a) The constant volume heat capacity of an ideal monatomic gas is 3R/2 but the constant volume heat capacity of an ideal diatomic gas is 5/2. Why are the values different? b) Give a simple, intuitive, short and non-mathematical explanation for why cp of an ideal gas is bigger than cv
The Correct Answer and Explanation is:
) The constant volume heat capacity of an ideal monatomic gas is 32R\frac{3}{2}R23R, but the constant volume heat capacity of an ideal diatomic gas is 52R\frac{5}{2}R25R. Why are the values different?
The values are different because monatomic and diatomic gases have different numbers of ways to store internal energy, known as degrees of freedom.
A monatomic gas (such as helium or argon) consists of single atoms that can only move in three directions — along the x, y, and z axes — so it has 3 translational degrees of freedom. According to the equipartition theorem, each degree of freedom contributes 12kT\frac{1}{2}kT21kT (or 12R\frac{1}{2}R21R per mole) to the internal energy. Therefore, the total internal energy per mole is 32RT\frac{3}{2}RT23RT, leading to a heat capacity at constant volume CV=32RC_V = \frac{3}{2}RCV=23R.
A diatomic gas (such as oxygen or nitrogen), in addition to the 3 translational degrees of freedom, also has rotational degrees of freedom. At room temperature, two rotational modes (rotation about two perpendicular axes) are active, adding 2 more degrees of freedom. This gives a total of 5 degrees of freedom (3 translational + 2 rotational), resulting in an internal energy per mole of 52RT\frac{5}{2}RT25RT, and thus CV=52RC_V = \frac{5}{2}RCV=25R.
Vibrational modes can also contribute at higher temperatures but are typically inactive at room temperature for most diatomic gases.
b) Give a simple, intuitive, short and non-mathematical explanation for why CPC_PCP of an ideal gas is bigger than CVC_VCV:
At constant volume, all the heat added goes into increasing the internal energy (heating the gas).
At constant pressure, part of the heat not only increases the internal energy but also does work to expand the gas against external pressure. Therefore, more heat is needed to achieve the same temperature change, making CPC_PCP larger than CVC_VCV.
