Chapter 5: Gases – Properties, Laws, and Molecular Theory

Chapter 5: Gases – Properties, Laws, and Molecular Theory

Outline of Topics

• Result of Molecular Collisions

• Gas Laws: Boyle's Law, Charles's Law, and Avogadro's Law

• Ideal Gas Law: Molar Volume, Density, and Molar Mass

• Gases and Partial Pressures

• Chemical Reactions: Stoichiometry Revisited

• Kinetic Molecular Theory: A Model for Gases

• Mean Free Path, Diffusion, and Effusion of Gases

• Effects of Size and Intermolecular Forces

Breathing: Putting Pressure to Work

How Pressure Drives Respiration

Breathing is a practical application of gas laws, where pressure differences move air in and out of the lungs. Muscles change the volume of the chest cavity, affecting the pressure inside the lungs.

• **Diaphragm and chest muscles** expand the chest cavity, increasing lung volume.

• As lung volume increases, **pressure inside the lung decreases**.

• Air flows into the lungs to balance the pressure difference.

• When the lungs are compressed, the process reverses and air is expelled.

Pressure: The Result of Molecular Collisions

Definition and Origin of Pressure

Pressure in gases arises from countless collisions of gas molecules with the walls of their container. The cumulative effect of these collisions produces a measurable force over a given area.

• **Pressure ($P$)** is defined as force per unit area:

$P = \frac{F}{A}$

• Where $F$ is force and $A$ is area.

• Gas molecules move rapidly and randomly, colliding with container walls.

Units of Pressure

Pressure can be measured in several units, often based on the height of a column of mercury (mmHg or Torr). Standard conditions for gases use specific pressure values.

Unit	Standard Pressure
bar	1.01 bar
Pa	101,325 Pa
kPa	101.3 kPa
Torr or mmHg	760 Torr
atm	1.00 atm
psi	14.7 psi

Gas Laws: Boyle's Law, Charles's Law, and Avogadro's Law

Boyle's Law

Boyle's Law describes the inverse relationship between pressure and volume for a fixed amount of gas at constant temperature.

• **Formula:** $PV = \text{constant}$

• As pressure increases, volume decreases, and vice versa.

Example: Compressing a gas in a syringe increases its pressure and decreases its volume.

Charles's Law

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

• **Formula:** $\frac{V}{T} = \text{constant}$

• As temperature increases, volume increases.

Example: Heating a balloon causes it to expand.

Avogadro's Law

Avogadro's Law relates the volume of a gas to the number of moles present, at constant temperature and pressure.

• **Formula:** $\frac{V}{n} = \text{constant}$

• Increasing the amount of gas increases its volume.

Example: Adding more gas to a tire increases its volume.

Combined Gas Law

The combined gas law merges Boyle's, Charles's, and Avogadro's laws to relate pressure, volume, temperature, and amount of gas.

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

The Ideal Gas Law

General Equation

The ideal gas law combines all the simple gas laws into one equation, relating pressure, volume, temperature, and moles of gas.

• **Formula:** $PV = nRT$

• $R$ is the universal gas constant ($0.08206$ L·atm·mol−1·K−1 or $8.314$ J·mol−1·K−1).

Example: Calculate the volume occupied by 1 mole of gas at standard temperature and pressure (STP).

Molar Volume

At STP (0°C and 1 atm), one mole of any ideal gas occupies 22.4 L.

• **Formula:** $V_m = \frac{RT}{P}$

Density and Molar Mass of a Gas

The density and molar mass of a gas can be determined using the ideal gas law.

• **Density Formula:** $d = \frac{PM}{RT}$

• **Molar Mass Formula:** $M = \frac{dRT}{P}$

Mixtures of Gases and Partial Pressures

Dalton's Law of Partial Pressures

In a mixture of gases, each gas exerts a pressure independently of the others. The total pressure is the sum of the partial pressures.

• **Formula:** $P_{\text{total}} = P_A + P_B + P_C + \ldots$

• Partial pressure of component $A$: $P_A = x_A P_{\text{total}}$

• Mole fraction: $x_A = \frac{n_A}{n_{\text{total}}}$

Example: Air is a mixture of N2, O2, and other gases. The partial pressure of N2 is about 0.78 atm at STP.

Collecting Gases Over Water

When collecting gases 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.

• **Formula:** $P_{\text{gas}} = P_{\text{total}} - P_{\text{H}_2O}$

Temperature (°C)	Vapor Pressure of Water (Torr)
55	118
60	146
65	187
70	233
75	286
80	355
85	433
90	525
95	634
100	760

Gases in Chemical Reactions: Stoichiometry

Using Gas Laws in Stoichiometry

Gas volumes can be related to moles in chemical reactions using the ideal gas law. This is useful for calculating reactant or product quantities in reactions involving gases.

• **Example:** 2 H2(g) + O2(g) → 2 H2O(l)

• Use $PV = nRT$ to find moles, then use stoichiometry to find mass of product.

Kinetic Molecular Theory: A Model for Gases

Postulates of Kinetic Molecular Theory (KMT)

KMT explains the behavior of gases by modeling them as small particles in constant, random motion.

• Gas particles are negligibly small compared to the distances between them.

• Collisions between particles and container walls are perfectly elastic.

• The average kinetic energy of particles is proportional to temperature (in Kelvin).

Equation for average kinetic energy:

$KE_{avg} = \frac{3}{2}RT$

Molecular Velocities

At a given temperature, all gases have the same average kinetic energy, but lighter molecules move faster.

• **Root-mean-square velocity ($u_{rms}$):**

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

• $M$ is molar mass in kg/mol.

Mean Free Path, Diffusion, and Effusion

Definitions

• **Mean free path:** Average distance a molecule travels between collisions.

• **Diffusion:** Spread of gas molecules throughout a space or another gas.

• **Effusion:** Escape of gas molecules through a small hole into a vacuum.

Graham's Law of Effusion:

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

• Lighter gases effuse faster than heavier gases.

Effects of Size and Intermolecular Forces

Real Gases vs. Ideal Gases

Ideal gas law assumes particles have no volume and no intermolecular forces. Real gases deviate from ideal behavior at high pressures and low temperatures.

• At high pressure, particle volume becomes significant.

• At low temperature, intermolecular attractions reduce pressure.

Van der Waals Equation

Van der Waals equation corrects for particle volume and intermolecular forces:

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

• $a$ corrects for intermolecular attractions.

• $b$ corrects for particle volume.

Gas	a (L2·atm·mol−2)	b (L·mol−1)
CO2	3.610	0.0429
H2S	4.481	0.0542
NO	1.351	0.0371
N2O	3.799	0.0371
NO2	5.29	0.0305
O2	1.363	0.0319
HCl	3.648	0.0320

Conditions for Greatest Deviation from Ideal Behavior

• Low temperature and high pressure cause the most deviation due to increased intermolecular forces and reduced free volume.

Additional info: Some equations and table entries were inferred and expanded for completeness and clarity.