Chapter 10: Gases – Properties, Laws, and Applications

Gases: An Overview

Introduction to Gases

Gases are one of the fundamental phases of matter, characterized by particles that move randomly and rapidly within their container. This random motion leads to unique physical properties that distinguish gases from solids and liquids.

• Phase of Matter: Gases have neither a fixed shape nor a fixed volume and will expand to fill any container.

• Particle Motion: Gas particles are in constant, random motion, moving very quickly compared to particles in other phases.

Pressure: The Fundamental Gas Property

Definition and Origin of Pressure

Pressure is a key property of gases, arising from collisions of gas particles with the walls of their container. Each collision exerts a force on the container's surface.

• Pressure Formula: $\text{Pressure} = \frac{\text{Force}}{\text{Area}} = \frac{F}{A}$

• Factors Affecting Pressure:

  • Concentration of Gas Molecules: $\text{Concentration} = \frac{\text{Number of particles}}{\text{Volume}}$. Fewer particles or larger volume results in lower pressure.

  • Average Speed of Particles: Faster-moving particles exert greater force upon collision, increasing pressure.

Pressure Imbalance in Biological Systems

Pressure differences across membranes, such as the eardrum, can cause discomfort or pain as the membrane is pushed outward or inward.

Atmospheric Pressure and Its Effects

• Atmospheric pressure varies with altitude and weather conditions.

• As altitude increases, the number of gas particles per volume decreases, leading to lower pressure.

• Pressure differences in the atmosphere drive wind and influence weather patterns.

Units of Pressure

Pressure can be measured in several units, all of which are interconvertible. The following table summarizes common units:

Unit	Abbreviation	Avg. Air Pressure at Sea Level
Atmosphere	atm	1.00 atm
Pascal (N/m2)	Pa	101,325 Pa
Pounds per square inch	psi	14.7 psi
Torr	torr	760 torr
mm Hg	mmHg	760 mm Hg
Bar	bar	1.013 bar

All these values represent the same pressure at sea level: $1\ \text{atm} = 760\ \text{torr} = 14.7\ \text{psi}$.

Measuring Gas Pressure

• Barometer: Measures atmospheric pressure using a column of mercury.

• Manometer: Measures the pressure of a gas sample in a container, comparing it to atmospheric pressure using a U-shaped tube filled with liquid.

• Other Sensors: Devices may measure absolute or gauge pressure, with different reference points and units.

Basic Properties of Gases

Four Fundamental Properties

• Pressure (P): Measured in atmospheres (atm).

• Volume (V): Measured in liters (L).

• Temperature (T): Measured in Kelvin (K), where $K = ^\circ C + 273$.

• Number of Particles (n): Measured in moles (mol).

These properties are interrelated; changing one affects the others. The simple gas laws describe the relationships between pairs of these properties.

The Simple Gas Laws

Boyle's Law: Pressure-Volume Relationship

Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature and the amount of gas are held constant.

• Mathematical Form: $P \times V = \text{constant}$

• Relationship: $P_1 V_1 = P_2 V_2$

• Example: If a lung contains 2.75 L of air at 1.02 atm, what is the pressure if the volume increases to 3.25 L (without inhaling)?

Boyle's Law in Real Life: Diving

• Every 10 m of water depth adds ~1 atm of pressure.

• Rapid ascent without exhaling can cause lung overexpansion due to pressure drop.

Charles's Law: Volume-Temperature Relationship

Charles's Law states that the volume of a fixed amount of gas at constant pressure increases linearly with increasing temperature (in Kelvin).

• Mathematical Form: $V = \text{constant} \times T$

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

• Absolute Zero: Extrapolation of volume vs. temperature lines gives $-273.15^\circ$C ($0$ K), the theoretical minimum temperature.

Charles's Law: Molecular Perspective

• Increasing temperature increases kinetic energy and speed of gas particles.

• More frequent and forceful collisions require a larger volume to maintain constant pressure.

Avogadro's Law: Volume-Mole Relationship

Avogadro's Law states that the volume of a gas is directly proportional to the number of moles (n) when pressure and temperature are constant.

• Mathematical Form: $V = \text{constant} \times n$

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

• Equal Volumes Principle: Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules, regardless of type.

The Ideal Gas Law

Combining the Simple Gas Laws

The relationships described by Boyle's, Charles's, and Avogadro's laws are combined in the Ideal Gas Law:

• Equation: $PV = nRT$

• Variables:

  • $P$ = pressure (atm)

  • $V$ = volume (L)

  • $n$ = moles

  • $R$ = ideal gas constant, $0.08206\ (L\cdot atm)/(mol\cdot K)$

  • $T$ = temperature (K)

• Allows calculation of any one variable if the other three are known.

Using the Ideal Gas Law

• Static Conditions: $PV = nRT$

• Changing Conditions: $\frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_2}$

• Can be used in place of other gas laws when multiple variables change.

Standard Conditions and Molar Volume

Standard Temperature and Pressure (STP)

• Standard Pressure: 1 atm

• Standard Temperature: 273.15 K (0°C)

• Standard Amount: 1 mol

• Molar Volume at STP: $V = \frac{nRT}{P} = 22.4\ L$ for 1 mol of any ideal gas

Molar Volume Visualization

• 1 mol of any ideal gas occupies 22.4 L at STP, regardless of its identity.

• Different gases have different masses but the same volume at STP.

Density of Gases

Calculating Gas Density

• Definition: Density is mass per unit volume.

• Formula: $\text{Density} = \frac{\text{mass}}{\text{volume}}$

• For gases at STP: $\text{Density} = \frac{\text{molar mass}}{22.4\ L}$

• General formula: $d = \frac{PM}{RT}$, where $M$ is molar mass.

Gas	Molar Mass (g/mol)	Density at STP (g/L)
Helium (He)	4.00	0.179
Nitrogen (N2)	28.02	1.25