Gas Laws Overview
States of Matter
Four States: Gas, Liquid, Solid, Plasma
Phase Transition States: Gas, Liquid, Solid (interconvertible at constant temperature)
Plasma: Formed from gas with continuously increasing temperature (not phase transition)
Gas Properties: Simplest state, no definite shape/volume, high compressibility
Historical Timeline
1662: Robert Boyle – Boyle’s Law (P-V relationship)
1787: Jacques Charles – Charles’s Law (V-T relationship)
1802: Joseph Louis Gay-Lussac – Pressure-Temperature Law
1811: Amedeo Avogadro – Avogadro’s Law (V-n relationship)
1834: Émile Clapeyron – Ideal Gas Law (PV=nRT)
1873: Johannes van der Waals – Real Gas Equation
Key Concepts
• Pressure-Volume-Temperature relationships
• Kinetic Molecular Theory (KMT) postulates
• Root mean square velocity (Crms)
• Real vs Ideal gas behavior
• Van der Waals corrections
• Compressibility factor (Z)
• Absolute zero and Kelvin scale
Learning Objectives
✓ Understand gas laws mathematically and graphically
✓ Apply PV = nRT to solve problems
✓ Explain gas behavior using Kinetic Theory
✓ Differentiate real vs ideal gases
✓ Use Van der Waals equation for real gases
✓ Calculate gas density and RMS velocity
✓ Interpret compressibility factor graphs
Properties of Gases
General Properties
Mass: Definite mass
Shape: No definite shape (takes container shape)
Volume: No definite volume (fills entire container)
Forces: Negligible intermolecular forces
Density: Very low compared to solids/liquids
Motion: High translational, rotational, vibrational motion
Packing: No proper packing, large empty spaces
Energetic Properties
Kinetic Energy: Very high (molecules move rapidly)
Thermal Expansion: High coefficient of expansion
Compressibility: Highly compressible (large empty spaces)
Diffusion: Spontaneous intermixing (Graham’s Law)
Effusion: Escape through small openings
Pressure: Exert pressure on container walls due to collisions
Temperature Dependence: Properties change significantly with temperature
Comparison with Solids & Liquids
| Property | Gas | Liquid | Solid |
|---|---|---|---|
| Shape | Indefinite | Definite (container) | Definite |
| Volume | Indefinite | Definite | Definite |
| Density | Very low | High | Highest |
| Compressibility | High | Very low | Negligible |
Units & Conversions
Pressure Conversions:
1 atm = 760 mmHg = 760 torr = 101325 Pa = 1.01325 bar = 14.7 psi
Volume Conversions:
1 m³ = 1000 dm³ = 1,000,000 cm³
1 dm³ = 1 L = 1000 cm³ = 0.001 m³
1 cm³ = 1 mL = 0.001 dm³ = 10⁻⁶ m³
Energy Conversions:
1 cal = 4.184 J
1 J = 0.239 cal = 10⁷ ergs
1 Nm = 1 J
Fundamental Gas Laws
Boyle’s Law (1662)
Statement: Volume of given mass of ideal gas is inversely proportional to pressure at constant temperature.
Mathematical Form: P₁V₁ = P₂V₂ = k (constant)
Formula: P ∝ 1/V (T constant)
Experimental Setup: Weight on movable piston decreases volume
Graphs:
- P vs V: Rectangular hyperbola
- P vs 1/V: Straight line through origin
- PV vs P: Straight line parallel to P-axis
Application: Breathing, syringe operation, soda cans
Charles’s Law (1787)
Statement: Volume of given mass of ideal gas is directly proportional to absolute temperature at constant pressure.
Mathematical Form: V₁/T₁ = V₂/T₂ = k
Formula: V ∝ T (P constant)
Experimental Setup: Heating gas in cylinder with movable piston
Key Point: Volume increases by 1/273 of original volume at 0°C for every 1°C rise
Absolute Zero: -273.15°C (0 K) where volume becomes zero
Graphs: V vs T: Straight line intersecting T-axis at -273.15°C
Avogadro’s Law (1811)
Statement: Equal volumes of all ideal gases contain equal number of molecules at same temperature and pressure.
