🔹 Kinetic Theory of Gases – Stage 1 (Page 9)

1️⃣ Maxwell–Boltzmann Distribution of Speeds

At a given temperature, gas molecules do not have the same speed. Instead, their speeds are distributed according to Maxwell’s distribution law.

The number of molecules having speed between v and v + dv is given by:

dN ∝ v² e^−mv²/2kT dv

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2️⃣ Maxwell Speed Distribution Curve

• Y-axis → Number of molecules
• X-axis → Speed (v)
• Curve starts from origin
• Rises to a maximum and then falls asymptotically

Area under the curve = Total number of molecules

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3️⃣ Characteristic Speeds of Gas Molecules

Speed	Expression
Most Probable Speed (vmp)	√(2RT / M)
Average Speed (vavg)	√(8RT / πM)
Root Mean Square Speed (vrms)	√(3RT / M)

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4️⃣ Relation Between Speeds

v_mp < v_avg < v_rms

Ratio:

v_mp : v_avg : v_rms = √2 : √(8/π) : √3

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5️⃣ Effect of Temperature on Distribution

• Increase in temperature → curve flattens
• Peak shifts towards higher speed
• Area under curve remains constant

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6️⃣ Effect of Molecular Mass

• Lighter gas → higher speeds
• Heavier gas → lower speeds
• At same temperature, all gases have same average kinetic energy

⟨KE⟩ = (3/2)kT

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7️⃣ Important Graph-Based Results

• Area under curve does NOT depend on temperature
• Higher T → broader curve
• For same T, lighter gas curve shifts right

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8️⃣ JEE Advanced Tricks

• Speeds ∝ √T
• Speeds ∝ 1 / √M
• Ratio questions are faster than absolute calculations
• Always compare rms speeds in energy-based questions

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9️⃣ Common JEE Traps

• Confusing v_avg with v_rms
• Assuming all molecules have same speed
• Wrong interpretation of graph area

✔ Stage 1 – Page 9 COMPLETE

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🔥 Kinetic Theory of Gases (KTG) – Complete Library

1. KTG – Part 1
2. KTG – Part 2
3. KTG – Part 3
4. KTG – Part 4
5. KTG – Part 5
6. KTG – Part 6
7. KTG – Part 7
8. KTG – Special Notes
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10. KTG – Part 10
11. KTG – Part 11
12. KTG – Part 12
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14. KTG – Part 14
15. KTG – Part 15
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17. KTG – Part 17
18. KTG – Part 18
19. KTG – Part 19
20. KTG – Part 20
21. KTG – Part 21
22. KTG – Part 22
23. KTG – Part 23
24. KTG – Part 24
25. KTG – Part 25
26. KTG – Part 26
27. KTG – Part 27
28. KTG – Part 28
29. KTG – Part 29
30. KTG – Part 30

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2. Properties – Part 2
3. Properties – Part 3
4. Properties – Part 4
5. Properties – Part 5
6. Properties – Part 6
7. Properties – Part 7
8. Properties – Part 8
9. Properties – Part 9
10. Properties – Part 10
11. Properties – Part 11
12. Properties – Part 12
13. Properties – Part 13
14. Properties – Part 14
15. Properties – Part 15
16. Properties – Part 16
17. Properties – Part 17
18. Properties – Part 18
19. Properties – Part 19
20. Properties – Part 20
21. Properties – Part 21
22. Properties – Part 22
23. Properties – Part 23
24. Properties – Part 24
25. Properties – Part 25

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