🔹 Kinetic Theory of Gases – Stage 1 (Page 3)

1️⃣ Root Mean Square (RMS) Speed

The RMS speed is defined as the square root of the mean of the squares of molecular speeds.

v_rms = √⟨v²⟩

From pressure equation:

P = (1/3)(Nm/V)⟨v²⟩

Rearranging:

⟨v²⟩ = 3P / ρ

where ρ = mass density = Nm / V

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2️⃣ RMS Speed in Terms of Temperature

Using ideal gas equation:

PV = nRT

For one molecule:

P = (NkT) / V

Substituting into pressure equation:

(1/3) m⟨v²⟩ = kT

Therefore,

v_rms = √(3kT / m)

In molar form:

v_rms = √(3RT / M)

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3️⃣ Translational Kinetic Energy of a Gas Molecule

Average kinetic energy per molecule:

⟨E⟩ = (1/2)m⟨v²⟩

Using ⟨v²⟩ = 3kT / m:

⟨E⟩ = (3/2)kT

Important: Average kinetic energy depends only on temperature, not on mass or nature of gas.

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4️⃣ Kinetic Energy per Mole of Gas

For one mole:

E = (3/2)RT

This result is valid for all ideal gases.

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5️⃣ Key Comparisons (JEE Favourite)

• At same temperature: lighter gas → higher RMS speed
• v_rms ∝ √T
• v_rms ∝ 1 / √M

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 JEE Exam Notes

• Kinetic energy depends only on temperature
• Speed distribution differs, but ⟨E⟩ is same
• Used heavily in comparative MCQs

✔ Stage 1 – Page 3 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
9. KTG – Part 9
10. KTG – Part 10
11. KTG – Part 11
12. KTG – Part 12
13. KTG – Part 13
14. KTG – Part 14
15. KTG – Part 15
16. KTG – Part 16
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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1. Properties – Part 1
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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