Chapter 6: Gases - Comprehensive Notes

Gas Pressure

• Pressure is the force exerted per unit area by gas molecules as they strike surfaces.
• Analogous to a ball bouncing against a wall, gas molecules exert force upon collision with a surface.
• $Pressure = \frac{Force}{Area}$
• $P = \frac{F}{A}$

Molecular Collision and Pressure

• Gas pressure results from constant movement and collisions of gas molecules.
• Pressure depends on:
  • Number of gas particles in a given volume.
  • Volume of the container.
  • Average speed of gas particles.

Gas Concentration and Pressure

• Total pressure depends on the concentration of gas molecules.
• Higher concentration leads to greater pressure.
• As volume increases, concentration decreases, leading to fewer collisions and lower pressure.

Particle Density and Gas Pressure

• Pressure is dependent on the number of gas particles in a given volume.
• Fewer particles result in lower force per unit area and lower pressure.
• Low density = low pressure; High density = high pressure.

Barometer

• A barometer is an evacuated glass tube submerged in mercury (Hg).
• Atmospheric pressure forces mercury up into the tube.
• Mercury's high density (13.5x more than water) allows atmospheric pressure to support a column of Hg only about 0.760 meters (760 mm or 30 inches) tall.
• Average atmospheric pressure at sea level supports a 760 mm column of mercury.

Common Pressure Units

• Pascal (Pa): 1 Newton per square meter (1 N/m²), Average Air Pressure at Sea Level: $101,325 Pa$
• Pounds per square inch (psi): Average Air Pressure at Sea Level: $14.7 psi$
• Torr (1 mmHg): Average Air Pressure at Sea Level: $760 torr$ (exact)
• Millimeters of mercury (mm Hg): Average Air Pressure at Sea Level: $760 mm Hg$
• Atmosphere (atm): Average Air Pressure at Sea Level: $1 atm$

Simple Gas Laws

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

Boyle's Law

• Pressure and volume are inversely proportional at constant temperature and amount of gas.
• $P \propto \frac{1}{V}$
• $P<em>1V</em>1 = P<em>2V</em>2$
• Graph of P versus V is a curve; P versus 1/V is a straight line.
• As P increases, V decreases by the same factor.

Molecular Interpretation of Boyle's Law

• Decreasing gas sample volume increases the frequency of molecule collisions, resulting in greater pressure.

Charles's Law

• Volume of a fixed amount of gas at constant pressure increases linearly with increasing temperature in kelvins.
• Volume increases with increasing temperature.
• $Kelvin \;T = Celsius\; T + 273$
• $\frac{V}{T} = constant$
• $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$

Absolute Zero

• Extrapolating volume-temperature lines back to zero volume yields absolute zero.
• Absolute zero: $-273.15\;^\circ C$ or $0\; K$
• Gases condense into liquids before reaching absolute zero experimentally.

Molecular View of Charles’s Law

• Increasing temperature causes gas particles to move faster and occupy more space, expanding volume.

Charles’s Law Explanation

• Increased temperature leads to faster-moving gas particles.
• More frequent and forceful collisions occur with the walls.
• To maintain constant pressure, gas must occupy a larger volume, reducing collision frequency and increasing the area over which collisions occur.

Avogadro's Law

• Volume is directly proportional to the number of gas molecules (moles).
• Constant P and T; More gas molecules = larger volume.
• Equal volumes of gases contain equal numbers of molecules at constant pressure and temperature.
• $\frac{V}{n} = constant$
• $\frac{V<em>1}{n</em>1} = \frac{V<em>2}{n</em>2}$

Avogadro’s Law Explanation

• Increasing the amount of gas at constant temperature and pressure increases volume proportionally because more particles fill more space.
• The volume of a gas sample increases linearly with the number of moles.

Gay-Lussac's Law

• The pressure exerted by a gas is directly related to the Kelvin temperature of the gas.
• Volume and amount of gas are constant.

