Chapter 6: Gases – Properties, Laws, and Real Gas Behavior

Gases: Properties and Behavior

Introduction to Gases

Gases are one of the fundamental states of matter, characterized by their ability to expand and fill any container. Their behavior is governed by several physical laws that relate pressure, volume, temperature, and the amount of gas present.

Gas Pressure

Definition and Origin

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

• Gas pressure results from the constant, random motion of gas particles.

• Atmospheric pressure varies with altitude; it decreases as altitude increases due to fewer gas particles in a given volume.

Factors Affecting Gas Pressure

• Number of gas particles in a given volume

• Volume of the container

• Average speed (temperature) of the gas particles

As the volume increases, the concentration of gas molecules decreases, leading to fewer collisions and lower pressure.

Gas Pressure and Particle Density

• Pressure is directly proportional to the density of gas particles.

• Low density = low pressure; high density = high pressure.

Pressure Imbalance in the Ear

• A difference in pressure across the eardrum can cause pain due to the membrane being pushed out.

Measuring Gas Pressure

The Barometer

• A barometer measures atmospheric pressure using a column of mercury.

• Atmospheric pressure supports a column of mercury 760 mm high at sea level (1 atm).

The Manometer

• A manometer measures the pressure of a gas in a container.

• It consists of a U-shaped tube with one side open to the atmosphere and the other connected to the gas sample.

• The difference in liquid levels indicates the pressure difference between the gas and the atmosphere.

Blood Pressure

• Measured using a sphygmomanometer, which uses an inflatable cuff and a pressure gauge.

The Simple Gas Laws

Overview

Four basic properties of gases are interrelated: pressure (P), volume (V), temperature (T), and amount in moles (n). The simple gas laws describe the relationships between pairs of these properties.

Boyle’s Law: Pressure and Volume

• At constant temperature and amount, the pressure of a gas is inversely proportional to its volume.

• Mathematically: $P \propto \frac{1}{V}$ or $PV = \text{constant}$

• As pressure increases, volume decreases by the same factor.

Example: If a diver ascends quickly, the pressure decreases and the volume of air in the lungs increases, which can cause injury.

Charles’s Law: Volume and Temperature

• At constant pressure and amount, the volume of a gas is directly proportional to its temperature (in kelvins).

• Mathematically: $V \propto T$ or $\frac{V}{T} = \text{constant}$

• Absolute zero (0 K) is the theoretical temperature at which a gas would have zero volume.

Example: A balloon expands when moved from ice water to boiling water due to increased particle motion.

Avogadro’s Law: Volume and Amount (Moles)

• At constant temperature and pressure, the volume of a gas is directly proportional to the number of moles.

• Mathematically: $V \propto n$ or $\frac{V}{n} = \text{constant}$

• Equal volumes of gases at the same temperature and pressure contain equal numbers of molecules.

The Ideal Gas Law

Combining the Gas Laws

• The relationships from the simple gas laws are combined into the ideal gas law:

$PV = nRT$

• Where R is the gas constant (0.0821 L·atm·mol−1·K−1 when P is in atm and V in liters).

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

Standard Temperature and Pressure (STP)

• Standard pressure: 1 atm

• Standard temperature: 273 K (0°C)

Molar Volume at STP

• At STP, 1 mole of any ideal gas occupies 22.4 L.

• This is called the molar volume.

Density of a Gas at STP

• Density ($d$) is mass divided by volume.

• For a gas at STP: $d = \frac{\text{molar mass}}{22.4\ \text{L}}$

• Density is directly proportional to molar mass.

Determining Molar Mass of a Gas

• Measure mass and volume at known P and T.

• Calculate moles using the ideal gas law, then molar mass = mass (g) / moles.

Mixtures of Gases and Partial Pressures

Mixtures of Gases

• Many gases, such as air, are mixtures of different gases.

• Each component behaves independently and contributes to the total pressure.

Partial Pressure and Dalton’s Law

• The partial pressure of a gas is the pressure it would exert if it were alone in the container.

• Dalton’s Law: The total pressure is the sum of the partial pressures of all components.

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

Mole Fraction

• The mole fraction ($X$) is the ratio of moles of a component to the total moles in the mixture.

• The partial pressure of a component: $P_A = X_A \times P_{\text{total}}$

Kinetic Molecular Theory (KMT)

Basic Postulates

• Gas particles are in constant, random motion.

• The size of gas particles is negligibly small compared to the distances between them.

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

• The average kinetic energy is proportional to the temperature in kelvins.

• There are negligible attractive or repulsive forces between particles.

Explaining Gas Laws with KMT

• Boyle’s Law: Decreasing volume increases collision frequency, raising pressure.

• Charles’s Law: Increasing temperature increases particle speed, requiring greater volume to maintain constant pressure.

• Avogadro’s Law: More particles mean more collisions; volume must increase to keep pressure constant.

• Dalton’s Law: Each gas acts independently; total pressure is the sum of individual pressures.

Temperature and Molecular Velocities

• At the same temperature, all gases have the same average kinetic energy.

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

The root mean square velocity ($u_{\text{rms}}$) is given by:

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

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

Diffusion and Effusion

Definitions

• Diffusion: The spread of gas molecules from high to low concentration.

• Effusion: The escape of gas molecules through a small hole into a vacuum.

Graham’s Law of Effusion

• The rate of effusion is inversely proportional to the square root of the molar mass:

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

Real Gases and Deviations from Ideal Behavior

Limitations of the Ideal Gas Law

• At high pressures and low temperatures, real gases deviate from ideal behavior.

• Assumptions of negligible volume and no intermolecular forces break down.

Van der Waals Equation

• Accounts for molecular volume and intermolecular attractions:

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

• a and b are van der Waals constants specific to each gas.

Summary of Real vs. Ideal Gas Behavior

• At low pressures, real gases behave nearly ideally.

• At high pressures, molecular volume increases the measured volume above ideal predictions.

• At low temperatures, intermolecular attractions reduce the measured pressure below ideal predictions.

Tables

Common Pressure Units

Unit	Symbol	Equivalent to 1 atm
Atmosphere	atm	1
Millimeter of mercury	mmHg	760
Torr	torr	760
Pascals	Pa	101,325
Bar	bar	1.01325

Relevant Image

Additional info: The image above is the cover of the referenced textbook, which is directly relevant as it identifies the source of the material and the context for the study notes.