Question

For an N-atom nonlinear polyatomic gas, the constant volume molar heat capacity Cv,m has the expected value of 3(N-1)R, based on the principle of equipartition of energy. The correct statement(s) about the measured value of Cv,m is/are

• A. The measured $C_{v,m}$ is independent of temperature.
• B. The measured $C_{v,m}$ is dependent on temperature.
• C. The measured $C_{v,m}$ is typically lower than the expected value.
• D. The measured $C_{v,m}$ is typically higher than the expected value.

Answer 1

The Correct Option is B, C

The question is about the constant volume molar heat capacity, \(C_{v,m}\), for an N-atom nonlinear polyatomic gas, which theoretically should be \(3(N-1)R\) based on the equipartition theorem. We need to analyze the behavior of the measured \(C_{v,m}\).

• Equipartition Principle: According to this principle, each degree of freedom contributes \(\frac{1}{2}RT\) per mole to the internal energy of a gas. A nonlinear molecule with \(N\) atoms theoretically has \(3(N-1)\) vibrational degrees of freedom. Each of these adds C_{v,m} = 3(N-1)R.
• Temperature Dependency: However, due to quantum considerations, especially at low temperatures, not all vibrational modes are fully excited. This means the full expected value based on classical physics is not achieved. As the temperature increases, more modes become accessible, hence making \(C_{v,m}\) temperature dependent.
• The actual natural availability of vibrational modes varies with temperature, causing the measured \(C_{v,m}\) to often be lower than the classical expectation, especially at lower temperatures. This indicates the correct statement that the measured \(C_{v,m}\) is typically lower than expected at lower temperatures.

Given this understanding, we can conclude:

• The option stating "The measured \(C_{v,m}\) is dependent on temperature" is correct as the heat capacity indeed varies with temperature due to the reasons stated above.
• The option stating "The measured \(C_{v,m}\) is typically lower than the expected value" is correct, especially at lower temperatures where not all vibrational modes are accessible.

Thus, the correct statements about the measured \(C_{v,m}\) are:

• The measured \(C_{v,m}\) is dependent on temperature.
• The measured \(C_{v,m}\) is typically lower than the expected value.