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Thermodynamics

๐ŸฅตThermodynamics Unit 7 โ€“ The Third Law of Thermodynamics

The Third Law of Thermodynamics explores the behavior of systems as they approach absolute zero temperature. It establishes that a perfect crystal's entropy becomes zero at this point, providing a reference for entropy calculations and insights into low-temperature phenomena. This law has far-reaching implications, from cryogenic technologies to chemical equilibria. It sets limits on refrigeration efficiency, helps predict reaction spontaneity, and plays a crucial role in understanding extreme environments in astrophysics and condensed matter physics.

Key Concepts and Definitions

  • Entropy is a measure of the disorder or randomness in a system and increases as temperature increases
    • At absolute zero (0 K), a perfect crystal has zero entropy since there is no disorder
  • Absolute zero is the lowest possible temperature where all molecular motion ceases
    • Corresponds to -273.15ยฐC or -459.67ยฐF
  • Heat capacity is the amount of heat required to raise the temperature of a substance by one degree
    • Approaches zero as temperature approaches absolute zero
  • Residual entropy refers to the entropy that remains in a system even at absolute zero due to degeneracy or disorder
  • Thermal equilibrium occurs when two systems in contact have the same temperature and no net heat flow between them

Historical Context and Development

  • The Third Law of Thermodynamics was developed in the early 20th century to address the behavior of systems at very low temperatures
  • Walther Nernst, a German chemist, proposed the heat theorem in 1906 which stated that entropy changes approach zero as temperature approaches absolute zero
    • This laid the groundwork for the Third Law
  • The Nernst-Simon statement, formulated by Franz Simon in 1937, provided a more rigorous mathematical formulation of the Third Law
  • Experimental work by Giauque and others in the 1920s and 1930s provided strong evidence supporting the Third Law
    • Measurements of heat capacities of various substances at low temperatures confirmed the predicted behavior
  • The development of the Third Law helped to establish the absolute temperature scale and provided insights into the quantum mechanical nature of matter

Statement of the Third Law

  • The entropy of a perfect crystal at absolute zero is exactly equal to zero
    • In other words, the entropy of a system approaches a constant value as the temperature approaches absolute zero
  • For systems with non-zero entropy at absolute zero (residual entropy), the change in entropy approaches zero as temperature approaches absolute zero
    • Mathematically, limโกTโ†’0ฮ”S=0\lim_{T \to 0} \Delta S = 0
  • The Third Law provides an absolute reference point for the determination of entropy
  • It is impossible to reach absolute zero in a finite number of steps or in a finite amount of time
    • Known as the unattainability principle

Mathematical Formulation

  • The mathematical statement of the Third Law is given by limโกTโ†’0(โˆ‚Sโˆ‚T)V=0\lim_{T \to 0} \left(\frac{\partial S}{\partial T}\right)_V = 0
    • This means that the change in entropy with respect to temperature approaches zero as temperature approaches absolute zero
  • The heat capacity of a system is related to the change in entropy with temperature by C=T(โˆ‚Sโˆ‚T)VC = T \left(\frac{\partial S}{\partial T}\right)_V
    • As Tโ†’0T \to 0, Cโ†’0C \to 0 according to the Third Law
  • For a perfect crystal, the entropy at absolute zero is given by S(0)=0S(0) = 0
    • This serves as a reference point for calculating absolute entropies
  • The Third Law can be used to calculate the absolute entropy of a substance at any temperature by integrating the heat capacity: S(T)=โˆซ0TC(Tโ€ฒ)Tโ€ฒdTโ€ฒS(T) = \int_0^T \frac{C(T')}{T'} dT'

Implications and Applications

  • The Third Law has important implications for the behavior of materials at low temperatures
    • Superconductivity and superfluidity are phenomena that occur near absolute zero
  • Understanding the Third Law is crucial for the development of cryogenic technologies and the study of condensed matter physics
  • The Third Law provides a fundamental limit on the efficiency of refrigeration and heat engines
    • It is impossible to achieve 100% efficiency in a heat engine or to reach absolute zero in a refrigerator
  • The Third Law is used in the calculation of chemical equilibria and reaction feasibility at low temperatures
    • Helps predict the spontaneity and direction of chemical reactions
  • The Third Law has applications in astrophysics, particularly in understanding the behavior of matter in extreme environments like neutron stars and white dwarfs

Experimental Verification

  • Experimental measurements of heat capacities at low temperatures have provided strong support for the Third Law
    • Heat capacities of various substances (metals, dielectrics, superconductors) approach zero as Tโ†’0T \to 0
  • Measurements of the entropy changes in chemical reactions and phase transitions near absolute zero are consistent with the Third Law
    • Entropy changes become very small at low temperatures
  • Spectroscopic studies of materials at low temperatures have confirmed the decrease in molecular motion and vibrational modes predicted by the Third Law
  • Investigations of the magnetic properties of materials at low temperatures (magnetization, susceptibility) have also verified the Third Law
    • Magnetic entropy contributions vanish as Tโ†’0T \to 0

Limitations and Exceptions

  • The Third Law applies strictly to perfect crystalline solids at absolute zero
    • Real materials may have defects, impurities, or other sources of disorder that lead to non-zero entropy at T=0T = 0
  • Systems with degenerate ground states (multiple lowest energy configurations) can have residual entropy at absolute zero
    • Examples include ice, carbon monoxide, and certain alloys
  • Glasses and other amorphous materials do not have a unique ground state and may violate the Third Law
    • They have a non-zero residual entropy due to structural disorder
  • Quantum systems with strongly correlated electrons or frustrated interactions can exhibit unusual low-temperature behavior that deviates from the Third Law
    • Examples include spin liquids and quantum spin ice

Connection to Other Laws of Thermodynamics

  • The Third Law is closely related to the Second Law of Thermodynamics, which states that the entropy of an isolated system always increases
    • The Third Law provides an absolute reference point for entropy and sets a lower limit on the entropy of a system
  • The Third Law is consistent with the First Law of Thermodynamics (conservation of energy) and the Zeroth Law (thermal equilibrium)
    • It describes the behavior of entropy as a system approaches thermal equilibrium at absolute zero
  • The Third Law has implications for the efficiency of heat engines and refrigerators, which are governed by the Second Law
    • It sets a fundamental limit on the minimum temperature that can be achieved in a refrigeration cycle
  • The Third Law is important for understanding the microscopic basis of the other laws of thermodynamics
    • It relates the macroscopic properties of a system (entropy, temperature) to the microscopic behavior of its constituent particles


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APยฎ and SATยฎ are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.