Kinetic Theory | Class 11 Physics Notes

12.6 Specific heat capacity

This section applies the law of equipartition of energy to calculate the specific heat capacities of gases and solids. For monatomic gases like argon, molecules have only three translational degrees of freedom, so the average energy per molecule is (3/2) k_B T. For one mole of such gas, the internal energy U = (3/2) R T, where R is the universal gas constant. The molar specific heat at constant volume C_v = dU/dT = (3/2) R. Using the relation C_p - C_v = R for ideal gases, the molar specific heat at constant pressure C_p = (5/2) R, and the ratio of specific heats γ = C_p / C_v = 5/3. For diatomic gases treated as rigid rotators (like O_2), there are five degrees of freedom (3 translational + 2 rotational), so U = (5/2) R T, C_v = (5/2) R, C_p = (7/2) R, and γ = 7/5. If vibrational modes are excited, an additional k_B T per vibrational mode is added, increasing specific heats accordingly. Polyatomic gases have 3 translational, 3 rotational, and ℓ vibrational modes, leading to C_v = (3 + ℓ) R, C_p = (4 + ℓ) R, and γ = (4 + ℓ)/(3 + ℓ). The relation C_p - C_v = R holds for all ideal gases. Tables 12.1 and 12.2 summarize predicted and measured specific heats, showing good agreement for many gases. The section also extends the equipartition law to solids, where atoms vibrate about fixed positions. Each atom has three vibrational degrees of freedom, each contributing k_B T energy, so for one mole, U = 3 R T. Since volume change is negligible, heat capacity C = dU/dT = 3 R, consistent with experimental molar specific heats of many solids (Table 12.3). Carbon is an exception due to quantum effects. These results demonstrate the power of kinetic theory and equipartition in explaining thermal properties of matter.

📊 Diagram: Table on page 11 (4×5); Table on page 11 (9×6); Table on page 11 (7×3)

🔗 Connection: Having understood specific heat capacities, the chapter proceeds to explain the concept of mean free path in gases, which describes molecular collisions and transport properties.

Table on page 11 (4×5)

Table on page 11 (9×6)

Table on page 11 (7×3)