Gas Laws and the Kinetic Molecular Theory

Properties of Gases

• Concentration and Pressure: Higher gas concentration = greater pressure.
• Container Volume: Small volume + lots of gas = high pressure.
• Particle Speed and Temperature: Average speed of gas particles $\propto$ gas temperature ($KE \propto T$).
• Rigidity of Containers: Rigid containers resist expansion (e.g., glass).
• Free Space: Gases mostly consist of empty space; molecules are widely dispersed.
• Diffusion and Expansion: Gases diffuse rapidly and expand indefinitely until evenly distributed.

Intensive Variables for Gases

These do not depend on the amount of gas:

• Pressure ($P$)
• Volume ($V$)
• Temperature ($T$)
• Number of Moles ($n$)

The Ideal Gas Law

• Relates variables: $\mathbf{PV = nRT}$.
• $R$ is the universal gas constant.

Assumptions of the Kinetic Molecular Theory (KMT) for Ideal Gases

• No Volume: Gas particles have negligible volume ($V_{\text{particle}} \to 0$).
• Elastic Collisions: Collisions lose no kinetic energy (particles never slow down or interact).
• Kinetic Energy and Temperature: Average kinetic energy $\propto$ absolute temperature ($KE_{\text{avg}} \propto T$).

Pressure Measurement and Units

• Mercury Barometer: Measured atmospheric pressure; $1 \text{ atm} = 760 \text{ mmHg}$ (at sea level).
• Units: mmHg, atm, Pascals ($1 \text{ Pa} = 1 \text{ N/m}^2$).
• Water Pressure: $30 \text{ ft}$ of water $\approx 1 \text{ atm}$.

Gas Laws Derived from the Ideal Gas Law

(When specific variables are held constant)

• Boyle's Law (Pressure-Volume):

  • Constant $T, n$
  • $P \propto \frac{1}{V}$ or $P<em>1V</em>1 = P<em>2V</em>2$
  • Application: Scuba diver's lung volume changes inversely with depth/pressure.

• Charles's Law (Volume-Temperature):

  • Constant $P, n$
  • $V \propto T$ or $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$
  • Application: Hot air balloons rise because heated air expands and becomes less dense. Balloons shrink in cold water.
  • Absolute Zero: Theoretical volume of ideal gas is zero at $0 \text{ K}$ ($-273.15^{\circ}C$).

• Avogadro's Law (Volume-Moles):

  • Constant $P, T$
  • $V \propto n$ or $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$
  • Example: Double the gas amount, double the volume (at constant P, T).

• Gay-Lussac's Law (Pressure-Temperature):

  • Constant $V, n$
  • $P \propto T$ or $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2}$
  • Application: Bicycle pump heats up due to increased pressure and temperature. Expanding gas from a tank cools as pressure drops.
  • Boiling Point: Water boils at lower temperatures at higher elevations due to lower atmospheric pressure.

Ideal Gas Law Calculations and Standard Molar Volume

• Calculations: Rearrange $PV=nRT$ as needed (e.g., $n = \frac{PV}{RT}$).
• Standard Temperature and Pressure (STP):
  • Temperature: $0^{\circ}C$ ($273.15 \text{ K}$)
  • Pressure: $1 \text{ atm}$
• Molar Volume at STP: One mole of any ideal gas occupies $22.4 \text{ L}$ at STP ($V = 22.4 \text{ L/mol}$).

Gas Density

• Density ($\rho$) = Mass ($m$) / Volume ($V$).
• From Ideal Gas Law: $\rho = \frac{PM}{RT}$, where $M$ is molecular weight.
• Example: Nitrogen ($M \approx 28 \text{ g/mol}$) is denser than Helium ($M \approx 4 \text{ g/mol}$) at the same P, V, T.

Mixtures of Gases and Dalton's Law of Partial Pressures

• Atmosphere: Mixture of N, O, Ar, trace gases.
• Dalton's Law: Total pressure ($P<em>{\text{total}}$) of a gas mixture is the sum of partial pressures ($P</em>i$) of individual gases: $\mathbf{P<em>{\text{total}} = P</em>A + P<em>B + P</em>C + \text{…}}$.
• Partial Pressure: Pressure an individual gas would exert if it filled the container alone ($P<em>i = \frac{n</em>i RT}{V}$).
• Example: $H<em>2O</em>2$ decomposition produces $H_2O$ and $$O