Gases: Kinetic Molecular Theory, Effusion & Diffusion, and Real Gas Behavior

Chapter 5: Gases

Overview

This study guide covers the properties and behavior of gases, focusing on the kinetic molecular theory, molecular velocities, effusion and diffusion, and deviations from ideal gas behavior. These concepts are essential for understanding the physical behavior of gases in chemical systems.

Kinetic Molecular Theory (KMT)

Fundamental Assumptions

• Gas molecules occupy negligible space compared to the overall volume of the container.

• Intermolecular forces are negligible under most conditions.

• Gas molecules are in constant, random motion and collisions are perfectly elastic.

Note: These assumptions are not always accurate, especially under high pressure or low temperature.

Average Kinetic Energy

According to KMT, the average kinetic energy of a gas at a given temperature is the same for all gases and is independent of the identity of the gas.

• At a given temperature (e.g., 273.15 K), helium and argon gases have the same average kinetic energy.

• However, their molecular masses differ, so their velocities differ.

Root Mean Square (RMS) Velocity

Definition and Formula

The root mean square velocity ($u_{rms}$) is the speed a molecule, atom, or ion would have if it possessed the average kinetic energy.

• Formula: $u_{rms} = \sqrt{\frac{3RT}{M}}$ Where: $R$ = 8.3145 J K-1 mol-1 (gas constant) $T$ = temperature in Kelvin $M$ = molar mass in kg mol-1

• Example: For N2 at 25°C ($T = 298$ K, $M = 0.028$ kg mol-1): $u_{rms} = 515$ m s-1

Distribution of Molecular Velocities

Not all molecules move at the same speed. The distribution depends on temperature and molar mass.

• Higher temperature increases the spread and average of molecular velocities.

• Lighter molecules (e.g., H2, He) move faster than heavier ones (e.g., H2O, Ar).

Effusion and Diffusion

Effusion

Effusion is the process by which gas molecules escape through a small aperture into a vacuum.

• The rate of effusion is directly proportional to the RMS velocity of the gas molecules.

• Faster-moving (lighter) molecules effuse more rapidly.

Diffusion

Diffusion is the mixing of one substance through another, such as the spread of perfume through air.

• Occurs by random molecular motion and collisions.

• Despite high molecular speeds, diffusion is slow due to frequent collisions (mean free path).

Mean Free Path

The mean free path is the average distance a molecule travels before colliding with another molecule.

• Higher gas density results in a shorter mean free path.

• Collisions slow the net movement of molecules across a room.

Graham's Law of Effusion

Graham's Law relates the rates of effusion (or diffusion) of two gases to their molar masses:

• Formula: $\frac{\text{rate of effusion for gas 1}}{\text{rate of effusion for gas 2}} = \sqrt{\frac{M_2}{M_1}}$ Where $M_1$ and $M_2$ are the molar masses of gases 1 and 2.

• Example: Calculate the ratio of effusion rates for helium (He, $M_1 = 4.003$ g mol-1) and methane (CH4, $M_2 = 16.04$ g mol-1): $\frac{\text{Rate He}}{\text{Rate CH}_4} = \sqrt{\frac{16.04}{4.003}} = 2.002$

Ideal Gas Law and Deviations

Ideal Gas Law

The ideal gas law relates pressure, volume, temperature, and amount of gas:

• Formula: $PV = nRT$

• For one mole of an ideal gas: $\frac{PV}{RT} = n$ (should equal 1 for 1 mol)

Deviations from Ideal Behavior

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

• At high pressure, the volume is higher than predicted due to finite molecular size.

• At low temperature, pressure is lower than predicted due to intermolecular attractions.

• Assumptions of negligible volume and no intermolecular forces become inaccurate.

Van der Waals Equation

The van der Waals equation introduces corrections for real gases:

• Formula: $\left[P + a\left(\frac{n}{V}\right)^2\right]\left[V - nb\right] = nRT$ Where: $a$ = correction for intermolecular attractions (L2 bar mol-2) $b$ = correction for finite molecular volume (L mol-1)

• Each gas has specific $a$ and $b$ values.

Van der Waals Constants Table

The following table lists $a$ and $b$ constants for selected gases:

Gas	a (L2 bar mol-2)	b (L mol-1)
He	0.0341	0.0237
Ne	0.213	0.0171
Ar	1.345	0.0320
Xe	4.250	0.0510
H2	0.244	0.0266
O2	1.360	0.0318
N2	1.390	0.0391
CO2	3.640	0.0427
NH3	4.170	0.0371
CH4	2.253	0.0428
H2O	5.536	0.0305
CCl4	19.75	0.1383

Summary Table: Ideal vs. Real Gas Behavior

Condition	Ideal Gas Prediction	Real Gas Observation
High Pressure	Volume matches prediction	Volume higher than predicted
Low Temperature	Pressure matches prediction	Pressure lower than predicted
Low Density	Assumptions accurate	Assumptions hold
High Density	Assumptions fail	Deviations significant

Key Takeaways

• Kinetic molecular theory explains gas behavior and the basis of the ideal gas law.

• Root mean square velocity depends on temperature and molar mass.

• Effusion and diffusion rates are governed by molecular velocities and Graham's Law.

• Real gases deviate from ideal behavior due to molecular size and intermolecular forces, especially at high pressure and low temperature.

• Van der Waals equation provides corrections for real gas behavior.

Additional info: The notes omit details on manometers and derivation of the ideal gas law, as per the syllabus guidance.