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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: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 18 - Kinetic Theory of GasesProblem 77a
Chapter 18, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
Textbook Question

At room temperature, it takes approximately 2.45 x 10³ J to evaporate 1.00 g of water. Estimate the average speed of evaporating molecules. What multiple of vrms (at 20°C) for water molecules is this? (Assume Eq. 18–4 holds.)

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Textbook Question

Using the ideal gas law, find an expression for the mean free path ℓM that involves pressure and temperature instead of (N/V). Use this expression to find the mean free path for nitrogen molecules at a pressure of 7.5 atm and 300 K.

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Textbook Question

A sample of cesium vapor is in an oven at 400°C. The volume of the oven is 75 cm³, the vapor pressure of Cs at 400°C is 17 mm-Hg, and the diameter of cesium atoms in the vapor is 0.33 nm. Determine the number of collisions a single Cs atom undergoes with other cesium atoms per second.

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