Chapter 10: Gases – The Ideal Gas Law, Gas Properties, and Mixtures

Gases and the Ideal Gas Law

Introduction to Gas Laws

The behavior of gases is described by several fundamental laws that relate pressure, volume, temperature, and amount of gas. The Ideal Gas Law is a central equation in general chemistry, allowing prediction and calculation of gas properties under various conditions.

• Ideal Gas Law: Relates pressure (P), volume (V), temperature (T), and number of moles (n) of a gas.

• Equation: $PV = nRT$ where R is the gas constant.

• Standard Pressure and Temperature (STP): 1 atm and 273 K.

• Gas Constant (R): Common values include:

  • 0.0821 L·atm/mol·K

  • 62.4 L·mm Hg/mol·K

  • 8.314 L·kPa/mol·K

Calculating Gas Properties

Determining Moles of Gas

To find the number of moles required to fill a container at a specific pressure and temperature, rearrange the ideal gas law:

• Formula: $n = \frac{PV}{RT}$

• Example: How many moles of a gas at 100°C does it take to fill a 1.00 L flask to a pressure of 1.50 atm?

Volume of Gas at Given Conditions

To determine the volume needed to contain a given mass of gas at specified temperature and pressure:

• First, convert mass to moles using molar mass.

• Apply the ideal gas law to solve for volume: $V = \frac{nRT}{P}$

• Example: A camping stove holds 3000.0 g of liquid C5H12. How large a container is needed to hold the gas at 25°C and 3.0 atm?

Pressure Exerted by a Gas

To find the pressure exerted by a known amount of gas in a cylinder:

• Use the ideal gas law, solving for pressure: $P = \frac{nRT}{V}$

• Convert units as needed (e.g., to kPa).

• Example: A 10.0 L gas cylinder contains 55 moles nitrogen. What pressure, in kPa, is exerted at 25°C?

Gas Mixtures and Partial Pressures

In a mixture of gases, each gas exerts a partial pressure proportional to its mole fraction.

• Dalton's Law of Partial Pressures: $P_{\text{total}} = P_1 + P_2 + P_3 + \ldots$

• Partial Pressure: $P_i = X_i \cdot P_{\text{total}}$ where $X_i$ is the mole fraction of gas i.

• Example: Calculate the partial pressure of O2 in air outside an airplane at 10 km altitude, given atmospheric pressure and mole fraction.

Gas Law Applications and Calculations

Changing Conditions: Combined Gas Law

When a gas sample undergoes changes in pressure, volume, and temperature, use the combined gas law:

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

• Example: If you have a gas at 12 atm, 23 L, and 200 K, and then raise the pressure to 14 atm and temperature to 300 K, what is the new volume?

Density and Molar Mass of Gases

The density of a gas can be related to its molar mass using the ideal gas law:

• Equation: $d = \frac{PM}{RT}$ where d is density, P is pressure, M is molar mass, R is the gas constant, and T is temperature.

• Example: What is the density of a gas with molar mass 100 g/mol at 0.5 atm and 27°C?

Determining Molar Mass from Gas Data

If the mass, volume, pressure, and temperature of a gas are known, its molar mass can be calculated:

• Steps:

  1. Calculate moles using the ideal gas law.

  2. Divide the mass by moles to get molar mass.

• Example: You have 6.3 mg of a boron hydride in a flask of 385 mL at 25°C and 11 torr. Determine the molar mass.

Gas Mixtures: Mole Fractions and Applications

Mole Fractions in Gas Mixtures

The mole fraction of a component in a mixture is the ratio of its moles to the total moles in the mixture.

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

• Application: Used to calculate partial pressures and composition of gas mixtures.

• Example: A gas mixture called Trimix contains 11.7% He, 56.2% N2, and 32.1% O2. Calculate the mole fractions of each gas.

Summary Table: Key Gas Law Equations

Law/Equation	Formula	Variables	Application
Ideal Gas Law	$PV = nRT$	P, V, n, R, T	Relates all major gas properties
Combined Gas Law	$\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}$	P, V, T (initial and final)	Changing conditions
Density of a Gas	$d = \frac{PM}{RT}$	d, P, M, R, T	Find density or molar mass
Dalton's Law	$P_{\text{total}} = \sum P_i$	Ptotal, Pi	Partial pressures in mixtures
Mole Fraction	$X_i = \frac{n_i}{n_{\text{total}}}$	Xi, ni, ntotal	Composition of mixtures

Additional info:

• All calculations require careful unit conversion (e.g., temperature to Kelvin, pressure to atm or torr).

• Gas law problems often involve rearranging equations to solve for the unknown variable.

• Understanding the relationships between variables is key to mastering gas law applications.