Lecture 20

Chapter 9: The Gaseous State

Introduction to Gases

Gases are one of the fundamental states of matter, characterized by their ability to fill any container and their unique physical properties. Understanding the behavior of gases is essential in general chemistry, as it underpins many laboratory and real-world phenomena.

• Gas: Normally in the gaseous state at ordinary temperatures and pressures.

• Vapor: The gaseous form of any substance that is normally a liquid or a solid.

Characteristics of a Gas

• Highly Compressible: Gases can be compressed much more than solids or liquids due to the large distances between molecules.

• Thermally Expandable: Gases expand significantly when heated.

• Low Viscosity: Gases flow easily and have low resistance to flow.

• Low Densities: Typical gas densities are much lower than those of solids or liquids, often measured in g/L.

• Infinitely Miscible: Gases mix completely with each other in any proportion.

I. The Pressure and Temperature of Gases

Pressure

Pressure is a measure of the force exerted by gas molecules as they collide with the surfaces of their container. It is a fundamental property used to describe the state of a gas.

• Definition: Pressure is the force per unit area exerted by a gas.

• Formula: $P = \frac{F}{A}$, where P is pressure, F is force, and A is area.

Measuring Gas Pressure: The Barometer

A barometer is an instrument used to measure atmospheric pressure. It typically consists of a column of mercury whose height is determined by the pressure exerted by the atmosphere.

• Principle: The weight of the mercury column is counterbalanced by the atmospheric pressure exerted on the mercury in the dish.

• Relationship: For a given pressure, the ratio of heights of liquid columns is proportional to the ratio of the densities of the liquids.

Units of Pressure

• Standard Atmospheric Pressure: The pressure which supports a column of mercury in a barometer exactly 760 mm high at 0°C at sea level.

• mm Hg: Millimeters of mercury, a common unit for pressure.

• Torr: 1 torr = 1 mm Hg.

• SI Unit: The SI unit of pressure is the pascal (Pa), where $1\ \text{atm} = 101,325\ \text{Pa}$.

• Other Units: Atmospheres (atm), kilopascals (kPa).

Example: Meteorologists often speak of pressure in inches of mercury. If the barometric pressure is 29.12 in, how many (a) mm Hg, (b) torr, (c) atm, (d) kPa are we talking about?

Unit	Conversion Factor
mm Hg	1 in = 25.4 mm
torr	1 torr = 1 mm Hg
atm	1 atm = 760 mm Hg
kPa	1 atm = 101.325 kPa

II. The Gas Laws

The gas laws describe the relationships between pressure, volume, temperature, and amount of gas. They are fundamental to understanding gas behavior.

a) Boyle's Law

Boyle's Law states that the pressure of a fixed amount of gas is inversely proportional to its volume at constant temperature.

• Formula: $P_1V_1 = P_2V_2$

• Application: Used to calculate changes in volume or pressure when temperature and amount of gas are constant.

Example: If the pressure on a gas is doubled, its volume is halved (at constant temperature).

b) 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: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$

• Application: Used to calculate changes in volume or temperature when pressure and amount of gas are constant.

Example: Heating a balloon causes it to expand.

c) Gay-Lussac's Law

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

• Formula: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Example: Increasing the temperature of a sealed container increases its pressure.

d) Avogadro's Law

Avogadro's Law states that the volume of a gas is directly proportional to the number of moles of gas at constant temperature and pressure.

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

Application: Used to relate volume and amount of gas.

Standard Temperature and Pressure (STP)

• STP: Standard conditions are 0°C (273.15 K) and 1 atm pressure.

• Molar Volume: At STP, one mole of any ideal gas occupies 22.414 L.

Example: How many grams of CO are contained in 10 L at STP?

• Use molar volume and molar mass to solve.

e) Combined Gas Law

The combined gas law combines Boyle's, Charles's, and Gay-Lussac's laws to relate pressure, volume, and temperature for a fixed amount of gas.

• Formula: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$

III. The Ideal Gas Equation

a) General

By combining the relationships described in the gas laws, we arrive at the ideal gas law, which relates pressure, volume, temperature, and amount of gas.

• Formula: $PV = nRT$

• R: The constant of proportionality, called the ideal gas constant.

• Value of R: $R = 0.08206\ \text{L atm mol}^{-1}\text{K}^{-1}$

Ideal Gas: A hypothetical gas that obeys the ideal gas law exactly. Real gases approximate ideal behavior under many conditions.

• Assumptions: Gas molecules have negligible volume and no intermolecular forces.

Note: Review Kelvin Temperature Scale. All gas law calculations use temperature in Kelvin.

General Combined Gas Law

When the amount of gas changes, or when comparing two sets of conditions, the general combined gas law is used:

• Formula: $\frac{P_1V_1}{n_1T_1} = \frac{P_2V_2}{n_2T_2}$

Example: A flashbulb of volume 2.0 cm3 contains O2 gas at a pressure of 2.3 atm and a temperature of 26°C. How many moles of oxygen are in the bulb?

Example (Variation): Suppose you put the bulb in the freezer. When you take it out it is at a temperature of -40°C. What is the new pressure of oxygen?

Misconception

• Pressure is not the amount of material: Pressure depends on the number of collisions of gas molecules with the container walls, not directly on the mass of the gas.

b) Gas Density

The ideal gas law can be rearranged to solve for the density of a gas.

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

• Units: Gas density is usually reported in g/L.

Example: What is the density of CO2 at 745 mm Hg and 65°C?

c) Molar Masses of Gaseous Materials

The density equation can be rearranged to solve for the molar mass of an unknown gas.

• Formula: $M = \frac{dRT}{P}$

• Application: By measuring the temperature, pressure, and density of a pure gas in a container of known volume, its molar mass can be determined.

IV. Reaction Stoichiometry and the Gas Laws

The ideal gas law allows us to relate the number of moles of a gas to its pressure, volume, and temperature. This is useful in chemical reactions involving gases.

• Formula: $PV = nRT$

• Application: Use stoichiometry to relate volumes of gases to moles in chemical equations.

Example:

Reaction	Application
3NO2(g) + H2O(l) → 2HNO3(aq) + NO(g)	How many moles of HNO3 can be made using 450 L of NO2 gas at a pressure of 5.0 atm at 295 K?

Use the ideal gas law to find moles of NO2, then use stoichiometry to find moles of HNO3.

Additional info: Where examples or equations were missing, standard textbook equations and applications have been provided for completeness.