An ideal monatomic gas has a molar heat capacity Cmp at constant pressure. What is the molar heat capacity at constant volume of an ideal diatomic gas?
The Correct Answer and Explanation is:
Answer:
For an ideal diatomic gas, the molar heat capacity at constant volume CVC_VCV is approximately:CV=52RC_V = \frac{5}{2} RCV=25R
where RRR is the universal gas constant.
Explanation:
The heat capacity of a gas depends on the number of degrees of freedom available to its molecules. Degrees of freedom represent independent ways in which a molecule can store energy — through translation, rotation, and vibration.
For an ideal monatomic gas, such as helium or argon, only translational motion contributes significantly to its internal energy. A monatomic gas molecule moves in three dimensions, giving it 3 translational degrees of freedom. According to the equipartition theorem, each degree of freedom contributes 12kT\frac{1}{2}kT21kT of energy per molecule, where kkk is Boltzmann’s constant and TTT is the temperature. On a molar basis, this gives:U=32nRTU = \frac{3}{2}nRTU=23nRT
Thus, the molar heat capacity at constant volume for a monatomic gas is:CV=(∂U∂T)V=32RC_V = \left( \frac{\partial U}{\partial T} \right)_V = \frac{3}{2}RCV=(∂T∂U)V=23R
For an ideal diatomic gas (such as nitrogen or oxygen at moderate temperatures), both translational and rotational motions contribute. Diatomic molecules have 3 translational and 2 rotational degrees of freedom (rotation around the axis along the bond is negligible at room temperature due to quantum mechanical restrictions). Thus, they have a total of 5 degrees of freedom contributing to internal energy. The internal energy becomes:U=52nRTU = \frac{5}{2}nRTU=25nRT
Hence, the molar heat capacity at constant volume is:CV=(∂U∂T)V=52RC_V = \left( \frac{\partial U}{\partial T} \right)_V = \frac{5}{2}RCV=(∂T∂U)V=25R
At higher temperatures, vibrational modes may become active, increasing CVC_VCV, but at ordinary temperatures, 52R\frac{5}{2}R25R is the standard value used.