Combined Gas Law

• Combines Boyle’s, Charles’s, and Gay-Lussac’s laws (n is constant).
• Boyle’s law: $P<em>1V</em>1 = P<em>2V</em>2$
• Charles’s law: $\frac{V<em>1}{T</em>1} = \frac{V<em>2}{T</em>2}$
• Gay-Lussac’s law: $\frac{P<em>1}{T</em>1} = \frac{P<em>2}{T</em>2}$
• Combined gas law: $\frac{P<em>1V</em>1}{T<em>1} = \frac{P</em>2V<em>2}{T</em>2}$

Ideal Gas Law

• Combination of gas laws into a single encompassing law.
• Boyle's Law: $V \propto \frac{1}{P}$
• Charles's Law: $V \propto T$
• Avogadro's Law: $V \propto n$
• $V \propto \frac{nT}{P}$

Ideal Gas Law Equation

• General equation: $PV = nRT$
• R is the gas constant; its value depends on the units of P and V.
• When P is in atm and V is in liters, use $PV = nRT$
• Other gas laws are derived from the ideal gas law by holding two variables constant.
• The ideal gas law allows finding one variable if the other three are known.
• $R = 0.08206 \frac{L \cdot atm}{mol \cdot K}$

Standard Conditions

• Standard temperature and pressure (STP) are often specified for gas volumes.
• Standard pressure = 1 atm.
• Standard temperature = 273 K ($0^\circ C$).

Molar Volume

• The volume occupied by one mole of a substance at STP (T = 273 K, P = 1 atm).
• $V = \frac{nRT}{P} = \frac{1.00 \;mol \times 0.08206 \frac{L \cdot atm}{mol \cdot K} \times 273 \;K}{1.00 \;atm} = 22.4 \;L$

Molar Volume at STP

• The volume of 1 mol of gas at STP is 22.4 L (molar volume).
• The identity of the gas is immaterial.
• One mole measurements of different gases have different masses, even though they have the same volume.
• 1 mole contains $6.022 \times 10^{23}$ molecules of gas.

Density of a Gas at STP

• Density is the ratio of mass to volume, generally in g/L.
• Mass of 1 mole = molar mass.
• Volume of 1 mole at STP = 22.4 L.
• $Density = \frac{molar \; mass}{molar \; volume}$

Examples of Density at STP

• Density of Helium (He) at STP: $\frac{4.00 \;g/mol}{22.4 \;L/mol} = 0.179 \;g/L$
• Density of Nitrogen (N2) at STP: $\frac{28.02 \;g/mol}{22.4 \;L/mol} = 1.25 \;g/L$
• A He balloon will float because it is less dense.

Gas Density and Molar Mass

• Density is directly proportional to molar mass.
• From the ideal gas law: $PV = nRT$
• $\frac{n}{V} = \frac{P}{RT}$

Molar Mass of a Gas

• Molar mass can be determined by measuring mass and volume under known pressure and temperature.
• The ideal gas law is used to find the amount in moles.
• Molar mass is calculated by dividing mass (in grams) by amount (in moles).
• $Molar \; mass = \frac{mass}{moles} = \frac{m}{n}$

Mixtures of Gases

• Many gas samples are mixtures.
• Dry air is a mixture of nitrogen (~78%), oxygen (~21%), argon (~0.9%), carbon dioxide (~0.04%), and trace gases.

Treating Gas Mixtures

• In some applications, mixtures can be treated as one gas.
• Even though air is a mixture, pressure, volume, and temperature can be measured as if it were a pure substance.
• The total moles of molecules can be calculated using P, V, and T.

Partial Pressure

• The pressure of an individual component in a gas mixture is its partial pressure.
• Partial pressure can be calculated using the ideal gas law, assuming each gas acts independently.
• $P<em>n = n</em>n \frac{RT}{V}$

Calculating Partial Pressure

• Partial pressure can be calculated if:
  • The fraction of the mixture it composes and the total pressure are known.
  • The number of moles of the gas in a container of known volume and temperature are known.
• The sum of partial pressures equals the total pressure (Dalton’s law of partial pressures).
  • $P<em>{total} = P</em>a + P<em>b + P</em>c + …$
• Gases behave independently.

Dalton's Law of Partial Pressures

• Partial pressure of each component is calculated from the ideal gas law and the number of moles of that component.
• $P<em>a = n</em>a \frac{RT}{V}$, $P<em>b = n</em>b \frac{RT}{V}$, $P<em>c = n</em>c \frac{RT}{V}$, …
• The sum of partial pressures equals the total pressure.
• $P<em>{total} = P</em>a + P<em>b + P</em>c + …$

Mole Fraction

• The ratio of the partial pressure a single gas contributes, and the total pressure is equal to the mole fraction.
• The number of moles of a component in a mixture divided by the total number of moles in the mixture is the mole fraction.
• $\chi<em>a = \frac{n</em>a}{n<em>{total}}$, where $\chi</em>a$ is the mole fraction of component a, $n<em>a$ is the number of moles of component a, and $n</em>{total}$ is the total number of moles in the mixture
• $\chi<em>a = \frac{P</em>a}{P_{total}}$

Mole Fraction and Partial Pressure

• Partial pressure of a component is its mole fraction multiplied by the total pressure.
• For gases, the mole fraction of a component is its percent by volume divided by 100%.

