15 October 1996

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So z_trans = z_x z_y z_z = (2mkT/h²)^3/2 L_x L_y L_z = (2mkT/h²)^3/2 V T^3/2.

None of the other factors depend upon T, so

E_trans = RT²[{ln(z_trans)}/T]_V
= RT²[{ln(T^3/2)}/T]_V
= (3/2)RT²[{ln(T)}/T]<sub>V
= (3/2)RT, exactly the value we needed to satisfy the Equipartition Theorem.

See Nash (pink pp. 68-72) for convincing derivations of other results like P = -[A/V]T = RT/V (1 mole) and the Sackur-Tetrode equation for Strans which gives monotomic gas entropies more credible than experimental values! All these and more from the Particle-in-the-Box eigenvalues from P.Chem. II.</sub>

Rotation

Rotation occurs about the three principle axes of any molecule unless it's linear! Because "rotation" about a linear molecule's line-of-nuclei, isn't a rotation at all; no nuclear mass is in motion! So linear molecules have only two rotation axes, perpendicular to the molecule (through its center of mass, of course), and the moments of inertia of these two rotations are identical.

You remember moment of inertia. It plays the role in rotation that mass plays to linear motion (translation). It is given as I = m_i r_i² where r is the perpendicular distance of any atom to the rotation axis.

While the classical rotational energy is E=(1/2)I², the quantal one is _rot = BJ(J+1) where B = ²/2I and is h/2. And the rotational degeneracy g_rot = 2J+1.

For the diatomic molecule (obviously linear), I = m₁ m₂ r_e² / (m₁ + m₂).

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Chris Parr
University of Texas at Dallas
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