Which type of ideal gas will have the largest value for Cp – Cv?
(A) Monoatomic
(B) Diatomic
(C) Polyatomic
(D) The value will be the same for all.
The correct answer and explanation is :
The correct answer is:
(C) Polyatomic
Explanation:
The difference between the heat capacity at constant pressure ((C_p)) and the heat capacity at constant volume ((C_v)) for an ideal gas is given by:
[
C_p – C_v = R
]
where (R) is the universal gas constant. This relationship holds for all ideal gases. However, the absolute values of (C_p) and (C_v) depend on the number of degrees of freedom available to the gas molecules.
In thermodynamics, the degrees of freedom of a molecule refer to the ways in which it can store energy. For example:
- Monoatomic gases (such as helium or neon) have only translational degrees of freedom, meaning the molecules can only move in three directions (x, y, z), contributing three degrees of freedom.
- Diatomic gases (like oxygen or nitrogen) have translational degrees of freedom (3), rotational degrees of freedom (2), and, at high temperatures, vibrational degrees of freedom (which require both translational and rotational motions).
- Polyatomic gases (such as methane or carbon dioxide) have all the degrees of freedom present in diatomic molecules, but they also possess additional vibrational degrees of freedom due to the more complex molecular structure.
As the number of degrees of freedom increases, the internal energy available to the gas also increases. This results in a higher specific heat at constant pressure ((C_p)) for polyatomic gases compared to diatomic and monoatomic gases.
Thus, polyatomic gases, due to their greater complexity and more degrees of freedom (especially vibrational modes), will generally exhibit the largest difference between (C_p) and (C_v). This difference increases because polyatomic gases can store more energy in additional degrees of freedom, causing (C_p) to be significantly higher than (C_v).
Therefore, the gas type with the largest value for (C_p – C_v) is polyatomic.