Unit 8

Properties of Gases Class Outline

• 1. Pressure, Volume, Temperature

• 2. Simple Gas Laws

• 3. The Ideal Gas Law

• 4. Applications of the combined gas law

• 5. Molar Volume, Density, and Molar Mass of a Gas

• 6. Mixtures of Gases and Partial Pressures

• 7. The kinetic molecular theory for gases

• 8. Diffusion, Effusions, and Real Gases

Properties that Describe Gases

• Four Basic Properties of Gases: - These properties include Pressure, Volume, Temperature, and the number of moles.

  • These properties are interrelated; a change in one affects the others.

Gas Pressure

• Definition and Characteristics: - Pressure is defined as the force per unit area, measured in SI units of N/m² or Pa.

  • Pressure is a characteristic property of all gases (and liquids).

• Pressure Units: - Atmospheres (atm)

  • Torr

  • mmHg

  • Definitions and Equivalence:

  • 1 atm = 760 mmHg = 760 torr = 14.7 lbs/in² = 101 kPa

Exerting Pressure

• Analogy for Understanding Pressure: - Even with a heavy weight (like an elephant), a figure skater exerts a higher pressure due to the smaller surface area of her skates.

The Simple Gas Laws

• Overview: - The simple gas laws describe relationships between pairs of properties while keeping the others constant.

• Key Laws: - Boyle’s Law:

  • $P \times V = \text{constant}$

  • States that as pressure increases on a fixed amount of gas at constant temperature, the volume decreases (inversely related).

  • Charles’s Law:

  • $V = \text{constant} \times T$

  • States that as temperature increases on a fixed amount of gas at constant pressure, the volume increases (directly related).

  • Gay-Lussac’s Law:

  • $P = \text{constant} \times T$

  • Pressure is directly proportional to temperature at constant volume.

  • Avogadro’s Law:

  • $V = \text{constant} \times n$

  • States that volume is directly related to the number of moles of gas when pressure and temperature are constant.

Boyle’s Law Example

• Example Calculation: - Initial volume = 2.5 L, Final volume = 5.0 L, Initial pressure = 3.8 atm, Final pressure = ?

• Equation: - $P<em>1V</em>1 = P<em>2V</em>2$

Charles’s Law Example

• Example Calculation: - A 0.100 L volume of gas is heated from 25°C to 100°C. Calculate new volume.

• Equation: - $V<em>1 / T</em>1 = V<em>2 / T</em>2$

Gay-Lussac’s Law

• Pressure and Temperature Relationship: - Pressure of a gas is directly proportional to its temperature in Kelvin when volume and the number of moles remain constant.

• Example Context: - When the hot plate is off, the pressure of gas is low; heating increases pressure.

Combined Gas Law Example

• Example Calculation: - A sample of fluorine gas, $0.180 L$, at $0.800 atm$ and $29°C$. Determine the new temperature at a volume of $90.0 mL$ and $3.20 atm$.

• Equation: - $P<em>1V</em>1 / T<em>1 = P</em>2V<em>2 / T</em>2$

Avogadro’s Law Example

• Example Calculation: - A $4.8 L$ sample of helium with $0.22$ moles. Determine additional moles to reach $6.4 L$.

• Equation: - $V<em>1 / n</em>1 = V<em>2 / n</em>2$

Ideal Gas Law

• Formulation: - The ideal gas law is given by $PV = nRT$

  • Alternatively written as:

  • $P<em>1V</em>1 / n<em>1T</em>1 = P<em>2V</em>2 / n<em>2T</em>2 = R$

• Value of R: - R is derived under standard conditions:

  • At standard temperature (STP) of $1 atm$ and $0°C$, $R = 0.0821 L \times atm / (mol \times K)$ or alternatively $R = 8.314 J/(mol \times K)$

Standard Conditions

• Definition of Standard Conditions: - A set of reference conditions for reporting measurements:

  • Standard pressure = $1 atm$

  • Standard temperature = $273.15 K$ (or $0°C$)

  • Standard volume = $22.4 L$ for one mole of any gas at STP.

Calculating Mass & Molar Mass (M) of a Gas

• The mass, molar mass, and density can be determined by manipulating the ideal gas equation.

• Procedure: - Heat a weighed sample until it becomes a gas, measure the temperature, pressure, and volume, then use the ideal gas law to find molar mass:

  • Rearranged:

  • $M = \frac{dPV}{RT}$

Ideal Gas Law Examples

• Example 1: - Cylinder of argon contains 58.0 L at 180 atm at 25°C. Calculate moles.

