Chapter 5.3: Gases – Ideal Gas Law, Applications, and Properties

Chapter 5 - Gases

Introduction

This chapter explores the fundamental properties of gases, the laws that govern their behavior, and the mathematical relationships used to describe and compare different gases. The Ideal Gas Law and its applications are central to understanding gas properties such as molar volume, density, and molar mass.

Gas Laws

Fundamental Gas Laws

Several empirical laws describe the relationships between pressure, volume, temperature, and amount of gas. These laws are the foundation for the Ideal Gas Law.

• Boyle's Law: At constant temperature and amount of gas, the volume of a gas is inversely proportional to its pressure. $V \propto \frac{1}{P}$

• Charles' Law: At constant pressure and amount of gas, the volume of a gas is directly proportional to its temperature (in Kelvin). $V \propto T$

• Gay-Lussac's Law: At constant volume and amount of gas, the pressure of a gas is directly proportional to its temperature (in Kelvin). $P \propto T$

• Avogadro's Law: At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles. $V \propto n$

The Ideal Gas Law

Definition and Equation

The Ideal Gas Law combines the above relationships into a single equation that describes the state of an ideal gas:

• Equation: $PV = nRT$

• Variables:

  • P: Pressure (in units such as Pa, kPa, bar, atm)

  • V: Volume (L, m3)

  • n: Amount of substance (moles)

  • R: Universal gas constant

  • T: Temperature (Kelvin)

• Proportionality: $V \propto \frac{nT}{P}$

The Ideal Gas Law can be rearranged to solve for any variable, depending on the known quantities.

Gas Constant (R)

The value of the gas constant R depends on the units used for pressure and volume:

Units	Value of R
Pa·m3·mol−1·K−1	8.314
J·mol−1·K−1	8.314
kPa·L·mol−1·K−1	8.314
bar·L·mol−1·K−1	0.08314

Applications of the Ideal Gas Law

The Ideal Gas Law is used to solve a variety of problems involving gases, such as determining the maximum temperature a container can withstand, predicting changes in pressure, volume, or temperature, and calculating the properties of gases under different conditions.

• Example 1: Calculating the maximum temperature before a paint can explodes. Given: $V = 0.5\,L$, $P_{max} = 4\,bar$, $n = 0.05\,mol$, $R = 0.08314\,bar\cdot L\cdot mol^{-1}\cdot K^{-1}$ $T = \frac{PV}{nR} = \frac{4\,bar \times 0.5\,L}{0.05\,mol \times 0.08314\,bar\cdot L\cdot mol^{-1}\cdot K^{-1}} = 481.12\,K = 208^{\circ}C$

• Example 2: Predicting pressure changes when volume and temperature are doubled. If both volume and temperature are doubled, the pressure remains the same (from $PV = nRT$).

• Example 3: Calculating the volume of a balloon at different pressures and temperatures. Use the combined gas law: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ Rearranged to solve for $V_2$: $V_2 = \frac{P_1V_1T_2}{T_1P_2}$

Comparing Different Gases

Molar Volume, Density, and Molar Mass

At standard temperature and pressure (STP: $T = 273.15\,K$, $P = 1\,bar$), one mole of any ideal gas occupies the same volume, but the mass and density differ depending on the gas.

Gas	Molar Mass (g/mol)	Volume (L at STP)	Density (g/L)
He	4.00	22.71	0.1763
Xe	131.29	22.71	5.781
CH4	16.04	22.71	0.7064

Molar Volume: The volume occupied by one mole of gas at STP is 22.71 L.

Density: Density ($d$) of a gas can be calculated using the ideal gas law: $d = \frac{PM}{RT}$ where $M$ is the molar mass.

Key Concepts

• Use the Ideal Gas Law to relate pressure, volume, temperature, and amount of gas.

• Express physical properties such as density and molar mass in terms of the Ideal Gas Law.

• At STP, all ideal gases have the same molar volume but different densities and masses.

Summary Table: Properties of Gases at STP

Property	Definition	Formula
Molar Volume	Volume occupied by 1 mole of gas at STP	$V_m = \frac{RT}{P}$
Density	Mass per unit volume of a gas	$d = \frac{PM}{RT}$
Molar Mass	Mass of 1 mole of a substance	$M = \frac{dRT}{P}$

Practice and Application

• Apply the Ideal Gas Law to solve for unknown variables in gas problems.

• Compare properties of different gases using molar volume, density, and molar mass.

• Use the combined gas law for changes in conditions (pressure, volume, temperature).

Additional info: The notes reference exercises from "Chemistry: A Molecular Approach, Chapter 5" for further practice (e.g., 5.13, 5.38, 5.48, 5.52, 5.54, 5.56, 5.62, 6.146).