Gases

Properties of Gases

Gases assume the shape and volume of their container, are compressible, have lower densities compared to liquids and solids, and form homogeneous mixtures in any proportion.

Pressure is defined as force per unit area: $pressure = \frac{force}{area}$

SI unit of force is the Newton (N): $1 N = 1 kg \cdot \frac{m}{s^2}$

SI unit of pressure is the Pascal (Pa): $1 Pa = 1 \frac{N}{m^2}$

Units of Pressure

• Standard Atmosphere (atm): 1 atm = 101,325 Pa
• mmHg: 1 mmHg = 133.322 Pa
• torr: 1 torr = 133.322 Pa
• bar: 1 bar = 1 x $10^5$ Pa

$1 atm = 101,325 Pa = 760 mmHg = 760 torr = 1.01325 bar = 14.7 psi$

Gas Laws

• Gay-Lussac's Law: The pressure of a gas at constant volume is directly proportional to temperature: $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2}$
• Boyle's Law: The pressure of a fixed amount of gas at constant temperature is inversely proportional to the volume: $P<em>1V</em>1 = P<em>2V</em>2$
• Charles' Law: The volume of a gas at constant pressure is directly proportional to the absolute temperature: $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$
• Avogadro's Law: The volume of a gas is directly proportional to the number of moles at constant temperature and pressure: $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$
• 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

The ideal gas equation is: $PV = nRT$, where R is the gas constant.

• R = 0.08206 L⋅atm/(mol⋅K)
• R = 8.314 J/(mol⋅K)
• R = 0.08314 L⋅bar/(mol⋅K)
• R = 1.987 cal/(mol⋅K)

Standard Temperature and Pressure (STP): Pressure is 1 atm, and Temperature is 0 °C (273.15 K).

Applications of the Ideal Gas Equation

• $d = \frac{P \cdot MM}{R \cdot T}$, where d is density and MM is molar mass.

Gas Mixtures: Dalton’s Law of Partial Pressures

The total pressure of a gas mixture is the sum of the partial pressures of each component: $P<em>{total} = \Sigma P</em>i$

Mole Fractions

$\chi<em>i = \frac{n</em>i}{n<em>{total}}$, where $\chi</em>i$ is the mole fraction of component i.

$\chi<em>i = \frac{P</em>i}{P_{total}}$

Reactions with Gaseous Reactants and Products

Use stoichiometry and the ideal gas equation to calculate volumes of gaseous reactants or products.

Collecting Gas over Water

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

Kinetic Molecular Theory

1. Gases are composed of particles separated by large distances.
2. Gas molecules are in constant, random motion and undergo elastic collisions.
3. Gas molecules do not exert attractive or repulsive forces on one another.
4. The average kinetic energy of gas molecules is proportional to the absolute temperature: $E_k \propto T$

Root-Mean-Square (rms) Speed

$u_{rms} = \sqrt{\frac{3RT}{M}}$, where R is the gas constant, T is temperature, and M is molar mass.

$\frac{u<em>{rms1}}{u</em>{rms2}} = \sqrt{\frac{M<em>2}{M</em>1}}$

Diffusion and Effusion

Graham’s Law: $Rate \propto \frac{1}{\sqrt{M}}$

Real Gases: Deviation from Ideal Behavior

At high pressure and low temperatures, real gases deviate from ideal behavior due to molecular volume and intermolecular forces.

Van der Waals Equation

$\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$