Cp/Cv Ratio from Degrees of Freedom

 

❓ Question

In Kinetic Theory of Gases, how do we quickly determine the value of
$$\gamma = \frac{C_p}{C_v}$$
for different types of gases?

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🖼 Concept Image

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✍️ Short Solution

This is a direct Degrees of Freedom → γ mapping concept.

Golden relation:

$$\gamma = \frac{C_p}{C_v} = \frac{f+2}{f}$$

👉 If you know f, you know everything.
No derivation needed in exam.

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🔷 Step 1 — KTG ka Golden Rule 💯

For any ideal gas:

$$\boxed{\gamma = \frac{f+2}{f}}$$

Where:

$$f = \text{degrees of freedom}$$

👉 Bas f pata ho, answer automatic.

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🔷 Step 2 — Degrees of Freedom (Core Memory Table)

🔹 Monatomic Gas

$$f = 3$$

(Only translational motion)

$$\gamma = \frac{5}{3}$$

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🔹 Diatomic Rigid Gas

$$f = 5$$

(3 translational + 2 rotational)

$$\gamma = \frac{7}{5}$$

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🔹 Diatomic Non-Rigid Gas

$$f = 7$$

(Vibrational mode ON)

$$\gamma = \frac{9}{7}$$

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🔹 Triatomic Rigid Gas

$$f = 6$$
$$\gamma = \frac{4}{3}$$

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🔷 Step 3 — Why Rigid vs Non-Rigid Matters

Rigid → vibration ignored

Non-rigid → vibration adds 2 extra DOF

⚠️ JEE trap exactly yahin hota hai.

Temperature high → vibrations active.

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🔷 Step 4 — Cp/Cv Trend Logic

As degrees of freedom increase:

$$\gamma \downarrow$$

So:

• Monatomic → highest γ

• More complex gas → lower γ

👉 Light gases → higher γ
👉 Complex molecules → lower γ

Concept clarity > formula memorization.

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🔷 Step 5 — Matching-Type Question Strategy

Exam hall trick:

1️⃣ Identify gas type
2️⃣ Instantly write f
3️⃣ Mentally map γ

❌ Formula solve karna = waste of time
✅ Concept mapping = fastest approach

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✅ Final Takeaway

$$\boxed{\gamma = \frac{f+2}{f}}$$

Gas type → f → γ
Bas itna yaad rakho.

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⭐ Golden JEE Insight

In KTG:

$$C_v = \frac{f}{2}R$$
$$C_p = C_v + R$$

From here automatically:

$$\gamma = \frac{f+2}{f}$$

🔥 Everything comes from degrees of freedom.