15.1 Critical point behavior and properties

Critical point behavior is crucial in understanding phase transitions and fluid properties. At the critical point, liquid and vapor phases become indistinguishable, leading to unique phenomena like critical opalescence and infinite compressibility.

Universality in critical phenomena reveals that diverse substances exhibit similar behavior near their critical points. This concept simplifies the study of critical behavior and has applications in various fields, from physics to materials science.

Critical Point Properties

Definition and Characteristics of the Critical Point

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• Critical point represents the highest temperature and pressure at which a substance can exist in vapor-liquid equilibrium
• Occurs at the end of the vapor pressure curve where the properties of the liquid and vapor phases become identical
• Above the critical point, distinct liquid and vapor phases do not exist, and the substance becomes a supercritical fluid
• Critical point is characterized by the critical temperature ($T_c$), critical pressure ($P_c$), and critical density ($\rho_c$)

Critical Temperature, Pressure, and Density

• Critical temperature ($T_c$) is the highest temperature at which a substance can exhibit vapor-liquid equilibrium
  • Above $T_c$, the substance exists as a single phase regardless of the applied pressure
  • Example: The critical temperature of water is 647.1 K (373.9°C)
• Critical pressure ($P_c$) is the vapor pressure at the critical temperature
  • Represents the highest pressure at which vapor-liquid equilibrium can occur
  • Example: The critical pressure of water is 22.06 MPa
• Critical density ($\rho_c$) is the density of the substance at the critical point
  • Corresponds to the average density of the coexisting liquid and vapor phases
  • Example: The critical density of water is 322 kg/m³

Critical Isotherm and Its Significance

• Critical isotherm is the isotherm on a pressure-volume (P-V) diagram that passes through the critical point
• Represents the boundary between the vapor-liquid region and the supercritical region
• On the critical isotherm, the compressibility of the substance becomes infinite, and the distinction between liquid and vapor phases disappears
• The shape of the critical isotherm is characterized by a horizontal inflection point at the critical point

Phase Behavior Near the Critical Point

Phase Coexistence and Critical Opalescence

• Near the critical point, the properties of the coexisting liquid and vapor phases become more similar
• The density difference between the phases decreases, and the interfacial tension approaches zero
• Critical opalescence occurs near the critical point due to increased density fluctuations
  • Causes the substance to appear cloudy or milky due to the scattering of light
  • Example: Near the critical point, a transparent fluid may exhibit a bluish haze or opalescence

Compressibility and Density Fluctuations

• Compressibility, which is the change in volume with respect to pressure, diverges near the critical point
• The isothermal compressibility becomes infinite at the critical point, indicating large density fluctuations
• Density fluctuations near the critical point lead to the formation of local regions with higher or lower densities compared to the average density
• These density fluctuations contribute to the observed critical opalescence and the unique properties of the substance near the critical point

Universality of Critical Phenomena

Concept of Universality

• Universality refers to the observation that many substances exhibit similar behavior near their critical points, regardless of their specific chemical nature
• The critical exponents, which describe the power-law dependence of various properties near the critical point, are found to be universal for a wide range of substances
• Universality allows for the grouping of substances into universality classes based on their critical behavior
• Examples of universality classes include the Ising model, the Heisenberg model, and the XY model

Significance and Applications of Universality

• Universality simplifies the study of critical phenomena by reducing the number of independent variables needed to describe the system
• It allows for the development of generalized theories and models that can be applied to a wide range of substances
• Universality has implications in various fields, such as condensed matter physics, statistical mechanics, and materials science
• Understanding universality helps in predicting the behavior of substances near their critical points and in designing processes that exploit the unique properties of supercritical fluids (e.g., supercritical fluid extraction, supercritical fluid chromatography)