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Symmetry and Statistics of Molecules

Module description

Previously, in NSC1003 Foundations in Natural Sciences, we have taken a largely classical approach to the laws of thermodynamics and their applications in chemistry, noting that classical thermodynamics relies upon no microscopic model of matter. We have also in NSC1003 and then further in NSC2002 Physical Chemistry, studied the quantum mechanics of atoms and molecules, which is our microscopic model of matter. We looked at the quantum mechanics behind atomic structure, periodicity, bonding and spectroscopy. We now must provide the link between the quantum mechanics of individual atoms and molecules and the laws of classical thermodynamics that apply to large collections of these particles. This is the realm of statistical thermodynamics and it reconciles the classical and quantum worlds, provided we also recognise that, deeply rooted in molecular quantum mechanics, is molecular symmetry. The existence or not of either chemical bonds or of spectroscopic transitions depends upon an overlap of wavefunctions between initial and final states. This, ultimately, is a matter of molecular symmetry, as is the degeneracy of molecular quantum states linked by rotation, which appears directly in the results of statistical thermodynamics. We will look at the symmetry and statistics of molecules to provide a deep understanding of how these topics underpin the structure of chemistry.

Module aims - intentions of the module

The aim of this module is to build on the chemistry covered in NSC1003 Foundations in Natural Sciences and NSC2002 Physical Chemistry. The specific aims are to reconcile classical thermodynamics to quantum mechanics through statistical thermodynamics and to examine the deep link between molecular quantum mechanics and molecular symmetry. Together, these topics explain many of the properties of atoms and molecules that underpin the theory and application of chemistry in the natural sciences.

You will develop the following graduate attributes:

• People skills in communicating with peers and discussing scientific ideas
• Independent research skills related to further reading around the topic
• Applied thinking and problem-solving – applying the knowledge you have gained to solve problems related to aspects of physical chemistry

Intended Learning Outcomes (ILOs)

ILO: Module-specific skills

On successfully completing the module you will be able to...

• 1. Describe in detail the consequences of degeneracy in terms of quantum states and levels in statistical thermodynamics, including the dilute limit.
• 2. Provide a physical interpretation of the molecular partition function and, given spectroscopic and other data, employ the equations relating the molecular partition function to thermodynamic functions, including the equilibrium constant.
• 3. Identify molecular symmetry elements and operations and so assign a point group to a molecule.
• 4. Derive the reducible and irreducible representations for a molecule and apply these to molecular structure and bonding.
• 5. Predict and explain orbital overlap in LCAO-MO theory and the number of IR and Raman active vibrations for simple molecules.

ILO: Discipline-specific skills

On successfully completing the module you will be able to...

• 6. Demonstrate and apply a knowledge and understanding of molecular statistics and symmetry as part of the sub-discipline of chemistry.
• 7. Describe and begin to evaluate aspects of current research in chemistry and chemistry-related areas (e.g. climate change, functional materials and medicine) with reference to textbooks and other literature sources.

ILO: Personal and key skills

On successfully completing the module you will be able to...

• 8. Communicate ideas effectively and professionally by written means.
• 9. Participate and interact effectively and professionally in discussion of scientific ideas.
• 10. With some guidance, begin to develop the skills for independent study.

Syllabus plan

Whilst the module’s precise content may vary from year to year, it is envisaged that the syllabus will cover some or all of the following topics:

• The identification of molecular symmetry elements and operations.
• Assigning a point group to a molecule.
• Deriving the reducible and irreducible representations for a molecule and applying these to molecular structure and bonding.
• Predicting orbital overlap in LCAO-MO theory.
• Predicting the number of IR and Raman active vibrations for simple molecules.
• A summary of combinatorial statistics and the Legendre method of undetermined multipliers.
• Derivation of the Fermi-Dirac and Bose-Einstein distributions, followed by their conflation in the dilute limit, within which the molecular partition function is introduced.
• The formulation of the functions of thermodynamics, including the equilibrium constant, in terms of the molecular partition function.
• A reprise of the quantisation of translational, rotational, vibrational and electronic energies of atoms and molecules, leading to equations for the contribution of these energy modes to the molecular partition function.
• The calculation of thermodynamic parameters, including the equilibrium constant, using the equations of statistical thermodynamics.

Learning activities and teaching methods (given in hours of study time)

Details of learning activities and teaching methods

Formative assessment

Summative assessment (% of credit)

Details of summative assessment

Details of re-assessment (where required by referral or deferral)

Re-assessment notes

Deferral – if you have been deferred for any assessment you will be expected to submit the relevant assessment. The mark given for a re-assessment taken as a result of deferral will not be capped and will be treated as it would be if it were your first attempt at the assessment.

Referral – if you have failed the module overall (i.e. a final overall module mark of less than 40%) you will be required to sit a further examination. The mark given for a re-assessment taken as a result of referral will count for 100% of the final mark and will be capped at 40%.

Indicative learning resources - Basic reading

• A. Vincent, “Molecular Symmetry and Group Theory”, 2^nd Edition, Wiley
• N.M. Laurendeau, “Statistical Thermodynamics: Fundamentals and Applications”, Cambridge University Press.

Indicative learning resources - Other resources

• Primary literature

Key words search

Molecular symmetry, group theory, statistical thermodynamics