Unit 3

CH301 Unit 3: Gases, States of Matter, and Intermolecular Forces

Eponymous Gas Laws – Relationships and Calculations

• 1. Boyle’s Law

  • Formula: $P<em>1V</em>1 = P<em>2V</em>2$

  • This law describes the inverse relationship between pressure (P) and volume (V) of a gas at constant temperature.

• 2. Avogadro’s Law

  • Formula: $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$

  • States that at constant temperature and pressure, equal volumes of gas contain equal number of moles (n).

• 3. Charles’ Law

  • Formula: $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$

  • Demonstrates the direct relationship between volume (V) and temperature (T) of a gas at constant pressure.

• 4. Gay-Lussac’s Law

  • Formula: $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2}$

  • Shows the direct relationship between pressure (P) and temperature (T) of a gas at constant volume. Also fits into stoichiometry of reactions with gases, allowing for volume ratios.

Combined Gas Law – Calculation

• Combined Gas Law Formula: $\frac{P<em>1V</em>1}{T<em>1} = \frac{P</em>2V<em>2}{T</em>2}$

• Example Problem:

  • Given:

  • Initial volume (V1) = 40 L

  • Initial temperature (T1) = 300 K

  • Initial pressure (P1) = 1200 torr

  • Final pressure (P2) = 800 torr

  • Final temperature (T2) = 150 K

  • Rearranging for $V<em>2$: $V</em>2 = \frac{P<em>1V</em>1T<em>2}{T</em>1P_2} = \frac{(1200)(40)(150)}{(300)(800)}$

  • Detailed calculation:

  • Simplifying:
    $\frac{1200}{800} = 1.5$
    $\frac{150}{300} = 0.5$

  • Result: $V_2 = 30 L$

Ideal Gas Law: Molecular Weight and Density – Derivation

• Ideal Gas Law Formula: $PV = nRT$

  • Deriving Molecular Weight (MW):

  • $MW = \frac{mass}{moles} = \frac{m}{n}$

  • From the gas law: $P V = \frac{m}{MW} RT$

  • Rearranging gives: $MW = \frac{mRT}{PV}$

  • Density (ρ):

  • $\rho = \frac{mass}{volume} = \frac{m}{V}$

  • Thus: $PV = \frac{m}{\rho}RT$

  • Which gives: $\rho = \frac{PMW}{RT}$

  • Key Relationships:

  • If pressure (P) increases, density (ρ) increases.

  • If molecular weight (MW) increases, density (ρ) increases.

  • If temperature (T) increases, density (ρ) decreases.

Dalton’s Law of Partial Pressures – Calculation

• Main Idea: Calculate the partial pressures of gases in a mixture based on mole fraction.

  • Example: Given a system with gases 3 moles Ne, 9 moles O2, 6 moles Ar and total pressure = 30 atm:

  • Calculate mole fractions:

  • Total moles = 3 + 9 + 6 = 18 moles

  • $P_{Ne} = \frac{3}{18} \times 30 = 5 \text{ atm}$

  • $P_{O2} = \frac{9}{18} \times 30 = 15 \text{ atm}$

  • $P_{Ar} = \frac{6}{18} \times 30 = 10 \text{ atm}$

Balancing a Combustion Reaction

• General Form of Combustion Reactions:

  • Hydrocarbon + Oxygen $\rightarrow$ Carbon Dioxide + Water

  • Simplest combustion reaction:

  • $H<em>2 + O</em>2 \rightarrow 2 H_2O$

  • Example for Methane (CH4):

  • $CH<em>4 + 2 O</em>2 \rightarrow CO<em>2 + 2 H</em>2O$

Gas Reaction Stoichiometry

• Calculating Volumes at STP:

  • Ideal Gas Volume: 22.4 L = 1 mol at STP.

  • Example: 92 g of ethanol (MW = 46 g/mol) reacts with 96 g of O2 at STP.

Kinetic Molecular Theory (KMT)

• Five Assumptions of Kinetic Molecular Theory:

  1. Gases are composed of particles in continuous random motion.

  2. The volume of gas particles is negligible compared to the volume occupied by gas.

  3. Gas particles collide elastically, with no intermolecular forces present.

  4. Average kinetic energy (KE) is proportional to temperature (T): $KE \propto T$

     • $KE = \frac{1}{2}mv^2$

  5. The relation between kinetic energy and speed: $KE = \frac{1}{2}kT$

Maxwell-Boltzmann Distributions of Gases

• Maxwell Distribution:

  • At constant temperature, gas particles exhibit a range of velocities.

  • High temperature = broader distribution of velocities.

  • Low temperature = narrower distribution of velocities.

Graham's Law: Diffusion, Effusion, and Gas Velocities

• Graham's Law formula: $\frac{v<em>1}{v</em>2} = \sqrt{\frac{MW<em>2}{MW</em>1}}$

• Example Calculation:

  • If dealing with Ar (MW=40 g/mol) and Br2 (MW=160 g/mol):

  • $\frac{v<em>{Ar}}{v</em>{Br2}} = \sqrt{\frac{MW<em>{Br2}}{MW</em>{Ar}}} = \sqrt{\frac{160}{40}} = 2$

Gas Non-Ideality – Conditions and van der Waals Equation

• Non-Ideal Gas Behavior:

  • Gases behave ideally at low pressures, large volumes, small n, and high temperatures.

  • van der Waals equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$

  • a accounts for intermolecular attractions.

  • b accounts for particle volume.

Distinguishing Intra- and Intermolecular Forces

• Intermolecular Forces: Weaker forces between molecules (e.g., dipole-dipole, dispersion).

• Intramolecular Forces: Stronger forces within molecules (e.g., covalent bonds).

Physical Properties of Liquids

• Boiling Point: The temperature at which vapor pressure equals atmospheric pressure.

• Surface Tension: Inward forces reducing surface area of a liquid.

• Capillary Action: Liquid climbing up the walls of a container due to cohesive forces.

• Viscosity: Resistance to flow due to attractives forces within the liquid.

Types of Solids and their Properties

• Metallic Solids: Conductive solids made up of metals (e.g., Fe, Cu).

• Ionic Solids: Composed of ions, usually have high melting points (e.g., NaCl).

• Covalent Network Solids: Atoms held together by covalent bonds (e.g., diamond).

Melting Points of Solids – Ranking

• Ranking Method: Higher charge density leads to higher melting points.

  • Example: LiCl < NaCl < KCl <MgCl2