Gases: Their Properties and Behavior – Chapter 10 Study Notes

Gases: Their Properties and Behavior

Gases and Gas Pressure

Gases are a fundamental state of matter characterized by their ability to fill any container, mix homogeneously, and be easily compressed. The behavior of gases is explained by the movement and interactions of their particles.

• Homogeneous and Compressible: Gas mixtures are uniform throughout and can be compressed due to the large amount of empty space between particles.

• Gas Pressure: Pressure is the force exerted per unit area by gas particles colliding with the walls of a container. Formula: $P = \frac{F}{A}$

• Barometer: An instrument used to measure atmospheric pressure, typically using a column of mercury. Atmospheric pressure pushes mercury up the column to a height proportional to the pressure.

• Units of Pressure:

  • Pascal (Pa)

  • Torr

  • Millimeter of mercury (mmHg)

  • Atmosphere (atm)

  • Bar

• Conversions:

  • 1 atm = 760 mm Hg = 760 torr

  • 1 bar = $1 \times 10^5$ Pa

  • 1 atm = 101,325 Pa

• Manometer: Device used to measure the pressure of a gas in a container relative to atmospheric pressure.

The Gas Laws

The physical properties of gases are described by four variables: pressure (P), temperature (T), volume (V), and number of moles (n). Several laws relate these variables under specific conditions.

• Boyle's Law: At constant temperature and moles, the volume of a gas is inversely proportional to its pressure. $V \propto \frac{1}{P}$ $PV = k$ $P_{\text{initial}} V_{\text{initial}} = P_{\text{final}} V_{\text{final}}$ Example: Doubling the pressure halves the volume.

• Charles's Law: At constant pressure and moles, the volume of a gas is directly proportional to its absolute temperature. $V \propto T$ $\frac{V}{T} = k$ $\frac{V_{\text{initial}}}{T_{\text{initial}}} = \frac{V_{\text{final}}}{T_{\text{final}}}$ Example: Doubling the temperature doubles the volume.

• Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles. $V \propto n$ $\frac{V}{n} = k$ $\frac{V_{\text{initial}}}{n_{\text{initial}}} = \frac{V_{\text{final}}}{n_{\text{final}}}$ Example: Doubling the moles doubles the volume.

The Ideal Gas Law

The ideal gas law combines the relationships described by Boyle's, Charles's, and Avogadro's laws into a single equation.

• Equation: $PV = nRT$

• Gas Constant (R): $R = 0.08206 \frac{L \cdot atm}{K \cdot mol}$

• Standard Temperature and Pressure (STP):

  • $T = 0^\circ C = 273.15\ K$

  • $P = 1\ atm$

• Molar Volume at STP: 1 mol of an ideal gas occupies 22.41 L at STP.

• Example Calculation: $V = \frac{nRT}{P}$ For 1 mol at STP: $V = \frac{(1\ mol)(0.08206\ L\ atm / K\ mol)(273.15\ K)}{1\ atm} = 22.41\ L$

Density of Gases

Gas density is the ratio of mass to volume and is typically expressed in g/L.

• Formula: $\text{Density} = \frac{\text{molar mass}}{\text{molar volume}}$

• At STP, the molar volume is 22.4 L.

• Density is directly proportional to molar mass.

• Alternative Formula: $d = \frac{PM}{RT}$

Stoichiometric Relationships with Gases

Gas laws can be used to relate the quantities of reactants and products in chemical reactions involving gases.

• Example: Decomposition of sodium azide in airbags: $2\text{NaN}_3 (s) \rightarrow 2\text{Na}(s) + 3\text{N}_2 (g)$ Steps:

  1. Calculate moles of $\text{N}_2$ produced from mass of $\text{NaN}_3$.

  2. Use the ideal gas law to find the volume of $\text{N}_2$ at given conditions.

Mixtures of Gases: Partial Pressure and Dalton’s Law

In a mixture of gases, each gas exerts a pressure independently of the others. The total pressure is the sum of the partial pressures.

• Dalton's Law of Partial Pressures: $P_{\text{total}} = P_1 + P_2 + \cdots + P_N$

• Partial Pressure: $P_n = n_n \frac{RT}{V}$

• Mole Fraction: $x_i = \frac{n_i}{n_{\text{total}}}$ or $x_i = \frac{P_i}{P_{\text{total}}}$

The Kinetic-Molecular Theory of Gases

This theory explains the macroscopic properties of gases by considering their molecular composition and motion.

• Gases consist of tiny particles (atoms or molecules) moving randomly.

• The volume of the particles is negligible compared to the total volume.

• Particles act independently; no attractive or repulsive forces.

• Collisions are elastic, meaning kinetic energy is conserved.

• Average kinetic energy is proportional to temperature (Kelvin).

Temperature and Molecular Velocities

The speed and kinetic energy of gas molecules depend on temperature and mass.

• Kinetic Energy of a Particle: $KE = mv^2$

• Average Kinetic Energy: $KE_{\text{avg}} = \frac{3RT}{2N_A}$

• Root Mean Square Speed: $u_{rms} = \sqrt{\frac{3RT}{M}}$ where $M$ is molar mass in kg/mol.

• Lighter particles move faster at the same temperature.

Mean Free Path

The mean free path is the average distance a molecule travels between collisions.

• Decreases as pressure increases.

Gas Diffusion and Effusion: Graham's Law

Diffusion is the mixing of gases; effusion is the escape of gas through a small hole.

• Graham's Law: The rate of effusion is inversely proportional to the square root of the molar mass. $\text{Rate} \propto \frac{1}{\sqrt{M}}$ $\frac{\text{rate}_A}{\text{rate}_B} = \sqrt{\frac{M_B}{M_A}}$

The Behavior of Real Gases

Real gases deviate from ideal behavior at high pressures and low temperatures due to intermolecular forces and the finite volume of particles.

• At high pressure, the volume occupied by gas particles becomes significant.

• Attractive forces between particles become important.

• Van der Waals Equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$ where $a$ and $b$ are constants that correct for intermolecular attractions and molecular volume, respectively.

Tables

Molar Volumes of Some Real Gases at 0°C and 1 atm

Gas	Molar Volume (L)
H2	22.43
He	22.41
NH3	22.40
N2	22.40
F2	22.38
Ar	22.09
CO2	22.06
Cl2	22.06

Average Speeds (m/s) of Some Gas Molecules at 25°C

Gas	Average Speed (m/s)
H2	1960
He	1360
H2O	650
N2	520
O2	490
CO2	415

Additional info: These notes expand on the original slides by providing full academic context, definitions, and examples for each law and concept, ensuring a self-contained study guide for exam preparation.