Law of Equipartition of Energy 🔥

When a gas reaches thermal equilibrium at temperature \(T\), every quadratic term in its total energy grabs the same average energy—exactly \( \tfrac12 k_B T \) per term. That’s the heart of the law of equipartition of energy.:contentReference[oaicite:0]{index=0}

1. Kinetic energy of one molecule 🏃‍♂️

The translational kinetic energy is \[ \varepsilon_t=\tfrac12 m v_x^2+\tfrac12 m v_y^2+\tfrac12 m v_z^2 \]:contentReference[oaicite:1]{index=1}

At equilibrium the average becomes \[ \langle\varepsilon_t\rangle =\tfrac12 k_B T+\tfrac12 k_B T+\tfrac12 k_B T =\tfrac32 k_B T \]:contentReference[oaicite:2]{index=2}

Because the three directions are equivalent, each component follows \[ \tfrac12 m v_x^2=\tfrac12 k_B T,\; \tfrac12 m v_y^2=\tfrac12 k_B T,\; \tfrac12 m v_z^2=\tfrac12 k_B T \]:contentReference[oaicite:3]{index=3}

2. Degrees of freedom 🎯

  • Translation: 3 directions in space → 3 degrees of freedom.:contentReference[oaicite:4]{index=4}
  • Rotation: • Monatomic: none. • Diatomic rigid rotator: 2 perpendicular axes → 2 degrees.:contentReference[oaicite:5]{index=5}
  • Vibration: Each vibrational mode adds 2 quadratic terms (kinetic + potential).:contentReference[oaicite:6]{index=6}

3. Rotational & vibrational energies 🔄🎻

For a diatomic rigid rotator \[ \varepsilon_t+\varepsilon_r =\tfrac12 m v_x^2+\tfrac12 m v_y^2+\tfrac12 m v_z^2 +\tfrac12 I_1\omega_1^2+\tfrac12 I_2\omega_2^2 \]:contentReference[oaicite:7]{index=7}

A molecule that can also vibrate gains an extra \[ \varepsilon_v=\tfrac12 k y^2+\tfrac12 m\!\left(\tfrac{dy}{dt}\right)^{\!2} \]:contentReference[oaicite:8]{index=8}

Equipartition says:

  • \(\tfrac12 k_B T\) for each translational or rotational degree.
  • \(k_B T\) per vibrational frequency (two terms, so double the share!).:contentReference[oaicite:9]{index=9}

4. Specific heat capacities \(C_v\) & \(C_p\) 🔥

4.1 Monatomic gas (e.g., He, Ar)

Internal energy per mole: \(U=\tfrac32 R T\). Thus \( C_v=\tfrac32 R,\; C_p=\tfrac52 R,\; \gamma=\tfrac{C_p}{C_v}=\tfrac53 \).:contentReference[oaicite:10]{index=10}

4.2 Diatomic gas – rigid rotator (e.g., O2, N2)

\(U=\tfrac52 R T\) → \( C_v=\tfrac52 R,\; C_p=\tfrac72 R,\; \gamma=\tfrac75 \).:contentReference[oaicite:11]{index=11}

4.3 Diatomic with one vibrational mode active

A single vibrational frequency adds \(k_B T\) per molecule, so \(U=\tfrac72 R T\). Consequently \( C_v=\tfrac72 R,\; C_p=\tfrac92 R,\; \gamma=\tfrac{9}{7} \).:contentReference[oaicite:12]{index=12}

4.4 Polyatomic gas ( f vibrational modes)

\(U=(3+f) R T\). Therefore \( C_v=(3+f)R,\; C_p=(4+f)R,\; \gamma=\dfrac{4+f}{3+f} \).:contentReference[oaicite:13]{index=13}

Note: For any ideal gas, \(C_p-C_v=R\).:contentReference[oaicite:14]{index=14}

Key Takeaways for NEET 🌟

  • Each quadratic energy term carries \( \tfrac12 k_B T \) (rotations & translations) while a vibrational frequency carries \(k_B T\). This is the core of equipartition.:contentReference[oaicite:15]{index=15}
  • Monatomic gas values: \(C_v=\tfrac32 R,\;C_p=\tfrac52 R,\;\gamma=\tfrac53\). These numbers show up often in questions.:contentReference[oaicite:16]{index=16}
  • Diatomic rigid rotator numbers: \(C_v=\tfrac52 R,\;C_p=\tfrac72 R,\;\gamma=\tfrac75\). Memorize them!:contentReference[oaicite:17]{index=17}
  • Relation \(C_p-C_v=R\) holds for every ideal gas—easy scoring point.:contentReference[oaicite:18]{index=18}
  • \(\gamma\) depends on degrees of freedom: more vibrational modes → lower \(\gamma\). Remember the formula \(\gamma=\tfrac{4+f}{3+f}\).:contentReference[oaicite:19]{index=19}

Keep these gems handy, and solving thermodynamics questions will feel like a breeze 😎.