- #1

- 551

- 1

According to QM, a diatomic gas molecule possesses rotational energy levels given by En = (1/2I)(h^2)n(n + 1). h is meant to be h-bar, Planck's constant over 2π here and I = moment of inertia. Energy level n has a degeneracy of 2n + 1.

Find the partition function of the rotational motion of a single such molecule.

Z = Σ gn.exp(-En/kT) = Σ (2n + 1)exp[-(n(n + 1)h^2)/kT]

Suppose T is sufficently small so that only the first 2 energy levels need to be considered. Find an expression for the mean rotational energy.

Here's where I'm stuck as I don't know which equation to use. Some of my notes say E avg = (1/Z)Σ En.gn.exp(-En/kT). In some of my other notes, there's an example for finding out the mean vibrational energy. E avg = (1/Z)(kT^2)dZ/dT. Obviously for rotation, the Boltzmann factor will be different as there's a different expression for En, but can I use this equation anyway?

Edit: I worked out Z for the first 2 states as Z = 1 + 3exp[-(h^2)/IkT].

Find the partition function of the rotational motion of a single such molecule.

Z = Σ gn.exp(-En/kT) = Σ (2n + 1)exp[-(n(n + 1)h^2)/kT]

Suppose T is sufficently small so that only the first 2 energy levels need to be considered. Find an expression for the mean rotational energy.

Here's where I'm stuck as I don't know which equation to use. Some of my notes say E avg = (1/Z)Σ En.gn.exp(-En/kT). In some of my other notes, there's an example for finding out the mean vibrational energy. E avg = (1/Z)(kT^2)dZ/dT. Obviously for rotation, the Boltzmann factor will be different as there's a different expression for En, but can I use this equation anyway?

Edit: I worked out Z for the first 2 states as Z = 1 + 3exp[-(h^2)/IkT].

Last edited: