Diatomic Molecules

Helium atoms in monoatomic gases, like helium gas, have three degrees of freedom for translation. For molecules like O₂ or N₂, which have two atoms positioned along the x-axis, they also have three degrees of freedom for translation but can also rotate around the z-axis and y-axis.

A diatomic molecule, like O₂, has two additional degrees of freedom due to its two perpendicular rotational orientations around its center of mass. This means it has a total of five degrees of freedom, with two degrees of rotational freedom. The rotating kinetic energy for each rotational degree of freedom is influenced by the moments of inertia about the z and y axes and the corresponding angular speeds. The total energy due to the degrees of freedom for translation and rotation in a diatomic molecule can be calculated accordingly. As a result, the total energy owing to the degrees of freedom for translation and rotation in a diatomic molecule is,

E = ½mv_x² + ½mv_y² + ½mv_z² + ½I_zω_z² + ½I_yω_y²

The quadratic terms in the aforementioned expression are related to the several degrees of freedom that a diatomic molecule can have. Each of them adds ½k_B T to the molecule’s overall energy. It was implied in the explanation above that the rotating molecule is a rigid rotator. Real molecules, on the other hand, have covalent links between their atoms, which allows them to execute extra motion, namely atomic vibrations about their mean locations, similar to a one-dimensional harmonic oscillator. As a result, these molecules have an extra degree of freedom that corresponds to their various vibrational modes. Only along the internuclear axis may the atoms oscillate in diatomic molecules like O₂, N_2, and CO. The vibrational energy associated with this motion is added to the molecule’s overall energy.

E = E(translational) + E(rotational) + E(vibrational)

Both the kinetic energy term and the potential energy term contribute to the word E(vibrational), which is composed of two components.

E(vibrational) = ½mu² + ½kr²

Where u denotes the rate of vibration of the molecule’s atoms, r denotes the distance between the oscillating atoms, and k is the force constant. Each of the quadratic velocity and position terms in equation 1 will contribute ½k_B T. The total internal energy is thus increased by 2 × ½k_B T for each mode or degree of freedom for vibrational motion.

Thus, for a non-rigid diatomic gas in thermal equilibrium at a temperature T, the mean kinetic energy associated with molecular translation along three directions is 3 × ½kB T, and the mean kinetic energy associated with molecular rotation about two perpendicular axes is 2 × ½k_B T, and the total vibrational energy is 2 × ½k_B T, which corresponds to kinetic and potential energy terms. The average energy for each molecule associated with each quadratic term is ½k_B T when the law of energy partition is applied to gas in thermal equilibrium at a temperature T. In contrast to extremely low temperatures, where quantum effects are significant, the law of energy equilibria only applies to high temperatures.

Also, Read

• Stefan Boltzman Constant
• Entropy Change Formula
• Behaviour of Gas Molecules

Law of Equipartition of Energy has many names such as Equipartition Theorem, Equipartition Principle, Law of Equipartition, or simply Equipartition and it describes the distribution of energy among the particles in a system that is at thermal equilibrium. The law of Equipartition of Energy tells us about how each degree of freedom of a particle in a system contributes to the average energy of the system. The Equipartition Theorem holds key significance in a wide range of fields of study, from thermodynamics and statistical mechanics to materials science and chemistry. This article covers the topic of the Law of Equipartition of Energy in varying detail.