Gases

Properties of Gases

• Gases assume the shape and volume of their container.
• Gases are compressible.
• Gases have variable densities much smaller than liquids and solids.
• Gases form homogeneous mixtures in any proportion.
• Pressure is force per unit area: $\text{pressure} = \frac{\text{force}}{\text{area}}$
• SI unit of force is the newton (N): $1 \text{ N} = 1 \text{ kg} \cdot \frac{\text{m}}{\text{s}^2}$
• SI unit of pressure is the pascal (Pa): $1 \text{ Pa} = 1 \frac{\text{N}}{\text{m}^2}$

Units of Pressure

• Standard Atmosphere (atm): $1 \text{ atm} = 101,325 \text{ Pa}$
• mmHg: $1 \text{ mmHg} = 133.322 \text{ Pa}$
• torr: $1 \text{ torr} = 133.322 \text{ Pa}$
• bar: $1 \text{ bar} = 1 \times 10^5 \text{ Pa}$
• $1 \text{ atm} = 101,325 \text{ Pa} = 760 \text{ mm Hg} = 760 \text{ torr} = 1.01325 \text{ bar} = 14.7 \text{ psi}$

Gas Laws

• Gay-Lussac's Law: $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2}$ (constant volume)
• Boyle's Law: $P<em>1V</em>1 = P<em>2V</em>2$ (constant temperature)
• Charles's Law: $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$ (constant pressure)
• Avogadro's Law: $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$ (constant temperature and pressure)
• Combined Gas Law: $\frac{P<em>1V</em>1}{n<em>1T</em>1} = \frac{P<em>2V</em>2}{n<em>2T</em>2}$

Ideal Gas Equation

• $PV = nRT$
• R (gas constant) values: 0.08206 L⋅atm/mol⋅K, 8.314 J/mol⋅K, 0.08314 L⋅bar/mol⋅K, 1.987 cal/mol⋅K
• STP Conditions: 1 atm and 0 °C (273.15 K)

Applications of Ideal Gas Equation

• Density: $d = \frac{P \cdot MM}{R \cdot T}$

Gas Mixtures: Dalton’s Law of Partial Pressures

• $P<em>{\text{total}} = \Sigma P</em>i$

Mole Fractions

• $\chi<em>i = \frac{n</em>i}{n<em>{\text{total}}}$ and $\chi</em>i = \frac{P<em>i}{P</em>{\text{total}}}$

Reactions with Gaseous Reactants and Products

• Use stoichiometry and the ideal gas equation to find required volumes.

Collecting Gas Over Water

• $P<em>{\text{total}} = P</em>{\text{collected gas}} + P<em>{H</em>2O}$

Kinetic Molecular Theory

• Gas particles are separated by large distances and have negligible volume.
• Gas molecules are in constant, random motion with elastic collisions.
• Gas molecules do not exert attractive or repulsive forces on each other.
• Average kinetic energy is proportional to absolute temperature: $E_k \propto T$

Molecular Speed

• Root-mean-square speed: $u_{\text{rms}} = \sqrt{\frac{3RT}{M}}$

Diffusion and Effusion

• Graham's Law: $\text{Rate} \propto \frac{1}{\sqrt{M}}$

Real Gases: Deviations from Ideal Behavior

• High pressure and low temperatures cause deviations.
• Van der Waals Equation: $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$