# Chemical Engineering

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The following table reports the energy of the triplet and singlet states of CH2, in atomic units, derived from Hartree-Fock calculations with different basis sets.

1. Explain what is meant by the following terms, and identify one or more entries from the table above with each term:

1. Evaluate the energy difference (in kJ mol±1) between singlet and triplet states of CH2 with each basis set. Comment on how your values vary with basis set and state, with reasons, which should be the most reliable value.

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1. Explain the terms static correlation and dynamic correlation and hence suggest why the Hartree-Fock prediction of the singlet-triplet energy gap in

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1. Answer ALL parts a) ± f).

VO2 undergoes a metallic to insulator phase transition as a function of temperature. Below 340 K it is insulating. Above 340 K, it has the rutile structure, which is characterised by edge-sharing octahedra. TiO2 can also adopt the rutile structure and is semiconducting.

1. Explain why this system would distort, and what would be expected in terms of structure and electronic properties. Use a sketch of the DOS to illustrate

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1. Answer ALL parts a) ± c).

1. A general formula for the molecular partition function is given in the Appendix. With reference to this formula, describe and explain the relative magnitudes of the translational, rotational and vibrational partition functions for a diatomic molecule. As part of your answer explain the number of degrees of freedom

1. The change in entropy when a system moves from state R to state P at constant temperature can related to the canonical partition function, Q, via:

§ Q  ·

S                                       k ln ¨¨                P  ¸¸

1. qN N!

1. Using the above formulae, calculate the change in entropy when one mole of a monoatomic ideal gas undergoes an isothermal expansion in which the volume is doubled. Note that only the translational partition function need be

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Appendix 1: General Data Sheet

Physical constants and units conversions

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Appendix 2

The equations set out below may be useful in answering the questions for this paper. In each case the symbols used have their usual meanings.

Counting microstates

The number of microstates available to a system of N particles which can occupy M states can be calculated using:

Ǩ

Ǩ  Ǩ ଶǨǥǤெǨ

6WLUOLQJ¶V

IRUPXOD lnN!         N ln N N

Partition functions for molecular degrees of freedom

qR2

Rotational diatomic molecule:

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• §¨ h2E ·¸2¨© 2S m ¸¹

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EhcBV ,

Canonical partition function and entropy

The canonical partition function is related to the molecular partition function via:

1. qN N!

The canonical partition function is linked to the entropy via:

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