Study Notes: Gases (General Chemistry, Chapter 5)

Gases: Properties, Laws, and Theories

Pressure: The Result of Molecular Collisions

Definition and Molecular Perspective

• Pressure is defined as the force exerted per unit area by gas molecules as they collide with the surfaces of their container.

• In the gaseous state, molecules are far apart and move randomly in all directions, filling the container.

• Collisions of gas molecules with the container walls create pressure.

• The SI unit of pressure is the Pascal (Pa), defined as one newton per square meter.

Formula:

$P = \frac{F}{A}$

• Where P is pressure, F is force, and A is area.

• Pressure increases with greater force or smaller area.

• Heating a gas increases the kinetic energy of molecules, leading to more frequent and forceful collisions, thus increasing pressure.

Units of Pressure

Pressure can be measured in several units. The most common are:

Unit Name	Unit	Standard Value
bar	bar	1 bar
Pascal	Pa	100,000 Pa
Torr (mmHg)	Torr (mmHg)	750.06 Torr
Atmosphere	atm	0.98692 atm

Additional info: 1 atm is often approximated as 101,325 Pa in other sources.

The Gas Laws

Boyle's Law

Boyle's Law describes the inverse relationship between the pressure and volume of a gas at constant temperature and number of moles.

• As volume decreases, pressure increases, and vice versa, provided temperature and amount of gas remain constant.

Formula:

$P_1 V_1 = P_2 V_2$

• Where P is pressure and V is volume.

Example: Compressing a gas from 2.0 L to 1.0 L at constant temperature doubles the pressure.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its absolute temperature at constant pressure and number of moles.

• As temperature increases, volume increases (if pressure is constant).

Formula:

$\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Where V is volume and T is temperature in Kelvin.

Example: Heating a gas in a balloon causes it to expand.

Gay-Lussac's Law

Gay-Lussac's Law states that the pressure of a gas is directly proportional to its absolute temperature at constant volume and number of moles.

• As temperature increases, pressure increases (if volume is constant).

Formula:

$\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Avogadro's Law

Avogadro's Law states that the volume of a gas is directly proportional to the number of moles at constant temperature and pressure.

• Adding more gas (increasing moles) increases the volume.

Formula:

$\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• Where n is the number of moles.

Combined Gas Law

The combined gas law incorporates Boyle's, Charles's, and Gay-Lussac's laws:

$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$

The Ideal Gas Law

Equation and Gas Constant

The ideal gas law relates pressure, volume, temperature, and number of moles for an ideal gas:

$PV = nRT$

• P = pressure

• V = volume

• n = number of moles

• R = ideal gas constant

• T = temperature (Kelvin)

Values of R:

• 8.314 J mol-1 K-1 (when P in kPa, V in L)

• 0.0821 L atm mol-1 K-1 (when P in atm, V in L)

• 0.08314 L bar mol-1 K-1 (when P in bar, V in L)

Applications: Molar Volume, Density, and Molar Mass

• Standard Temperature and Pressure (STP): Defined as 1 bar and 273.15 K.

• At STP, 1 mole of an ideal gas occupies 22.7 L.

Calculating Molar Volume:

$V_m = \frac{RT}{P}$

• Where V_m is the molar volume.

Calculating Density:

$d = \frac{MP}{RT}$

• Where d is density, M is molar mass.

Calculating Molar Mass:

$M = \frac{mRT}{PV}$

• Where m is mass of the gas.

Mixtures of Gases and Partial Pressures

Dalton's Law of Partial Pressures

• In a mixture, each gas exerts a pressure as if it were alone in the container.

• The total pressure is the sum of the partial pressures of each component.

Formula:

$P_{total} = P_1 + P_2 + P_3 + \ldots$

• Partial pressure of a gas: $P_i = x_i P_{total}$, where $x_i$ is the mole fraction.

Mole Fraction:

$x_i = \frac{n_i}{n_{total}}$

Collecting Gases Over Water

• When a gas is collected over water, the total pressure includes both the gas and water vapor.

• To find the pressure of the dry gas, subtract the vapor pressure of water from the total pressure.

Formula:

$P_{total} = P_{gas} + P_{H_2O}$

Gases in Chemical Reactions: Stoichiometry Revisited

• The ideal gas law can be used to relate the volume, pressure, or temperature of a gaseous reactant or product to the amount in moles.

• Stoichiometric calculations can use $n = \frac{PV}{RT}$ to determine moles of gas involved in reactions.

Real Gases: Deviations from Ideal Behavior

Limitations of the Ideal Gas Law

• The ideal gas law is most accurate at low pressures and high temperatures.

• At high pressures or low temperatures, real gases deviate due to finite molecular volume and intermolecular forces.

van der Waals Equation

• The van der Waals equation corrects for non-ideal behavior by accounting for molecular size and intermolecular attractions.

Equation:

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

• a corrects for intermolecular forces; b corrects for molecular volume.

Kinetic Molecular Theory

Postulates and Implications

• Gases consist of many small particles in constant, random motion.

• Collisions between molecules and with container walls are elastic (no energy lost).

• The average kinetic energy of gas molecules is proportional to the absolute temperature.

Average Kinetic Energy:

$KE_{avg} = \frac{3}{2}RT$

• At a given temperature, all gases have the same average kinetic energy, regardless of molecular mass.

• Lighter molecules move faster than heavier ones at the same temperature.

Root Mean Square Speed:

$u_{rms} = \sqrt{\frac{3RT}{M}}$

• Where M is the molar mass in kg/mol.

Example: Hydrogen molecules move faster than oxygen molecules at the same temperature.

Additional info: The kinetic molecular theory explains why gases of different molar masses have the same molar volume at STP: their average kinetic energies are equal at a given temperature, even though their speeds differ.