Chapter 6: Gases – General Chemistry Study Notes

Gases

Pressure Units and Conversions

Pressure is a fundamental property of gases, defined as the force exerted per unit area. It can be measured in several units, including atmospheres (atm), millimeters of mercury (mm Hg), pounds per square inch (psi), and kilopascals (kPa).

• Common Pressure Units: atm, mm Hg, psi, kPa

• Conversion Relationships:

  • 1 atm = 14.7 psi

  • 760 mm Hg = 1 atm

• Conversion Example: To convert 132 psi to mm Hg, first convert psi to atm, then atm to mm Hg.

Example: A bicycle tire is inflated to 132 psi. To find the pressure in mm Hg:

• Convert 132 psi to atm: $\frac{132\ \text{psi}}{14.7\ \text{psi/atm}} = 8.98\ \text{atm}$

• Convert atm to mm Hg: $8.98\ \text{atm} \times 760\ \text{mm Hg/atm} = 6825\ \text{mm Hg}$

Gas Laws

Gas laws describe the relationships between pressure, volume, temperature, and amount of gas.

Boyle’s Law

Boyle’s Law states that the volume of a gas is inversely proportional to its pressure at constant temperature.

• Equation: $P_1 V_1 = P_2 V_2$

• Example: If lung volume increases from 2.75 L to 3.25 L at constant temperature, the pressure decreases accordingly.

Charles’s Law

Charles’s Law states that the volume of a gas is directly proportional to its temperature (in Kelvin) at constant pressure.

• Equation: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Example: A gas sample decreases in volume from 2.80 L to 2.57 L when cooled. To find the initial temperature, use the equation above.

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.

• Equation: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$

• Example: If a lung contains 0.254 mol of air at 6.15 L, and the volume decreases to 2.55 L, calculate the moles exhaled.

Ideal Gas Law

The ideal gas law combines the previous laws into a single equation relating pressure, volume, temperature, and moles.

• Equation: $PV = nRT$

• R: Ideal gas constant (0.0821 L·atm/mol·K)

• Example: Calculate the volume occupied by 0.845 mol of nitrogen gas at 1.37 atm and 315 K.

Density and Molar Mass of Gases

The density of a gas can be calculated using its molar mass, pressure, and temperature.

• Equation: $d = \frac{PM}{RT}$

• Example: Calculate the density of nitrogen gas at 755 mm Hg and a given temperature.

The molar mass of a gas can be determined from its mass, volume, pressure, and temperature.

• Equation: $M = \frac{mRT}{PV}$

• Example: Find the molar mass of a gas sample with mass 0.311 g, volume 0.225 L, temperature, and pressure.

Partial Pressures and Gas Mixtures

Dalton’s Law states that the total pressure of a gas mixture is the sum of the partial pressures of each component.

• Equation: $P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$

• Mole Fraction: $X_i = \frac{n_i}{n_{\text{total}}}$

• Partial Pressure: $P_i = X_i \times P_{\text{total}}$

• Example: Calculate the mass of argon in a mixture given partial pressures of helium and neon.

Collecting Gases Over Water

When gases are collected over water, the total pressure includes both the gas and water vapor. The partial pressure of the gas is found by subtracting the vapor pressure of water from the total pressure.

• Equation: $P_{\text{gas}} = P_{\text{total}} - P_{\text{water}}$

• Example: Find the mass of oxygen collected over water at a given temperature and pressure.

Gas Stoichiometry

Gas stoichiometry involves using the ideal gas law and balanced chemical equations to relate volumes, masses, and moles of reactants and products.

• Example: Calculate the volume of hydrogen gas needed to synthesize a given mass of methanol.

• Molar Volume at STP: 1 mol of an ideal gas occupies 22.4 L at standard temperature and pressure (273 K, 1 atm).

Kinetic Molecular Theory and Gas Properties

The kinetic molecular theory explains the behavior of gases in terms of molecular motion.

• Root Mean Square Velocity: The average speed of gas molecules, calculated as:

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

• Example: Calculate the root mean square velocity of oxygen molecules at a given temperature.

Graham’s Law of Effusion

Graham’s Law relates the rate of effusion of gases to their molar masses.

• Equation: $\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$

• Example: An unknown gas effuses at 0.462 times the rate of nitrogen. Calculate its molar mass.

Summary Table: Common Gas Laws and Relationships

Law	Equation	Variables
Boyle’s Law	$P_1 V_1 = P_2 V_2$	Pressure, Volume
Charles’s Law	$\frac{V_1}{T_1} = \frac{V_2}{T_2}$	Volume, Temperature
Avogadro’s Law	$\frac{V_1}{n_1} = \frac{V_2}{n_2}$	Volume, Moles
Ideal Gas Law	$PV = nRT$	Pressure, Volume, Moles, Temperature
Dalton’s Law	$P_{\text{total}} = \sum P_i$	Partial Pressures
Graham’s Law	$\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$	Effusion Rates, Molar Mass

Applications and Practice Problems

• Convert between different pressure units using provided relationships.

• Apply gas laws to solve for unknown variables in real-world scenarios (e.g., tire inflation, lung volume changes).

• Calculate density and molar mass of gases using the ideal gas law.

• Determine partial pressures and mole fractions in gas mixtures.

• Use stoichiometry and molar volume to relate gas volumes to chemical reactions.

• Calculate root mean square velocity and apply Graham’s Law to effusion problems.

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