Kinetic Molecular Theory and Molecular Speeds

• State and apply the postulates of Kinetic Molecular Theory (KMT)
• Derive the expression for pressure from molecular collisions
• Calculate root-mean-square, average, and most-probable speeds
• Know the ratio u_rms : u_avg : u_mp and what it means physically
• Interpret the Maxwell-Boltzmann speed distribution and how temperature shifts it
• Apply Graham's law of effusion to compare gas speeds and separation

Level 1 — Foundations

Postulates of KMT:

1. Gases consist of large numbers of molecules in continuous, random motion.
2. Volume of individual molecules is negligible compared to total gas volume.
3. Intermolecular forces are negligible (molecules move freely between collisions).
4. Collisions between molecules and container walls are perfectly elastic (no energy loss).
5. Average kinetic energy of molecules is directly proportional to absolute temperature (T in K).

Average KE per molecule: KE_avg = (3/2) kT

where k = Boltzmann constant = 1.38 × 10⁻²³ J/K

Per mole: KE_avg = (3/2) RT (R = 8.314 J/mol·K)

Level 2 — JEE depth

Derivation sketch — pressure from molecular motion: Consider N molecules of mass m in a cubic box of side L.

For one molecule moving in x-direction with speed vₓ:

• Momentum change per collision with wall = 2mvₓ
• Time between successive collisions = 2L/vₓ
• Force on wall by one molecule = mv²ₓ/L

Sum over all N molecules and all three directions (using v² = vₓ² + vy² + vz²): P = Nm⟨v²⟩ / 3V

This gives: PV = (1/3)mN⟨v²⟩ = (1/3)Mu²_rms (for 1 mol)

Comparing with PV = RT: u_rms = √(3RT/M)

This confirms KMT is consistent with the ideal gas equation.

Three characteristic speeds (for 1 mole of gas, molar mass M):

Ratio: u_rms : u_avg : u_mp = √3 : √(8/π) : √2 ≈ 1.732 : 1.596 : 1.414

In all formulas M must be in kg/mol when R = 8.314 J/mol·K.

Maxwell-Boltzmann distribution:

• The distribution of molecular speeds is a skewed bell curve.
• At higher T: curve shifts right (higher speeds), flattens, peak (u_mp) moves to higher value.
• At higher M: curve shifts left (lower speeds for heavier gas at same T).
• Area under curve is always 1 (total probability).

Graham's Law of Effusion: rate of effusion ∝ 1/√M (at same T and P)

r₁/r₂ = √(M₂/M₁) = √(d₂/d₁)

Applications: separating isotopes (e.g., UF₆ enrichment), detecting gas leaks.

JEE trap: Use M in kg/mol in speed formulas with R = 8.314 J/mol·K to get speed in m/s. If M is given in g/mol, divide by 1000.

JEE trap: Graham's law gives rate ratio, not speed ratio directly — but effusion rate ∝ average speed, so the ratio is the same.

Find u_rms for N₂ at 27°C

M(N₂) = 28 g/mol = 0.028 kg/mol
T = 27 + 273 = 300 K
R = 8.314 J mol⁻¹ K⁻¹

u_rms = √(3RT/M)
      = √(3 × 8.314 × 300 / 0.028)
      = √(7482.6 / 0.028)
      = √(267235.7)
      = 516.9 m/s ≈ 517 m/s

Answer: u_rms(N₂) at 27°C ≈ 517 m/s

Which effuses faster, H₂ or O₂? Find the rate ratio.

Graham's law: r(H₂)/r(O₂) = √(M(O₂)/M(H₂))
            = √(32/2)
            = √16
            = 4

H₂ effuses 4 times faster than O₂.

This is why hydrogen balloons deflate much faster than air-filled ones.

• Q1: Find u_mp for O₂ at 27°C (M = 32 g/mol).
• Q2: At what temperature will u_rms of H₂ equal u_rms of N₂ at 300 K?
• Q3: A gas X effuses 2 times faster than SO₂ (M = 64). Find molar mass of X.
• Q4: If KE_avg per molecule at 300 K = (3/2)kT, find KE_avg in joules (k = 1.38×10⁻²³ J/K).
• Stretch: Q5: Two gases He (M=4) and Xe (M=131) are in the same container at 25°C. Find the ratio of their (a) u_rms values and (b) average KE per molecule. What does part (b) tell you about equipartition?

NCERT Chapter 5 link: Chapter 5 (Class 11) covers KMT postulates, derivation of PV = (1/3)mu²_rms, the three speeds, Maxwell distribution diagrams, and Graham's law with examples. Study the diagram showing how the Maxwell curve changes with temperature — it is often tested as an MCQ in JEE Mains.

Exam connections: JEE Mains asks numerical comparisons of speeds between gases, temperature at which one gas has the same u_rms as another, and rate-ratio problems via Graham's law. JEE Advanced may ask conceptual questions about the Maxwell distribution shape or what happens to the distribution when a reaction removes fast molecules.

Study strategy: Draw the Maxwell distribution curve yourself multiple times for different temperatures and different gases. Then practise the three speed formulas by deriving them from the single expression for PV; once you see they all come from the same root, you only need to remember which factor (2, 8/π, or 3) goes under the square root.

• Use the platform-native live simulation or PhET-style tool for this topic.
• Mirror / body / home activity: blow up two balloons, one with breath (CO₂/N₂ mix) and one with a lighter gas if available; observe deflation rates and record for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.