) = 0$ for perfect crystals, you'd only ever know entropy differences, never absolute values.

Applications in statistical mechanics

The third law constrains how statistical mechanical quantities behave in the low-temperature limit, providing important checks on theoretical models.

Partition function behavior

The canonical partition function is $Z = \sum_i e^{-\beta E_i}$, where $\beta = 1/(k_B T)$.

As $T \to 0$ (so $\beta \to \infty$), the exponential factor $e^{-\beta E_i}$ suppresses every term except those with the lowest energy $E_0$. The partition function simplifies to:

$Z \xrightarrow{T \to 0} g_0 \, e^{-\beta E_0}$

where $g_0$ is the ground state degeneracy. This means:

• If $g_0 = 1$ (non-degenerate ground state), the system is in a single microstate and $S \to 0$
• If $g_0 > 1$, the system retains entropy $S_0 = k_B \ln g_0$

This low-temperature limit is often the easiest regime to check when building a new statistical mechanical model.

Free energy considerations

Both the Helmholtz free energy ($F = U - TS$) and Gibbs free energy ($G = H - TS$) simplify as $T \to 0$:

• Since $S \to 0$ (or a small constant) and $T \to 0$, the $TS$ term vanishes
• Free energy converges to the ground state energy: $F \to E_0$ and $G \to H_0$

This means that at very low temperatures, the spontaneity of a process is determined almost entirely by energy (enthalpy), not entropy. The entropic driving force disappears. This is why low-temperature phase transitions tend to favor the most energetically stable (most ordered) phase.

Experimental verification

Calorimetry at low temperatures

Verifying the third law requires measuring thermodynamic properties at temperatures approaching absolute zero. This demands specialized techniques:

• Dilution refrigerators can cool samples to millikelvin temperatures by exploiting the mixing of $^3\text{He}$ and $^4\text{He}$
• Adiabatic demagnetization uses the entropy change of paramagnetic salts in varying magnetic fields to reach microkelvin temperatures
• Precision calorimeters at these temperatures measure tiny heat inputs and the resulting temperature changes

These measurements let you integrate $C_p / T$ from near zero up to room temperature, giving absolute entropies that can be compared with third-law predictions.

Specific heat measurements

Specific heat data at low temperatures have been among the strongest confirmations of the third law:

• All measured substances show $C \to 0$ as $T \to 0$, consistent with the Nernst theorem
• Metals show $C = \gamma T + \alpha T^3$ at low $T$, where the linear term comes from conduction electrons and the cubic term from phonons (Debye behavior)
• Anomalies in specific heat curves reveal phase transitions (superconducting transitions, magnetic ordering) and help map out phase diagrams at low temperatures

Limitations and exceptions

Quantum effects

Quantum mechanics both underlies the third law and creates situations where naive classical reasoning breaks down:

• Zero-point energy means no system ever truly has zero kinetic energy, even at $T = 0$
• Quantum tunneling allows particles to access configurations that would be classically forbidden, affecting low-temperature thermodynamic properties
• Quantum phase transitions occur at $T = 0$ and are driven by quantum fluctuations rather than thermal fluctuations. Near a quantum critical point, the system's behavior deviates from standard third-law expectations, and the influence of the critical point extends to finite temperatures

Non-crystalline materials

The third law in its Planck form applies strictly to perfect crystals. Disordered systems complicate things:

• Glasses and amorphous solids are frozen in non-equilibrium configurations. They retain configurational entropy at low temperatures because the system can't explore all microstates on experimental timescales.
• The Kauzmann paradox arises when you extrapolate the entropy of a supercooled liquid below its glass transition: the liquid's entropy would apparently drop below the crystal's entropy at a finite temperature (the Kauzmann temperature $T_K$). This suggests either a thermodynamic phase transition must intervene or the extrapolation breaks down.
• Whether glasses truly violate the third law or simply can't equilibrate fast enough remains an active area of research.

Connection to other laws

First and second laws

The three laws of thermodynamics work together as a complete framework:

• The first law (energy conservation) tells you that energy is neither created nor destroyed, but it says nothing about the direction of processes
• The second law (entropy increase) tells you which direction processes go spontaneously, but it doesn't fix an absolute scale for entropy
• The third law completes the picture by anchoring entropy at $S = 0$ for a perfect crystal at $T = 0$, giving you absolute entropy values

A practical consequence: the third law sets an upper bound on the efficiency of refrigerators. Since you can't reach absolute zero in finite steps, no cooling process is perfectly efficient. The closer you try to get to $0 \, K$, the more work each incremental temperature drop requires.

Statistical interpretation

From a statistical mechanics perspective, the third law is almost a natural consequence of quantum mechanics:

• Boltzmann's formula $S = k_B \ln W$ directly connects entropy to the number of accessible microstates
• At $T = 0$, only the ground state is occupied, so $W = g_0$
• For a non-degenerate ground state, $W = 1$ and $S = 0$

The third law also connects to information theory: entropy measures the missing information about a system's microstate. At absolute zero with a unique ground state, you know exactly which state the system is in, so the information-theoretic entropy is zero.

