Chapter 10: Gases

What Is a Gas?

• Gases are a phase of matter composed of particles moving randomly and very fast in their container(s).

• Gas particles travel in straight lines until they collide with a container wall or another particle; collisions cause bounce-backs.

• A gaseous atom or molecule exerts a force when it collides with a surface or with another gaseous particle.

• Molecular collisions contribute to pressure.

Gas Pressure and the Kinetic Theory of Gases

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

• Gas pressure arises from the constant movement and collisions of gas molecules with surfaces.

• Pressure depends on:

  • Number of gas particles in a given volume

  • Volume of the container

  • Average speed of gas particles

• Pressure definition (conceptual):

  • $P = \frac{F}{A}$

• Kinetic Theory postulates:

  • Collisions between gas particles and/or surfaces are elastic (no energy exchange).

  • If energy exchange occurs, collisions are inelastic.

Factors Upon Which Gas Pressure is Dependent

• Higher concentration of gas molecules → higher pressure.

• Fewer gas particles in a given volume → lower pressure.

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

• As volume increases (with a constant number of molecules), concentration decreases, leading to fewer molecular collisions and lower pressure.

Atmospheric Pressure Effects

• Variations in Earth’s atmospheric pressure drive wind and weather prediction.

• High-pressure (H) regions typically indicate clear weather; low-pressure (L) regions indicate unstable weather.

• Pressure decreases with increasing altitude because the number of gas particles in a given volume decreases with altitude.

Pressure Imbalance in the Ear

• A pressure difference across the eardrum pushes the membrane outward, causing a “popped eardrum.”

Pressure Units

• Common units:

  • Millimeter of mercury (mmHg) or Torr: 1 mmHg = 1 Torr

  • Atmosphere (atm)

• Conversions:

  • 1 mmHg = 1 Torr

  • 760 mmHg = 1 atm

  • 760 Torr = 1 atm

A Table of Pressure Units

• Pa = Pascal (1 N/m^2); 1 atm = 101,325 Pa

• psi = pounds per square inch; 1 atm ≈ 14.7 psi

• Torr (mmHg): 1 torr = 1 mmHg; 760 torr = 1 atm

• InHg (inches of mercury): 29.92 inHg = 1 atm

• 1 atm = 1.00 atm; 1.013 bar ≈ 1 atm

• $8.98\ \text{atm} \times \frac{760\ \text{mmHg}}{1\ \text{atm}} \approx 6.82 \times 10^3\ \text{mmHg}$

The Manometer

• A manometer measures gas pressure using a U-shaped tube partially filled with a liquid.

• One side connects to the gas sample; the other side is open to air.

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

• Example: If gas pressure > atmospheric pressure, the left side liquid level is higher.

Basic Properties of Gases

• Four basic properties (P, V, T, n) with units:

  • Pressure (P) in atm

  • Volume (V) in L

  • Temperature (T) in Kelvin (K) = Celsius T + 273

  • Amount in moles (mol)

• Kelvin temperature relation: $T(K) = T(^{\circ}C) + 273.15$

• These properties are interrelated; simple gas laws relate pairs of properties while keeping the other two constant.

The Simple Gas Laws (overview)

• Boyle’s Law: pressure and volume with constant temperature and amount

• Charles’s Law: volume and temperature at constant pressure and amount

• Avogadro’s Law: volume and amount (moles) at constant temperature and pressure

• These relationships are combined into the Ideal Gas Law:

• Ideal Gas Law: $PV = nRT$

Boyle’s Law: Volume and Pressure

• History: Boyle (1627–1691) and Hooke used a J-tube; increasing pressure by adding mercury decreases volume, showing an inverse relationship.

• Law: When T and n are constant, $P \propto \frac{1}{V}$ or $P V = \text{constant}$

• Graphical note:

  • P vs V yields a curve; P vs (1/V) yields a straight line.

• Formula: $P1\cdot V1=P2\cdot V2$

Charles’s Law: Volume and Temperature (Direct Relationship)

• Statement: For a fixed amount of gas at constant pressure, volume increases linearly with temperature in Kelvin.

• Relation: V​ = constant ⇒ V1/T1=V2/T2

• Molecular view: Increasing temperature increases average kinetic energy, faster collisions against container walls; to keep pressure constant, volume expands to spread collisions over a larger area.

• Temperature scale: The extrapolated lines to V = 0 converge at −273.15 °C = 0 K (absolute zero).

• Molecular view: as temperature rises, particles move faster; to maintain constant pressure, volume must increase; balloon example: moving from ice water to boiling water causes volume expansion.

Avogadro’s Law: Volume and Moles

• Direct relationship: volume is directly proportional to the number of gas molecules when P and T are constant.

• Equal volumes of gases contain equal numbers of molecules (molar volume is the same for all gases at the same T and P).

• Relationship: V = constant × n (moles)

  • V/n = constant

  • $(\frac{V1}{n1})=\left(\frac{V2}{n2}\right)$

Ideal Gas Law: PV = nRT

• Can combine Boyle’s, Charles’s, and Avogadro’s laws into one equation:

  • $PV = nRT$

• Given n, P, T, solve for V: $V = \frac{nRT}{P}$

• Verification: under STP, 1 mol occupies 22.4 L; approximate checks are allowed for ballpark validation.

