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Ch. 18 - Kinetic Theory of Gases
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 18 - Kinetic Theory of GasesProblema 77a
Capitolo 18, Problema 77a

From the van der Waals equation of state, show that the critical temperature and pressure are given by Tcr = 8a / 27bR , Pcr = a / 27b². [Hint: Use the fact that the P versus V curve has an inflection point at the critical point so that the first and second derivatives are zero.]

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Start with the van der Waals equation of state: P=nV−aV2−nV

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van der Waals Equation of State

The van der Waals equation is a modified ideal gas law that accounts for the volume occupied by gas molecules and the attractive forces between them. It is expressed as (P + a(n/V)²)(V - nb) = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and a and b are substance-specific constants. This equation is crucial for understanding real gas behavior, especially near the critical point.
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Kinematics Equations

Critical Point

The critical point of a substance is the end point of the phase equilibrium curve, where distinct liquid and gas phases cease to exist. At this point, the properties of the liquid and gas phases become indistinguishable, leading to a unique critical temperature (Tᵣ) and critical pressure (Pᵣ). Understanding the critical point is essential for analyzing phase transitions and the behavior of substances under varying conditions.
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Phase Diagrams

Inflection Point and Derivatives

An inflection point on a curve is where the curvature changes, which in the context of the P-V diagram indicates a change in the phase behavior of a substance. At the critical point, both the first derivative (pressure with respect to volume) and the second derivative (change of pressure with respect to volume) are zero. This condition is used to derive the critical temperature and pressure from the van der Waals equation, highlighting the relationship between thermodynamic properties.
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