Kinetic Molecular Theory
Understanding gas behavior at the molecular level
Five Postulates of KMT
Particles in Constant Motion
Gas particles are in continuous, random, straight-line motion until they collide with each other or the container walls.
Negligible Volume
The volume of individual gas particles is negligible compared to the total volume of the container.
No Intermolecular Forces
There are no attractive or repulsive forces between gas particles (ideal gas assumption).
Elastic Collisions
All collisions between gas particles and between particles and walls are perfectly elastic (no energy loss).
Temperature Proportional to KE
The average kinetic energy of gas particles is directly proportional to absolute temperature (Kelvin).
Key Formulas
Average Kinetic Energy
KEavg = (3/2)RT
or
KEavg = (3/2)kBT
Where:
- • KEavg = average kinetic energy per mole (J/mol) or per particle (J)
- • R = 8.314 J/(mol·K) (per mole)
- • kB = 1.381 × 10⁻²³ J/K (Boltzmann constant, per particle)
- • T = temperature (K)
Root Mean Square (RMS) Speed
urms = √(3RT/M)
or
urms = √(3kBT/m)
Where:
- • urms = root mean square speed (m/s)
- • R = 8.314 J/(mol·K)
- • T = temperature (K)
- • M = molar mass (kg/mol) - MUST be in kg!
- • m = mass of one molecule (kg)
Relationship Between KE and Speed
KE = (1/2)mu²
Where:
- • m = mass of particle (kg)
- • u = speed of particle (m/s)
Worked Examples
Example 1: RMS Speed of O₂
Problem: Calculate the RMS speed of O₂ molecules at 25°C.
Solution:
Step 1: Convert temperature to Kelvin
T = 25 + 273.15 = 298.15 K
Step 2: Get molar mass in kg/mol
M(O₂) = 32.00 g/mol = 0.03200 kg/mol
Step 3: Apply formula
urms = √(3RT/M)
urms = √[(3)(8.314)(298.15) / 0.03200]
urms = √(232,214)
urms = 482 m/s
Example 2: Comparing Speeds
Problem: Which moves faster at the same temperature: H₂ or N₂?
Solution:
Since urms = √(3RT/M), and T is constant:
urms ∝ 1/√M
Lighter gas moves faster!
M(H₂) = 2.02 g/mol
M(N₂) = 28.02 g/mol
Speed ratio:
u(H₂)/u(N₂) = √[M(N₂)/M(H₂)]
u(H₂)/u(N₂) = √(28.02/2.02) = √13.87 = 3.72
H₂ moves 3.72× faster than N₂
Example 3: Average Kinetic Energy
Problem: Calculate the average KE per mole of gas at 300 K.
Solution:
KEavg = (3/2)RT
KEavg = (3/2)(8.314 J/mol·K)(300 K)
KEavg = 3741 J/mol
KEavg = 3.74 kJ/mol
Note: This is the same for ALL gases at 300 K!
Temperature Effects
Speed Distribution at Different Temperatures
Key Observations:
- • Higher T → Higher average speed
- • Higher T → Broader distribution (more variation in speeds)
- • Higher T → Lower peak (but wider curve)
- • At absolute zero (0 K), all molecular motion stops
Three Types of Molecular Speed:
1. Most probable speed (ump) = √(2RT/M)
2. Average speed (uavg) = √(8RT/πM)
3. RMS speed (urms) = √(3RT/M)
ump < uavg < urms
Common Mistakes
Wrong Units for Molar Mass
Must use kg/mol, not g/mol! M(O₂) = 0.032 kg/mol
Using Celsius Instead of Kelvin
Always use absolute temperature (K)!
Confusing KE per Mole vs per Particle
Use R for per mole, kB for per particle!
Memory Aid
Light gases (H₂, He) move FAST. Heavy gases (Xe, Rn) move SLOW.