Chapter 5.6: Gases – Properties, Laws, and Real Gas Behavior

Gases: Properties and Behavior

Gas Molecules in Movement

Gas molecules are in constant, random motion, and their behavior can be described statistically. The mean free path is the average distance a molecule travels between collisions.

• Mean Free Path: Increases as pressure decreases, since molecules are further apart.

• Collision Frequency: The number of collisions per unit time, given by:

$\text{Collision frequency} = \frac{u_{rms}}{\text{mean free path}}$

• Diffusion: Movement of molecules from high to low concentration. Rate of diffusion is proportional to the root mean square speed ($u_{rms}$).

• Effusion: Escape of gas molecules through a small hole into a vacuum. Rate of effusion is also proportional to $u_{rms}$.

Example: If pressure is reduced, the mean free path increases and molecules collide less frequently.

Graham’s Law of Effusion

Effusion Rate and Molar Mass

Graham’s Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass.

• Root Mean Square Speed: $u_{rms} = \sqrt{\frac{3RT}{M}}$

• Rate of Effusion: $\text{Rate} \propto u_{rms} \propto \sqrt{\frac{1}{M}}$

• Comparing Two Gases:

$\frac{\text{Rate}_A}{\text{Rate}_B} = \sqrt{\frac{M_B}{M_A}}$

• Application: Used to determine the molar mass of an unknown gas by comparing its effusion rate to a known gas.

Example: If a gas effuses at 0.342 times the rate of helium ($M_{He} = 4.00\ g\cdot mol^{-1}$), its molar mass is:

$M_{unknown} = \frac{4.00}{(0.342)^2} = 34.2\ g\cdot mol^{-1}$

Diffusion and Effusion

Definitions and Comparisons

Diffusion and effusion describe the movement of gas molecules, but differ in context.

• Diffusion: Mixing of gases due to random molecular motion.

• Effusion: Passage of gas through a tiny orifice into a vacuum.

• Rate of Diffusion/Effusion: Lighter molecules move and spread faster than heavier ones.

Example Table: Diffusion Rates by Molar Mass

Gas	Molar Mass (g/mol)	Relative Rate
He	4.00	Fastest
N2	28.01	Faster
CO2	46.00	Moderate
Cl2	70.91	Slower
I2	222	Slowest

• Key Point: The lower the molar mass, the higher the rate of diffusion/effusion.

Example: If a gas effuses twice as fast as another with $M_B = 131\ g\cdot mol^{-1}$, then $M_A = \frac{1}{4} M_B = 32.75\ g\cdot mol^{-1}$.

The Ideal Gas Law

Fundamental Equation

The Ideal Gas Law relates pressure, volume, temperature, and amount of gas:

$PV = nRT$

• P: Pressure (atm, bar, etc.)

• V: Volume (L)

• n: Moles of gas

• R: Universal gas constant

• T: Temperature (K)

Example: Calculate the pressure exerted by 1 mole of an ideal gas at 273 K in a 22.4 L container.

What Makes a Gas “Ideal”

Assumptions of the Ideal Gas Model

An ideal gas is a theoretical construct based on two main assumptions:

• Negligible Molecular Volume: The volume of individual gas molecules is much smaller than the volume of the container.

• No Intermolecular Forces: Gas molecules interact only through perfectly elastic collisions; no attractive or repulsive forces exist.

If either assumption fails, the gas exhibits real gas behavior.

Deviation from Ideality

Real Gas Behavior

Real gases deviate from ideal behavior under certain conditions, especially at high pressures and low temperatures.

• Observed Molar Volume: Real gases may have molar volumes different from the ideal value (22.4 L at STP).

• Graphical Comparison: Molar volume of real gases (e.g., CO2, NH3, N2, He, H2) can be slightly higher or lower than the ideal gas prediction.

Example Table: Molar Volumes at STP

Gas	Molar Volume (L)
Ideal Gas	22.70
CO2	22.34
NH3	22.59
N2	22.68
He	22.71
H2	22.71

Molecular Volume

Effect of High Pressure

At normal pressure, the free volume (space between molecules) is nearly equal to the container volume. At high pressure, the volume occupied by molecules becomes significant.

• Real Gas Volume: At high pressure, the volume occupied by a mole of gas is larger than predicted by the ideal gas law.

$\frac{PV}{RT} > 1$

Example: Argon at high pressure shows a larger molar volume than predicted by the ideal gas law.

Intermolecular Forces

Attractive Forces and Pressure

Real gases experience attractive forces between molecules, which affect their behavior.

• Non-Elastic Collisions: Collisions are not perfectly elastic due to intermolecular attractions.

• Effect on Pressure: Attractive forces reduce the pressure exerted by the gas compared to an ideal gas.

$\frac{PV}{RT} < 1$

Example: At low temperature and high pressure, the pressure measured is lower than predicted by the ideal gas law.

Formula for Kinetic Energy:

$KE = \frac{mu^2}{2}$

Van der Waals Equation: Correction to the Ideal Gas Law

Accounting for Real Gas Effects

The Van der Waals equation introduces correction factors for molecular volume and intermolecular forces:

• Correction for Volume: $V_{real} = \frac{nRT}{P} + nb$

• Correction for Pressure: $P_{real} = \frac{nRT}{V - nb} - a\left(\frac{n}{V}\right)^2$

• a and b: Constants specific to each gas, accounting for intermolecular attraction (a) and molecular volume (b).

Example: For 10.0 moles of a real gas ($a = 6.00\ \text{bar} \cdot \text{L}^2 \cdot \text{mol}^{-2}$, $b = 0.0500\ \text{L} \cdot \text{mol}^{-1}$) in a 1.00 L container at 500 bar, the temperature can be calculated using the Van der Waals equation.

Summary Table: Ideal vs. Real Gas Behavior

Condition	Ideal Gas	Real Gas
Low Pressure	Behaves ideally	Small deviations
High Pressure	Assumptions fail	Significant deviations
High Temperature	Behaves ideally	Small deviations
Low Temperature	Assumptions fail	Significant deviations

Additional info: Real gases approach ideal behavior at low pressure and high temperature, where molecular volume and intermolecular forces are negligible.