15.3 Near-critical behavior and anomalies

Near-critical behavior is a wild ride in thermodynamics. As fluids approach their critical point, they start acting weird. Properties like compressibility and heat capacity go crazy, getting super big or even infinite!

This weirdness happens because of huge fluctuations in density and energy. These fluctuations spread out over long distances, making the fluid super sensitive to tiny changes. It's like the whole system becomes one big, connected blob!

Critical Phenomena and Fluctuations

Critical Fluctuations and Correlation Length

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• Critical fluctuations are large-scale fluctuations in density and other properties that occur near the critical point
• These fluctuations become more pronounced and extend over larger distances as the critical point is approached
• Correlation length is a measure of the spatial extent of these fluctuations
• Near the critical point, the correlation length diverges, meaning it becomes very large (on the order of micrometers or even millimeters)
• This divergence indicates that the system becomes highly correlated over long distances, with fluctuations in one region affecting those in distant regions

Divergence and Critical Slowing Down

• Many thermodynamic properties exhibit divergent behavior near the critical point
• Divergence means that these properties tend toward infinity as the critical point is approached
• Examples of properties that diverge include compressibility, heat capacity, and thermal expansion coefficient
• Critical slowing down is another phenomenon observed near the critical point
• It refers to the slowing down of the system's response to perturbations or changes in external conditions
• Near the critical point, the system takes longer to relax back to equilibrium after a disturbance, as the fluctuations become more persistent and long-lived

Anomalous Properties Near Critical Point

Anomalous Compressibility

• Compressibility is a measure of how much a substance's volume changes in response to a change in pressure
• Near the critical point, the compressibility becomes anomalously large, meaning the substance becomes highly compressible
• This anomalous behavior is a consequence of the large-scale density fluctuations occurring near the critical point
• The divergence of compressibility near the critical point is related to the divergence of the correlation length
• Experimentally, this can be observed as a rapid change in density with small changes in pressure near the critical point

Anomalous Heat Capacity

• Heat capacity is a measure of the amount of heat required to change a substance's temperature by a certain amount
• Near the critical point, the heat capacity also exhibits anomalous behavior, becoming very large
• This means that a small addition of heat can cause a significant change in temperature near the critical point
• The anomalous heat capacity is related to the large-scale energy fluctuations occurring in the system
• Experimentally, this can be observed as a sharp peak in the heat capacity as a function of temperature near the critical point

Theoretical Approaches

Scaled Equations of State

• Scaled equations of state are theoretical models used to describe the behavior of fluids near the critical point
• These equations are based on the idea of scaling, which relates the behavior of a system at different scales (e.g., microscopic and macroscopic)
• Scaled equations of state take into account the universal features of critical phenomena, such as the divergence of correlation length and thermodynamic properties
• One example is the van der Waals equation, which can be rescaled to describe the behavior of fluids near the critical point
• These equations often involve scaling exponents, which characterize the power-law behavior of various properties near the critical point
• Scaled equations of state provide a framework for understanding and predicting the anomalous behavior of fluids in the critical region
• They have been successful in describing experimental data and have contributed to the development of a unified theory of critical phenomena