Specific Heat Capacity 🚀

1. Quick Recap of Heat & Work 🔥

When a gas pushes a piston, the work it does is \( \Delta W = P\Delta V \) :contentReference[oaicite:0]{index=0}. The energy balance (First Law) reads \( \Delta Q = \Delta U + P\Delta V \)  (11.3) :contentReference[oaicite:1]{index=1}.

2. Heat Capacity Basics 📏

  • Heat capacity: \( S = \dfrac{\Delta Q}{\Delta T} \)  (11.4) :contentReference[oaicite:2]{index=2}
  • Specific heat capacity (per kilogram): \( s = \dfrac{1}{m}\dfrac{\Delta Q}{\Delta T} = \dfrac{S}{m} \)  (11.5) :contentReference[oaicite:3]{index=3} Unit – J kg–1 K–1
  • Molar specific heat capacity (per mole): \( C = \dfrac{1}{\mu}\dfrac{\Delta Q}{\Delta T} \)  (11.6) :contentReference[oaicite:4]{index=4} Unit – J mol–1 K–1

3. Solids – the 3R Rule 🎯

Law of equipartition gives the neat result \( C = 3R \approx 25\;\text{J mol}^{-1}\text{K}^{-1} \)  (11.7) :contentReference[oaicite:5]{index=5}. Most solids match this at room temperature; carbon is a well-known exception.

4. Water: the Classic Benchmark 💧

  • 1 calorie raises 1 g of water from 14.5 °C to 15.5 °C :contentReference[oaicite:6]{index=6}.
  • In SI units: \( s_{\text{water}} = 4186\;\text{J kg}^{-1}\text{K}^{-1} \) (or 4.186 J g–1 K–1) :contentReference[oaicite:7]{index=7}.
  • “Mechanical equivalent of heat” is simply the conversion 1 cal = 4.186 J, so using joules keeps life easy 🤗.

5. Gases – Two Flavors of C 🍃

Because gases can expand, we define:

  • \( C_v \) – heat capacity at constant volume
  • \( C_p \) – heat capacity at constant pressure

For an ideal gas: \( C_p – C_v = R \)  (11.8)

6. Example Wow-Moment 🌈

Evaporating 1 g of water needs \( \Delta Q = 2256 \,\text{J} \). Work done against atmospheric pressure is \( \Delta W = 169.2 \,\text{J} \). So the rise in internal energy is \( \Delta U = 2086.8 \,\text{J} \) :contentReference[oaicite:9]{index=9}. Most of the heat goes into breaking molecular bonds rather than doing external work – cool, right? 😎

High-Yield Nuggets for NEET 🏆

  1. \( s = \dfrac{1}{m}\dfrac{\Delta Q}{\Delta T} \) definition & units.
  2. Ideal-gas relation \( C_p – C_v = R \).
  3. Dulong–Petit limit for solids: \( C \approx 3R \).
  4. Benchmark value \( s_{\text{water}} = 4186\;\text{J kg}^{-1}\text{K}^{-1} \) and calorie concept.
  5. First-law link \( \Delta Q = \Delta U + P\Delta V \) showing how heat, work, and energy connect.

Keep practicing, stay curious, and physics will feel warm & bright! 🌟