Answer ALL parts.

Expressions $($some approximate$)$ for the $1-$D translational, rotational, and vibrational

components of the molecular partition function for a diatomic molecule are given by

$q^x=(\frac{2mkT}{h^2}{)}^{\frac{1}{2}}x$

$q^R=\frac{kT}{hcB}$

$q^V=\frac{1}{1-e^{-(h\frac{c}{b}ar\frac{D}{k}T)}}$

The symbols all take their usual meanings and the relevant energy levels are treated as

being those of the quantum mechanical square well, rigid rotor and simple harmonic oscillator respectively.

$($a$)$ For a typical diatomic molecule at room temperature, state the order, from smallest to largest of $q^x,q^R$ and $q^V.$ Explain your reasoning for giving your answer in terms of the expected energy levels and typical quantum numbers populated in each case.

$4$

$($b$)$ Explain how the $1-$D partition function for translation, $q^x,$ is used to make the partition function for translation in a $3-$D cuboid container. Your answer should include the resultant form of the translational partition function $q^T,$ in $3$D$.$

$3$

Question $1$ continued.

$($c$)$ Starting with the expression for $q^R$ given above, show that the average energy stored in rotational motion of a diatomic molecule increases linearly with temperature.

$($d$)$ Explain the result from part c$)$ in terms of the principle of equipartition and the quantum mechanical behaviour of a rigid rotor.

quantum mechanical behaviour of a rigid rotor.

$($e$)$ What component of the partition function is shown in the graph below for a fictional molecular system? State information you can deduce about the molecule concerned.

$2$