Thermodynamic Properties of Ideal Gases of Diatomic Molecules Calculated Using Partition Functions and Sample Problems

Abstract

The material presented in this lecture is adapted from Chapter 4 of M&S. In Lecture 40, we presented expressions for the translational and electronic contributions to the partition function of a diatomic molecule. In this lecture, first, we will discuss the vibrational contribution to the partition function of a diatomic molecule, including deriving expressions for the average vibrational energy and vibrational contribution to the heat capacity at constant volume of an ideal gas of diatomic molecules. Second, we will introduce the characteristic vibrational temperature, whose value relative to the system temperature will determine the probability that a given vibrational energy level is populated. Further, we will solve Sample Problem 41.1 to calculate the probability that a vibrational energy level is populated in the case of nitrogen molecules, including discussing the effect of temperature. Third, we will discuss the rotational contribution to the partition function of a diatomic molecule, including deriving expressions for the average rotational energy and rotational contribution to the heat capacity at constant volume of an ideal gas of diatomic molecules. In addition, we will introduce the characteristic rotational temperature, whose value relative to the system temperature will determine the probability that a given rotational energy level is populated. Fourth, we will emphasize that the rotational partition function of a diatomic molecule contains a symmetry number, which is equal to one when the two atoms comprising the diatomic molecule are different (referred to as the heteronuclear case) and is equal to two when the two atoms are identical (referred to as the homonuclear case). Fifth, we will present an expression for the total partition function of a diatomic molecule, which includes translational, vibrational, rotational, and electronic contributions. Finally, we will solve Sample Problem 41.2 to calculate the average molar internal energy and molar heat capacity at constant volume of an ideal gas of diatomic molecules, including identifying the various contributions to each thermodynamic property.