Real Gases: Deviations from Ideal Gas Behavior and the van der Waals Equation

Real Gases and Deviations from Ideal Gas Behavior

Introduction to Ideal and Real Gases

The Kinetic Molecular Theory provides the foundation for understanding the behavior of gases. The Ideal Gas Law is derived from this theory and assumes that gas particles do not interact and occupy no volume. However, real gases deviate from ideal behavior under certain conditions, necessitating corrections to the Ideal Gas Law.

• Ideal Gas Law: $PV = nRT$

• Assumptions: Gas particles have negligible volume and experience no intermolecular forces.

• Real Gases: Deviate from ideal behavior at high pressures and low temperatures due to particle volume and intermolecular forces.

Corrections to the Ideal Gas Law: The van der Waals Equation

Volume Correction

In reality, gas particles occupy a finite volume. The volume available for particle movement is less than the container's volume. The van der Waals equation corrects for this by subtracting a term proportional to the number of moles:

• Corrected Volume: $(V - nb)$, where n is the number of moles and b is the van der Waals constant for particle size.

• Physical Meaning: b accounts for the excluded volume due to the finite size of gas molecules.

Pressure Correction

Gas particles experience intermolecular attractions, which reduce the force of collisions with the container walls, lowering the observed pressure. The van der Waals equation adds a correction term:

• Corrected Pressure: $P + a\left(\frac{n}{V}\right)^2$, where a is the van der Waals constant for intermolecular forces.

• Physical Meaning: a corrects for the reduction in pressure due to attractive forces between particles.

The van der Waals Equation

The complete van der Waals equation for real gases is:

$\left(P + a\left(\frac{n}{V}\right)^2\right)(V - nb) = nRT$

• a: Measures the strength of intermolecular attractions (IMF).

• b: Measures the effective size of gas particles.

When Do Gases Deviate from Ideal Behavior?

High Pressure

• At high pressures, gas particles are closer together, making the volume of the particles significant compared to the container volume.

• Volume correction ($b$) becomes important.

Low Temperature

• At low temperatures, intermolecular attractions become more significant as particles move more slowly and are more likely to interact.

• Pressure correction ($a$) becomes important.

Summary Table: Conditions Favoring Deviations

Condition	Deviation	Correction Needed
High Pressure	Particle volume significant	Volume correction ($b$)
Low Temperature	Intermolecular forces significant	Pressure correction ($a$)

van der Waals Constants for Common Gases

The values of a and b differ for each gas, reflecting differences in intermolecular forces and particle size. The table below lists these constants for several gases:

Substance	a (L2·atm/mol2)	b (L/mol)
He	0.034	0.0237
Ne	0.211	0.0171
Ar	1.34	0.0322
H2	0.244	0.0266
N2	1.39	0.0391
O2	1.36	0.0318
CO2	3.59	0.0427
CH4	2.25	0.0428
H2O	5.46	0.0305
CCl4	20.4	0.1383

Additional info: Table values inferred from the provided image and standard references.

Examples and Applications

Comparing Ideal and Real Gas Pressures

• For a sample of CO2 at 1.00 mol, 1.00 L, and 273 K:

• Ideal Gas Pressure: $P = \frac{nRT}{V} = 7.470$ atm

• Real Gas Pressure (van der Waals): $P = \frac{nRT}{V - nb} - a\left(\frac{n}{V}\right)^2 = 7.189$ atm

• The real gas pressure is lower due to intermolecular attractions.

Which Gases Behave Most Ideally?

• Most Ideal: Gases with small, nonpolar molecules and weak intermolecular forces (e.g., H2).

• Most Real: Gases with strong intermolecular forces or large molecular size (e.g., H2O).

Summary

• The Ideal Gas Law is an approximation; real gases deviate under high pressure and low temperature.

• The van der Waals equation introduces corrections for particle volume and intermolecular forces.

• Understanding these deviations is essential for accurate predictions of gas behavior in real-world conditions.