• Example 2: - 500 mL vessel with 2.00 g of an unknown compound at 127°C, unter pressure 0.871 atm. Calculate molar mass.

Ideal Gas Law Applications in Auto Air Bags

• Chemical Reaction: - $2 NaN<em>3 (s) \rightarrow 2 Na (s) + 3 N</em>2 (g)$

  • Combusting sodium and other materials to generate nitrogen gas.

• Example Calculation: - Using 85.0 g of NaN3 in a 25.0 L airbag at $25°C$, calculate final bag pressure.

Gas Stoichiometry

• Relationship: - Volumes of gases are proportional to their number of moles if measured at the same pressure and temperature.

• Example: - Determine volume of $NH<em>3$ produced from 50 L of $H</em>2$ based on reaction:

  • $N<em>2(g) + 3 H</em>2(g) \rightarrow 2 NH_3(g)$.

Dalton’s Law of Partial Pressures

• Definition: - The total pressure of a gas mixture equals the sum of the partial pressures of its components.

• Equation: - For two gases A and B under constant T & V:

  • $P<em>{\text{total}} = P</em>A + P_B$

• Example Problem: - Gases collected by water displacement, calculating collected volume and mass of hydrogen under specific conditions.

Mole Fraction (χ)

• Definition: - The mole fraction of a component is the ratio of the moles of that component to the total moles in the mixture:

  • $\text{Mole fraction of A} = \frac{n<em>A}{n</em>A + n<em>B + n</em>C}$

• Context: - The sum of all mole fractions equals to 1.

Kinetic Molecular Theory (KMT) of Gases

• Five Assumptions (Postulates): 1. Particles are in continuous random motion.

  2. Distance between particles is significantly greater than their size (valid at low pressure).

  3. Attraction between particles is negligible.

  4. Kinetic energy of the particles is directly proportional to Kelvin temperature.

  5. Collisions between particles are elastic, with no energy loss (theoretically true).

• Ideal vs Real Gases: - Ideal gases obey KMT; no gas is perfectly ideal, but helium is considered closest.

Pressure Experiments and KMT

• How Pressure Affects Gases:

• Effects on Pressure: - More moles (n) increase pressure due to more collisions.

  • Higher temperature (T) increases pressure due to increased kinetic energy.

  • Decreased volume (V) results in increased pressure due to concerted collisions.

Gas Velocity Example

• Determining Average Velocity: - $\text{u}_{\text{rms}} = \sqrt{\frac{3RT}{M}}$

• Constants: - Where M is molar mass in kg/mol.

• Unit Conversion: - Note: $R = 0.0821 L \times atm / (mol \times K)$ or $R = 8.314 J/(mol \times K)$

Diffusion & Effusion Explanations

• Definitions: - Diffusion: Random dispersion of molecules leading to equal distribution.

  • Effusion: The passage of gas through a small opening.

• Driving Force: - Both processes are driven by entropy.

  • KMT asserts all gases have the same kinetic energy at given temperatures.

• Graham's Law: - The rate of diffusion or effusion is inversely proportional to the square root of their molar masses for two gases, A and B:

  • $\frac{Rate<em>A}{Rate</em>B} = \sqrt{\frac{M<em>B}{M</em>A}}$

Graham’s Law of Diffusion or Effusion

• Key Relation: - $Rate \propto \frac{1}{\sqrt{M}}$

• Example Application: - Utilized in uranium enrichment processes for atomic bomb production.

Real Gases Behavior

• Ideal Gas Law Assumptions: - No interactions between gas molecules and they occupy no space.

• Real Gases: - Deviate from ideal assumptions at low temperatures and high pressures due to finite molecular volume and intermolecular forces.

• Molar Volume Deviations: - Real gases might not occupy the ideal gas volume of $22.41 L$ per mole at STP.

Real Gases and Deviations from Ideality

• Causes of Deviations: - Highly complex particles exhibit more attractive forces and collision inelasticity.

  • High pressures bring particles closer, while low temperatures enhance attractions.

• Adjustments to Ideal Gas Law: - Two corrections for particle volume and attraction:

  • $P<em>{\text{ideal}} = P</em>{\text{real}} + Y$

  • $V<em>{\text{ideal}} = V</em>{\text{real}} - X$

• Van der Waals Equation: - Used for real gases to correct behaviors.

Summary of Key Concepts for Unit VIII

• Familiarity with KMT, all gas laws including Boyle’s, Charles’, Combined, Avogadro’s, and Ideal

• Understanding Dalton’s law of partial pressures, mole fractions