Questions about ergodicity (whether a system actually explores all its accessible states) become especially important near $T = 0$, since relaxation times can diverge and systems may get trapped in metastable configurations.

Implications for phase transitions

Order-disorder transitions

As temperature drops toward zero, systems tend to become more ordered, consistent with entropy decreasing. The third law constrains how this ordering happens:

• Near a continuous (second-order) phase transition, thermodynamic quantities follow power laws characterized by critical exponents
• The concept of universality means that systems with very different microscopic details can share the same critical exponents if they belong to the same universality class
• Symmetry breaking is central: the ordered phase has lower symmetry than the disordered phase. For example, a ferromagnet above its Curie temperature has no preferred magnetization direction, but below it, the spins align and break rotational symmetry.

Quantum phase transitions

Unlike thermal phase transitions (driven by temperature), quantum phase transitions occur at $T = 0$ and are driven by changing a non-thermal parameter like pressure, magnetic field, or chemical composition.

Key features:

• At a quantum critical point, quantum fluctuations (not thermal fluctuations) destroy the ordered phase
• The influence of a quantum critical point fans out into a "quantum critical region" at finite temperatures, producing unusual thermodynamic and transport properties (non-Fermi liquid behavior in metals, for instance)
• Experimental probes include neutron scattering, NMR, and specific heat measurements under varying pressure or field
• Quantum phase transitions are central to understanding exotic states like quantum spin liquids, heavy fermion systems, and topological phases of matter

Third law in quantum systems

Ground state degeneracy

When a system has multiple ground states with exactly the same energy, the third law doesn't give $S = 0$ at absolute zero. Instead:

$S_0 = k_B \ln g_0$

This degeneracy can arise from:

• Geometric frustration: In frustrated magnets (e.g., antiferromagnets on a triangular lattice), not all pairwise interactions can be simultaneously satisfied, leading to a large number of degenerate ground states
• Topological degeneracy: Some quantum systems have ground state degeneracies that depend on the topology of the space they occupy, not on local properties. This is the basis for topological qubits in quantum computing, where information is stored in the global topological properties and is inherently protected from local perturbations.
• Quantum Hall systems exhibit topological degeneracy, with the number of ground states depending on the genus of the surface

Quantum entanglement effects

At very low temperatures, quantum entanglement becomes a dominant feature of many-body systems:

• Entanglement entropy quantifies how much quantum information is shared between subsystems. For a ground state of a local Hamiltonian, entanglement entropy typically scales with the boundary area of the subsystem (the "area law"), not its volume.
• Near quantum critical points, entanglement entropy can develop logarithmic corrections to the area law, signaling long-range quantum correlations
• Entanglement structure at $T = 0$ determines many physical properties and is deeply connected to the nature of the ground state
• Practical applications include quantum sensing and metrology, where entangled states at ultra-low temperatures can achieve measurement precision beyond classical limits

Technological applications

Cryogenics

Third-law principles directly inform the design and limits of cryogenic technology:

• Superconducting magnets in MRI machines and particle accelerators (like the LHC) operate at liquid helium temperatures (~4 K), where electrical resistance vanishes
• Dilution refrigerators used in quantum computing labs routinely reach ~10 mK
• The unattainability of absolute zero means every cryogenic system faces diminishing returns: each further temperature reduction requires disproportionately more work
• Cryopreservation of biological samples uses controlled cooling, though the physics there is more about avoiding ice crystal damage than about approaching absolute zero

Low-temperature physics

Ultra-low temperatures open the door to quantum phenomena that are invisible at room temperature:

• Superconductivity (zero electrical resistance) and superfluidity (zero viscosity in liquid helium) are macroscopic quantum effects observable only below critical temperatures
• Precision instruments like atomic clocks and gravitational wave detectors (LIGO) exploit low-temperature environments to minimize thermal noise
• Ultra-cold atom experiments use laser cooling and evaporative cooling to reach nanokelvin temperatures, creating Bose-Einstein condensates where thousands of atoms occupy a single quantum state
• These platforms serve as quantum simulators, allowing researchers to study many-body quantum physics in highly controlled settings

Historical development

Walther Nernst's contribution

Nernst formulated his heat theorem in 1906 while studying chemical equilibria at low temperatures. His key observation was that the entropy change for chemical reactions approaches zero as $T \to 0$.

This was initially controversial. Nernst's theorem was an empirical generalization, and its exact scope was debated for years. Planck later strengthened it by proposing that the entropy itself (not just entropy changes) goes to zero for perfect crystals at absolute zero. Nernst received the Nobel Prize in Chemistry in 1920, in part for this work.

Refinements and generalizations

The third law evolved significantly after Nernst:

• Planck (1911) proposed the strong form: $S \to 0$ as $T \to 0$ for perfect crystals
• Einstein and Debye developed quantum models of heat capacity that naturally produced $C \to 0$ as $T \to 0$, providing theoretical support
• Simon's formulation (1927) stated the unattainability of absolute zero as an equivalent statement of the third law
• Modern research continues to probe the third law's boundaries in quantum many-body systems, particularly in systems with topological order, frustration, or long-range entanglement, where the relationship between ground state structure and low-temperature thermodynamics remains an active frontier