Mathematical Form: V₁/n₁ = V₂/n₂ = k
Formula: V ∝ n (P, T constant)
Experimental Setup: Adding more gas moles increases volume in closed cylinder
Avogadro’s Number: 6.022 × 10²³ molecules/mole
Molar Volume: 22.414 L at STP (0°C, 1 atm)
Application: Stoichiometry of gases, determining molecular formulas
Ideal Gas Equation
Combining All Laws: PV = nRT
R Values:
- 0.0821 L·atm·K⁻¹·mol⁻¹
- 8.314 J·K⁻¹·mol⁻¹
- 62.36 L·torr·K⁻¹·mol⁻¹
- 1.987 cal·K⁻¹·mol⁻¹
Density Formula: d = PM/RT
Relationships:
- d ∝ P (T constant)
- d ∝ M (P, T constant)
- d ∝ 1/T (P constant)
Memory Aid: “PV = nRT” → “People Vote = Not Really True”
Gas Law Formulas Summary
Charles’s Law: V₁/T₁ = V₂/T₂ (P constant)
Avogadro’s Law: V₁/n₁ = V₂/n₂ (P, T constant)
Ideal Gas Law: PV = nRT
Combined Gas Law: P₁V₁/T₁ = P₂V₂/T₂
Density: d = PM/RT
Gas Constant (R) Values
| Pressure Units | Volume Units | Value of R |
|---|---|---|
| atm | L (dm³) | 0.0821 L·atm·K⁻¹·mol⁻¹ |
| atm | mL (cm³) | 82.05 cm³·atm·K⁻¹·mol⁻¹ |
| mmHg (torr) | L | 62.36 L·torr·K⁻¹·mol⁻¹ |
| Pa (SI) | m³ (SI) | 8.314 J·K⁻¹·mol⁻¹ |
| bar | L | 0.0831 L·bar·K⁻¹·mol⁻¹ |
Kinetic Molecular Theory
Key Scientists
Bernoulli (1738): Founder of KMT – proposed gas pressure results from molecular collisions
Clausius (1857): Derived kinetic equation and deduced all gas laws from KMT
Maxwell (1860): Presented Maxwell-Boltzmann distribution of molecular velocities
Boltzmann (1871): Studied distribution of energies among molecules
Van der Waals (1873): Corrected ideal gas equation for real gases
Perrin (1908): Experimental verification of KMT (Nobel Prize 1926)
Postulates of KMT
- Gases consist of tiny particles (atoms/molecules)
- Particles move randomly in straight lines with constant velocity
- Particle volume is negligible compared to container volume
- No intermolecular forces except during collisions
- Collisions are perfectly elastic (no energy loss)
- Average kinetic energy ∝ absolute temperature (K)
- Pressure results from molecular collisions with walls
- Gravity effects are negligible compared to collision effects
Molecular Velocities
Root Mean Square Velocity:
Crms = √(3RT/M)
Average Velocity: Cav = √(8RT/πM)
Most Probable Velocity: Cmp = √(2RT/M)
Relationship: Cmp : Cav : Crms = 1 : 1.128 : 1.224
Temperature Dependence: C ∝ √T
Molecular Mass Dependence: C ∝ 1/√M
Example (He at 300K): Crms ≈ 1360 m/s
Kinetic Energy & Temperature
Average Translational KE per molecule:
KEavg = (3/2)kT
where k = R/NA = 1.38 × 10⁻²³ J/K (Boltzmann constant)
Total KE for n moles: KEtotal = (3/2)nRT
Temperature Interpretation:
- Gases/Liquids: Measure of average translational KE
- Solids: Measure of average vibrational KE
Heat Flow: Transfer of KE from hotter to colder body
Absolute Zero (0 K): Molecular motion ceases (theoretically)
Kinetic Gas Equation
where: P = Pressure, V = Volume, N = Number of molecules
m = Mass of one molecule, Crms = Root mean square velocity
Real vs Ideal Gases
Ideal Gases
Definition: Gases that obey PV = nRT under all conditions
Assumptions:
- Molecules have negligible volume
- No intermolecular forces
- Perfectly elastic collisions
- Follow all gas laws strictly
Examples: No real gas is perfectly ideal
Best Approximations: H₂, He, Ne at high T, low P
Cannot be liquefied (no intermolecular forces)
Z = 1 (Compressibility factor)
Real Gases
Definition: Gases that deviate from ideal behavior
Deviations occur because:
- Molecules have actual volume
- Intermolecular forces exist (attractive/repulsive)
- Collisions may be inelastic
Conditions for Deviation:
- High pressure
- Low temperature
- Near liquefaction point
