The Kinetic Theory of Gases explains the behavior of gases based on the motion of their molecules. It connects microscopic properties (molecules) with macroscopic quantities (pressure, temperature, volume).
👉 Core Idea: Gas properties arise from the random motion of a large number of molecules.
1. Ideal Gas
Definition
An ideal gas is a hypothetical gas that perfectly follows gas laws under all conditions.
Ideal Gas Equation
PV = nRT
Where:
- P = pressure
- V = volume
- n = number of moles
- R = universal gas constant
- T = temperature (in Kelvin)
2. Gas Laws
Boyle’s Law (Constant Temperature)
P ∝ 1/V
PV = constant
Charles’ Law (Constant Pressure)
V ∝ T
Gay-Lussac’s Law
P ∝ T
Avogadro’s Law
V ∝ n
3. Kinetic Theory of Gases (Assumptions)
- Gas consists of a large number of molecules
- Molecules are in random motion
- Collisions are perfectly elastic
- Volume of molecules is negligible
- No intermolecular forces
4. Pressure of Gas (Derivation Idea)
Pressure arises due to collisions of gas molecules with container walls.
Formula
P = (1/3) ρ v²
Where:
- ρ = density
- v = RMS speed
5. RMS Speed (Very Important)
Definition
Root mean square speed is the square root of the average of squares of speeds.
Formula
v_rms = √(3RT/M)
Where:
- R = gas constant
- T = temperature
- M = molar mass
Concept Clarity
👉 Higher temperature → higher molecular speed
👉 Lighter gas → higher speed
6. Average and Most Probable Speeds
Most Probable Speed
v_mp = √(2RT/M)
Average Speed
v_avg = √(8RT/πM)
Relation
v_rms > v_avg > v_mp
7. Temperature (Microscopic View)
Temperature is a measure of average kinetic energy of molecules.
Formula
KE_avg = (3/2)kT
Where k = Boltzmann constant
8. Degrees of Freedom
Definition
Number of independent ways a molecule can store energy.
Examples
- Monoatomic gas → 3
- Diatomic gas → 5
- Polyatomic gas → 6
9. Law of Equipartition of Energy
Statement
Energy is equally distributed among all degrees of freedom.
Energy per Molecule
E = (f/2)kT
Where f = degrees of freedom
10. Specific Heat Capacity
Relation
γ = Cp/Cv
For Monoatomic Gas
γ = 5/3
11. Mean Free Path
Definition
Average distance travelled by molecule between collisions.
Formula
λ = 1 / (√2 π d² n)
Where:
- d = diameter of molecule
- n = number density
12. Real Gases vs Ideal Gases
Real Gases
- Intermolecular forces present
- Do not follow gas laws exactly
Ideal Gases
- No forces
- Follow gas laws perfectly
Important Numericals
Numerical 1
Find RMS speed at 300 K for gas with M = 0.028 kg/mol
v_rms = √(3RT/M)
Numerical 2
Find kinetic energy at 300 K
KE = (3/2)kT
Numerical 3
Find pressure using formula
P = (1/3)ρv²
Numerical 4
Find degrees of freedom of diatomic gas
Answer: 5
Important Formula Sheet
- PV = nRT
- v_rms = √(3RT/M)
- KE = (3/2)kT
- P = (1/3)ρv²
- γ = Cp/Cv
Concept Clarity (Important)
👉 WHY gases exert pressure?
Due to collisions of molecules with container walls.
👉 WHY temperature increases speed?
Because kinetic energy increases.
👉 WHY ideal gas is hypothetical?
Because real gases always have intermolecular forces.
Common Mistakes
- Using temperature in °C instead of Kelvin
- Confusing speeds (v_rms, v_avg, v_mp)
- Ignoring molar mass units
Conclusion
Kinetic Theory of Gases connects microscopic motion with macroscopic properties. Understanding this chapter helps in mastering thermodynamics and real gas behavior.
👉 Focus on concept clarity + formulas + numerical practice.