20.13:
Heat Capacities of an Ideal Gas III
The number of independent ways the gas molecules can move, rotate, or vibrate in three-dimensional space is called degrees of freedom.
According to the Equipartition theorem, the total energy of the gas molecules at equilibrium is equally distributed among their degrees of freedom.
The internal energy change can be determined by the temperature change, which can be equated to its molar heat capacity at constant volume.
From this expression, the molar heat capacity at constant volume can be written in terms of degrees of freedom.
By substituting the obtained value in the heat capacities relation, an expression for molar heat capacity at constant pressure is also determined.
The molar heat capacity at constant pressure over molar heat capacity at constant volume is the ratio of heat capacities.
For instance, at room temperature, a diatomic gas has five degrees of freedom. Substituting this, the ratio of heat capacities equals 1.40.
The ratio of molar heat capacities for any gas is always greater than unity, and the calculated values are listed.
20.13:
Heat Capacities of an Ideal Gas III
The number of independent ways a gas molecule can move along straight line, rotate, and vibrate is called its degrees of freedom. Supposing d represents the number of degrees of freedom of an ideal gas, the molar heat capacity at constant volume of an ideal gas in terms of d is
The molar heat capacity at constant pressure can also be written in terms of d using the relationship between the two heat capacities. Thus,
The ratio of molar heat capacity at constant pressure over that at constant volume is called the ratio of heat capacity or the ratio of specific heats. Since CP is always greater than CV, the ratio of specific heats is always greater than one.
A diatomic ideal gas has more degrees of freedom than a monatomic gas. In addition to the three degrees of freedom for translation, it has two degrees of freedom for rotation perpendicular to its axis. Furthermore, the molecule can vibrate along its axis.
Classically, if gas molecules had only translational kinetic energy, collisions between the molecules would soon make them rotate and vibrate. However, as explained, quantum mechanics controls which degrees of freedom are active. Both rotational and vibrational energies are limited to discrete values.
For temperatures below about 60 K, the energies of hydrogen molecules are too low for a collision to bring the rotational state or vibrational state of a molecule from the lowest energy to the second-lowest energy. So, the only form of energy is translational kinetic energy, and d = 3, as in a monatomic gas. Above that temperature, the two rotational degrees of freedom begin to contribute; some molecules are excited to the rotational state with the second-lowest energy. This temperature is much lower than the one at which rotations of monatomic gases contribute because diatomic molecules have much higher rotational inertias and, hence, much lower rotational energies. From about room temperature (a bit less than 300 K) to about 600 K, the rotational degrees of freedom are fully active but the vibrational ones are not, and d = 5. Then, finally, above about 3000 K, the vibrational degrees of freedom are fully active, and d = 7. as predicted by the classical theory. Thus, the ratio of specific heat for diatomic gas is 1.40 at room temperature but equals 1.28 at higher temperatures.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. 605-607, 638-639.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. 88-91, 127 Retrieved from https://openstax.org/books/university-physics-volume-2/pages/3-5-heat-capacities-of-an-ideal-gas; https://openstax.org/books/university-physics-volume-2/pages/2-3-heat-capacity-and-equipartition-of-energy