Third Law of Thermodynamics

The Third Law of Thermodynamics says that as a substance gets closer to absolute zero (0 Kelvin), the entropy—or disorder—of a perfect crystal drops to zero. At this point, the crystal reaches perfect order, with only one possible way its particles can be arranged. In simple terms, there’s no more randomness. This law also tells us that it’s impossible to actually reach absolute zero using any limited number of steps, because it would mean removing all thermal energy—something we just can't do. The Third Law plays a key role in helping scientists understand how materials behave at extremely low temperatures. It's also crucial for accurately measuring entropy and plays a big part in the design and limitations of cooling systems.

1.0Statements of Third Law of Thermodynamics

• As the temperature of a system approaches absolute zero, the entropy of a perfect crystalline substance approaches zero.

${\lim }_{\tau \to 0}S=0S=\text{Entropy},T=\text{Absolute Temperature}$

• This law is also known as the Nernst Heat Theorem

2.0Absolute Zero

• It is the lowest possible temperature, defined as 0 Kelvin (K), which is equivalent to:
• –273.15°C (Celsius)
• –459.67°F (Fahrenheit)

At absolute zero:

• All classical motion of particles theoretically ceases (they have minimal vibrational motion due to quantum mechanics).
• A system reaches its minimum internal energy.
• It's the point where entropy is at a minimum (for a perfect crystal, according to the third law of thermodynamics).

3.0Concept of Entropy

• It is a measure of disorder, randomness, or uncertainty in a system.
• In thermodynamics, entropy (usually denoted as S) measures the number of microscopic configurations (ways particles can be arranged) that correspond to a system's macroscopic state. The higher the entropy, the more disordered or random the system is,

$$\begin{gathered} S=k_B\ln \Omega \\ S=\text{Entropy},k_B=\text{Boltzman Constant},\Omega =\text{is the number of microstates} \\ \Delta S=\frac{q_{rev}}{T} \end{gathered}$$

• Entropy increases when heat is added to a system and decreases when heat is removed (assuming constant temperature).
• In irreversible processes, entropy increases overall—even if energy is conserved (thanks to the Second Law of Thermodynamics).
• In isolated systems, entropy is likely to increase over time, reflecting the natural tendency toward disorder.

4.0Perfect Crystal

• In a perfect crystal at absolute zero (0 K), the concept of microstates becomes very straightforward. A microstate states a specific arrangement of particles in a system. At 0 K, all atoms in a perfect crystal are in their lowest possible energy state and are arranged in a perfectly ordered, repeating lattice structure. Because the temperature is zero, there is no thermal motion—the atoms are completely stationary. As a result, the system has only one possible microstate, since there is no energy available to allow any variation in atomic positions or motions. This complete uniformity and absence of disorder mean that the system is in a state of maximum order. Consequently, the entropy, which is a measure of disorder or the number of microstates, is exactly zero for a perfect crystal at 0 K.
• A perfect crystal is a solid in which the atoms or molecules are ordered in a perfectly ordered, repeating pattern, without any defects such as dislocations or impurities. At absolute zero (0 K), the crystal exists in a single ground energy state, with no variations in atomic positions or motion. This idealized structure ensures maximum order and minimal entropy.

5.0Mathematical Formulation and Derivation

$$\begin{gathered} \text{Change in Entropy is given by,} \\ dS=\frac{C}{T}dT \\ \text{Integrate from T = 0 to a temperature T} \\ S(T)-S(0)={\int }_0^T\frac{C}{T}dT \\ \text{If we assume} \\ C\propto T^n\text{as }T\to 0 \\ \text{ For a Debye Solid }C\propto T^{3}as\frac{C}{T}\to T^2 \\ {\int }_0^T{T}^{2}dT=\frac{T^3}{3}\to 0\text{as }T\to 0 \\ {\lim }_{T\to 0}S(T)=S(0)=0 \end{gathered}$$

6.0Physical Interpretation

Entropy as a Measure of Disorder

•  Entropy (SS) represents the degree of disorder or randomness in a system, or equivalently, the number of ways its particles can be arranged (microstates). At high temperatures, particles possess more energy and can occupy a wide range of positions and energy states, resulting in greater disorder and higher entropy. In contrast, at absolute zero (0 K), a perfect crystal—with a perfectly ordered, defect-free atomic arrangement—achieves its lowest energy configuration. In this state, particles are completely ordered and stationary, allowing only one microstate, which corresponds to zero entropy.

Perfect Crystal Condition

• A perfect crystal has atoms arranged in a flawless, repeating lattice without any impurities or structural defects. At 0 K, thermal motion ceases entirely, and atoms are fixed in place. The system exists in a unique configuration, meaning only a single microstate is available.

Zero Thermal Motion:

•  As temperature approaches absolute zero, particles lose all thermal energy, and their motion slows to a complete stop. Since entropy is associated with randomness and molecular motion, the absence of motion leads to the absence of disorder—hence, entropy becomes zero.

Absolute Reference Point for Entropy:

•  The third law provides an absolute baseline for entropy, unlike many other thermodynamic properties. With S=0S = 0 at 0 K for a perfect crystal, entropy can be measured on an absolute scale from this point upward, making thermodynamic calculations more precise.

Residual Entropy (Practical Consideration):

• In real-world materials, perfect crystals do not exist. Imperfections such as impurities, defects, or the presence of multiple equivalent ground states can persist even at 0 K. These imperfections introduce a degree of disorder, resulting in a small but non-zero entropy at absolute zero—this is known as residual entropy.

7.0Applications Third Law of Thermodynamics

1. Calculation of Absolute Entropy

2. Cryogenics and Low-Temperature Physics

3.Residual Entropy and Material Science

4.Statistical Mechanics and Quantum Behavior

8.0Limitations of Third Law of Thermodynamics

1. Idealization of the Perfect Crystal

The law assumes a perfect crystal, which does not exist in reality.

Real substances have defects and impurities, causing residual entropy at 0 K.

2. Unattainability of Absolute Zero

The third law implies that absolute zero cannot be reached by any finite number of processes.

Practically, 0 K is unattainable, so the entropy at 0 K is theoretical.

3. Residual Entropy in Real Systems

Materials like glasses, random alloys, and some molecular crystals retain non-zero entropy at 0 K.

The law doesn't apply strictly to these systems without corrections.

4. Not Always Helpful in High-Temperature Regimes

The third law is most relevant at very low temperatures.

It has limited direct application in typical room-temperature thermodynamics.