Can be liquefied (have intermolecular forces)
Z ≠ 1 (Deviation from ideality)
Compressibility Factor (Z)
Definition: Z = PV/RT
For Ideal Gas: Z = 1 (always)
For Real Gases:
- Z < 1 at moderate P: Attractive forces dominant
- Z > 1 at high P: Repulsive forces (molecular volume) dominant
- Z → 1 at low P, high T: Approaches ideal behavior
Boyle Temperature (TB): Temperature where Z ≈ 1 over wide P range
Memory Aid: “Z < 1 → Attractive, Z > 1 → Repulsive”
Van der Waals Equation
For n moles:
[P + a(n/V)²] (V – nb) = nRT
For 1 mole: (P + a/V²)(V – b) = RT
Corrections:
- a: Correction for intermolecular attraction
- b: Correction for molecular volume
Units:
- a: atm·L²·mol⁻² or Pa·m⁶·mol⁻²
- b: L·mol⁻¹ or m³·mol⁻¹
For ideal gas: a = 0, b = 0 → PV = RT
Van der Waals Constants for Common Gases
| Gas | a (atm·L²·mol⁻²) | b (L·mol⁻¹) | Ease of Liquefaction |
|---|---|---|---|
| He | 0.034 | 0.0237 | Most difficult |
| H₂ | 0.244 | 0.0266 | Difficult |
| N₂ | 1.39 | 0.0391 | Moderate |
| O₂ | 1.36 | 0.0318 | Moderate |
| CO₂ | 3.59 | 0.0427 | Easier |
| NH₃ | 4.17 | 0.0371 | Easy |
| H₂O | 5.46 | 0.0305 | Easiest |
Interactive Gas Law Simulator
Explore Gas Behavior in Real-Time
Observations:
- Increase Pressure: Volume decreases (Boyle’s Law)
- Increase Temperature: Volume increases (Charles’s Law)
- Increase Moles: Volume increases (Avogadro’s Law)
- Heavier Gas: Lower RMS velocity, higher density
- High P, Low T: Z deviates from 1 (Real gas behavior)
Memory Aids & Problem-Solving Tips
Gas Laws Mnemonics
Boyle’s Law: “Pressure up, Volume down” (P-V inverse)
Charles’s Law: “Hot air rises” (V-T direct)
Avogadro’s Law: “More gas, more space” (V-n direct)
Ideal Gas Law: “PV = nRT” → “People Vote = Not Really True”
R Values: 0.0821 (for atm·L), 8.314 (for J), 62.4 (for torr·L)
Problem Solving Steps
1. Identify: Which variables change? Which are constant?
2. Choose Law: Constant T → Boyle’s; Constant P → Charles’s; All change → Combined or Ideal
3. Convert Units: T to Kelvin, P to consistent units, V to liters
4. Write Equation: P₁V₁/T₁ = P₂V₂/T₂ or PV = nRT
5. Solve Algebraically: Rearrange for unknown variable
6. Check: Does answer make sense? Units correct?
Common Mistakes to Avoid
❌ Using Celsius instead of Kelvin for temperature
❌ Forgetting to convert pressure units consistently
❌ Confusing STP (0°C, 1 atm) with SATP (25°C, 1 bar)
❌ Using wrong R value for given units
❌ Assuming real gases are ideal at high P/low T
❌ Forgetting molecular mass in RMS velocity calculation
❌ Confusing Crms, Cav, Cmp
Graph Interpretation Tips
Boyle’s Law: P vs V = hyperbola; P vs 1/V = straight line
Charles’s Law: V vs T = straight line intersecting T-axis at -273°C
Compressibility (Z):
• Z < 1: Attractive forces dominant (below ideal line)
• Z > 1: Repulsive forces dominant (above ideal line)
• Z = 1: Ideal behavior (straight line at Z=1)
Maxwell-Boltzmann: Distribution shifts right with T, lower peak for heavier gases
Experimental Connections
Boyle’s Experiment: Mercury column with trapped air
Charles’s Experiment: Gas thermometer with movable mercury plug
RMS Velocity Evidence: Brownian motion, diffusion rates
Real Gas Behavior: Liquefaction, critical phenomena
Absolute Zero: Extrapolation of volume vs temperature graphs
Avogadro’s Number: Determined from electrolysis, Brownian motion
Van der Waals Constants Logic
‘a’ (Attraction Correction):
• Larger for easily liquefiable gases (H₂O > NH₃ > CO₂ > N₂ > H₂ > He)
• Units: Pressure·Volume²·mol⁻²
‘b’ (Volume Correction):
• Larger for bigger molecules
• b ≈ 4 × actual molecular volume
• Units: Volume·mol⁻¹
Memory: “a for attraction, b for bulk (volume)”