Collecting Gases

• Gases are often collected by water displacement.
• Collected gas contains water vapor due to evaporation.
• The partial pressure of water vapor (vapor pressure) depends only on temperature.
• Use a table to find the vapor pressure of water at a given temperature.
• $P<em>{dry gas} = P</em>{total} - P<em>{H</em>2O}$

Gases in Chemical Reactions

• In reactions with gaseous reactants or products, the quantity of a gas is specified in terms of its volume at a given temperature and pressure.
• Stoichiometry involves relationships between amounts in moles.
• The ideal gas law is used to determine amounts in moles from volumes, or vice versa.
• $n = \frac{PV}{RT}$

Reactions Involving Gases

• Pressures can be partial pressures.
• At STP, use 1 mole = 22.4 L.

Kinetic Molecular Theory

• The simplest model for the behavior of gases.
• A gas is modeled as a collection of particles (molecules or atoms) in constant motion.

Basic Postulates of Kinetic Molecular Theory

• The size of gas molecules is negligibly small.
• The average kinetic energy of a particle is proportional to the temperature in kelvins.
• Collisions between particles or with the container walls are completely elastic.

More on Kinetic Molecular Theory

• Gas particles are constantly moving.
• Attraction between particles is negligible.
• Particles bounce off each other and the container walls without sticking.
• There is a lot of empty space between gas particles.

Average Kinetic Energy

• The average kinetic energy of gas particles is directly proportional to the Kelvin temperature.
• Increasing temperature increases the average speed of particles.
• Not all particles move at the same speed.

Elastic Collisions

• Collisions are completely elastic; energy may be exchanged, but there is no overall loss.
• Kinetic energy lost by one particle is completely gained by the other.

Nature of Pressure

• Constantly moving gas particles strike the container walls with a force.
• The collective force exerted by many particles results in constant pressure.
• $P = \frac{F}{A}$

Gas Laws Explained - Boyle's Law

• Volume is inversely proportional to pressure at constant number of particles and temperature.
• Decreasing volume forces molecules into a smaller space.
• More frequent collisions increase pressure.

Gas Laws Explained - Charles's Law

• Volume is directly proportional to absolute temperature at constant number of particles and pressure.
• Increasing temperature increases the average speed and kinetic energy of particles.
• The greater volume spreads collisions over a larger surface area, maintaining constant pressure.

Gas Laws Explained - Avogadro's Law

• Volume is directly proportional to the number of gas molecules at constant temperature and pressure.
• Increasing the number of gas molecules increases collisions on the walls.
• Volume must increase to keep the pressure constant.

Gas Laws Explained - Dalton's Law

• The total pressure of a gas mixture is the sum of the partial pressures of its components.
• Particles have negligible size and do not interact.
• Particles of different masses have the same average kinetic energy at a given temperature.
• The total pressure of collisions is the same.

Kinetic Molecular Theory and the Ideal Gas Law

• Kinetic molecular theory implies PV = nRT.
• Pressure on a container wall is the total force due to collisions divided by the wall area.
• $P = \frac{F_{total}}{A}$

Temperature and Molecular Velocities

• Average kinetic energy depends on average mass and velocity.
• Gases in the same container have the same temperature and average kinetic energy.
• Lighter particles have a faster average velocity than more massive particles.
• $\frac{1}{2}m u_{rms}^2$
• $u_{rms} = \sqrt{\frac{3RT}{M}}$

Root Mean Square Velocity and Temperature

• As temperature increases, the average velocity increases.

Molecular Speed Versus Molar Mass

• Heavier molecules must have a slower average speed to have the same average kinetic energy.

Molecular Velocities Versus Temperature

• As temperature increases, the velocity distribution shifts toward higher velocity.
• The distribution function spreads out, resulting in more molecules with faster speeds.

Mean Free Path

• Molecules travel in straight lines until they collide.
• The average distance a molecule travels between collisions is the mean free path.
• Mean free path decreases as pressure increases.

Diffusion and Effusion

• Diffusion: molecules spreading from high to low concentration.
• Effusion: molecules escaping through a small hole into a vacuum.
• Rates of diffusion and effusion are related to root means square average velocity.
• At the same temperature, gas movement rate is inversely proportional to the square root of its molar mass.

Graham’s Law of Effusion

• Relates the rates of effusion of two different gases at the same temperature.
• $\frac{rate<em>A}{rate</em>B} = \sqrt{\frac{M<em>B}{M</em>A}}$