Specific Heat Capacity – Quick & Friendly Notes 🌡️🔥

1. Why Specific Heat Matters

Specific heat tells us how much heat you must add to one mole of a substance to raise its temperature by 1 K. Bigger value ➡️ more heat needed.

2. Law of Equipartition of Energy 🎲

Every quadratic “energy slot” (translation, rotation) grabs an average energy of \( \tfrac{1}{2}k_B T \). A vibrational slot grabs twice that because it stores both kinetic and potential energy, giving \( k_B T \). This simple rule lets us predict the specific heats of gases and even solids. :contentReference[oaicite:0]{index=0}

3. Gases

3.1 Monatomic Gases (e.g., He, Ne) 🎈

• Energy-holding “slots”: 3 translations.
• Energy per mole: \( U = \tfrac{3}{2} R T \). :contentReference[oaicite:1]{index=1}
• Specific heats: \( C_v = \tfrac{3}{2} R \), \( C_p = \tfrac{5}{2} R \). :contentReference[oaicite:2]{index=2}
• Ratio \( \gamma = \dfrac{C_p}{C_v} = \dfrac{5}{3} \approx 1.67 \). :contentReference[oaicite:3]{index=3}

3.2 Diatomic Gases (e.g., N₂, O₂) 👫

• Rigid molecules bring 3 translations + 2 rotations = 5 slots.
• Energy per mole: \( U = \tfrac{5}{2} R T \). :contentReference[oaicite:4]{index=4}
• Specific heats: \( C_v = \tfrac{5}{2} R \), \( C_p = \tfrac{7}{2} R \). :contentReference[oaicite:5]{index=5}
• Ratio \( \gamma = \dfrac{7}{5} = 1.40 \). :contentReference[oaicite:6]{index=6}
• Add one vibrational slot ⇒ \( C_v = \tfrac{7}{2} R \), \( C_p = \tfrac{9}{2} R \), \( \gamma = \dfrac{9}{7} \). :contentReference[oaicite:7]{index=7}

3.3 Polyatomic Gases (≥3 atoms) 🍃

• Slots: 3 translations + 3 rotations + \( f \) vibrations.
• Specific heats: \( C_v = (3+f)R \), \( C_p = (4+f)R \). :contentReference[oaicite:8]{index=8}
• Ratio \( \gamma = \dfrac{4+f}{3+f} \). :contentReference[oaicite:9]{index=9}
• Always \( C_p – C_v = R \) for an ideal gas, whatever its crowd size. :contentReference[oaicite:10]{index=10}

3.4 Handy Reference Table 📊

Values come straight from the equipartition predictions and match experiments nicely at room temperature. :contentReference[oaicite:11]{index=11}

4. Solids 🪨

• Each atom jiggles in 3D, giving average energy \( 3k_B T \).
• Per mole this adds to \( U = 3RT \). :contentReference[oaicite:12]{index=12}
• Specific heat (at constant anything, since solids hardly expand): \( C = 3R \approx 25\; \text{J mol⁻¹ K⁻¹} \). :contentReference[oaicite:13]{index=13}
• Most solids follow this (Carbon is a quirky outlier). :contentReference[oaicite:14]{index=14}

5. Worked Example ✍️

A 44.8 L cylinder holds helium at STP (2 mol). Raise its temperature by \( 15^\circ\text{C} \) at constant volume.

Because He is monatomic, use \( C_v = \tfrac{3}{2}R \). Heat needed:

\[ Q = n\,C_v\,\Delta T = 2 \times 1.5R \times 15 = 45R = 374\;\text{J}. \] :contentReference[oaicite:15]{index=15}

6. High-Yield NEET Nuggets 🏆

1. \( C_p – C_v = R \) for any ideal gas—memorize this shortcut!
2. Fast recall: \( \gamma = 5/3 \) (monatomic) and \( \gamma = 7/5 \) (rigid diatomic) show up in adiabatic questions.
3. Equipartition links degrees of freedom to heat capacity—count the slots, get \( C_v \).
4. Dulong-Petit rule for solids: \( C \approx 3R \). Concept questions love it.
5. Use \( Q = nC\Delta T \) to find heat quickly—just pick \( C_v \) for fixed-volume or \( C_p \) for fixed-pressure situations.