Standard Conditions and Molar Volume

• Standard conditions (STP):

  • Standard pressure: $P = 1\ \text{atm}$

  • Standard temperature: $T = 273\ \text{K} = 0\,^\circ\text{C}$

  • Standard amount: $n = 1\ \text{mol}$

  • Standard volume: $V = 22.4\ \text{L}$

• The volume occupied by one mole of a substance at STP is its molar volume, independent of gas identity.

Molar Volume at STP

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

• Note: Different gases have different masses at the same molar volume due to differing molar masses.

Density of a Gas

• Definition: density d = mass/volume; for gases, typically in g/L.

• Using the ideal gas law, with n = m/M and P, T, V:

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

• Where M is molar mass in g/mol.

• At STP, 1 mol occupies 22.4 L, so density = M/22.4 g/L; example values:

  • He (M = 4.00 g/mol): $d_{\text{He}} = \frac{4.00}{22.4} \approx 0.179\ \text{g/L}$

Molar Mass of a Gas

• Determined by heating a weighed sample to gas, measuring T, P, V, applying the ideal gas law to find n, and using mass to calculate M = m/n.

Gas Density and Molar Mass (Relationship)

• Gas density is directly proportional to molar mass at fixed P and T.

• For any gas: $d = \frac{PM}{RT}$

• The molar density concept relates to the idea that density reflects molar mass per volume under the given conditions.

• Dalton’s Law of Partial Pressures states that the total pressure exerted by a mixture of non-reacting gases in a container is equal to the sum of the partial pressures that each gas would exert if it were alone in the container at the same temperature and volume.

  • Mathematically, Dalton's Law is expressed as: P total l=PA+PB+PC⋯

  • The Kinetic Molecular Theory provides the theoretical basis for Dalton's Law, as it postulates that gas components do not interact and have negligible size, meaning each gas independently contributes to the total pressure.

Partial Pressure: P gas

• The partial pressure ($P_i$) of a single gas in a mixture is the pressure that the gas would exert if it were the only gas present in the container.

• It can be calculated if either:

  • The fraction of the mixture it constitutes (mole fraction) and the total pressure are known, or

  • The number of moles (n) of the gas in the container (with given V and T) is known using the Ideal Gas Law.

Partial Pressure: P gas

• The pressure of a single gas in a mixture can be calculated if either:

  • The fraction of the mixture it constitutes and the total pressure are known, or

  • The number of moles of the gas in the container (with given V and T) is known.

O2 Kinetic Molecular Theory

• A gas is a collection of particles in constant motion.

• Postulates:

  • Particles are constantly moving.

  • Attractive forces between particles are negligible.

  • Collisions are elastic (no net loss of kinetic energy).

  • There is a lot of empty space between particles relative to their size.

Temperature and Molecular Velocities

• The average kinetic energy of gas particles is directly proportional to the Kelvin temperature:

  • $\text{KE}_{avg} = \frac{3}{2}RT$ (per mole)

• Heavier molecules must move more slowly to have the same KE as lighter molecules at the same T.

• RMS velocity, urms, relates to temperature and molar mass:

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

  • Here, M is the molar mass in kg/mol.

• Thus, heavier molecules have smaller rms at the same temperature.

Temperature and Molecular Velocities (continued)

• As temperature increases, the velocity distribution shifts toward higher speeds; more molecules move faster at higher T.

Mean Free Path and Diffusion/Effusion

• Mean free path is the average distance a molecule travels between collisions; it decreases as pressure increases.

• Diffusion: spreading of a collection of molecules from high to low concentration.

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

• For gases at the same T, rates of diffusion and effusion are related to rms velocity; faster gases diffuse/effuse more quickly.

Graham’s Law of Effusion

• Rates of effusion for two gases at the same temperature are inversely proportional to the square root of their molar masses:

  • $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$

Gases in Chemical Reactions: Stoichiometry Revisited

• In gas-phase reactions, the amounts are often given as volumes at a given T and P.

• Stoichiometric calculations with gases use PV = nRT to convert volumes to moles (and vice versa).

• When gases are at STP, 1 mol occupies 22.4 L.

• Pressures in a reaction could be partial pressures.

Reactions Involving Gases

• General plan: use PV = nRT to relate volumes and moles; apply stoichiometric ratios to convert between species.

• Example focus: use the mass of a gaseous product to determine the required volumes/pressures of reactants.

Van der Waals Equation and Real Gases

• Real gases deviate from ideal behavior at high pressures or low temperatures due to finite molecular size and intermolecular attractions.

• van der Waals modification introduces two constants a and b to account for these effects:

  • For n moles: $\left(P + a\left(\frac{n}{V}\right)^2\right)\left(V - nb\right) = nRT$

  • a accounts for intermolecular attractions (reduces pressure), and b accounts for finite molecular volume (reduces available volume).

• At low pressure, PV/RT for real gases is typically less than 1 due to attractions; at high pressure, PV/RT can be greater than 1 due to molecular volume effects.

VnReal Gases: PV/RT Plots

• PV/RT vs P plots show differences between real and ideal gases.

• For many real gases, PV/RT is below 1 at low P and above 1 at high P, illustrating the competing effects of attractions and finite volume.

• • $PV = nRT$