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Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2

USSC2001 Energy Lectures 4&5 Physical Chemistry Chemical Thermodynamics Bio-Organic Chemistry and Protein Folding. Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543. Email matwml@nus.edu.sg

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Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2

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  1. USSC2001 Energy Lectures 4&5 Physical ChemistryChemical ThermodynamicsBio-Organic Chemistry and Protein Folding Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2007/USC2001/ Tel (65) 6516-2749 1

  2. Topics Membrane Magic – power your mobile through the interaction of [not so] inert gases! Kinetic Theory of Gases – quantum effects lah? Entropy – and other state functions, the big picture. Mass Action – it’s the law! Entropy – Boltzmann’s “Itsy Bitsy Teeny Weeny” pic. Derivation of Boltzmann’s Distribution Derivation of Quantum Effects on Heat Capacity Derivation of Mass Action for Ideal Gases Entropy Driven Bioorganic Processes 2

  3. Membrane Magic Assume slow motion, thermal equilibrium temp. Helium-permeable membrane start n moles of each gas piston Argon+Helium Helium 3

  4. Ideal Gases and Kinetic Theory Ideal Gas Law Kinetic Theory dimensionless heat capacity at constant volume effective degrees of freedom molar heat capacity at constant volume R = 8.314472(15) J / mole-K for monotomic gases for linear-molecular gases for linear-molecular gas for nonlinear-mol. gas 4

  5. Quantum Effects on Heat Capacity 25 °C, 100 kPa Gas “Theoretical” Experimental Argon 3=3+0rot+0vib 3 Helium 3=3+0rot+0vib 3 Hydrogen 4.9325 5=3+2rot+0vib Oxygen 5=3+2rot+0vib 5.0672 Carbon Dioxide 6.8857 7=3+2rot+2vib Hydrogen Sulfide 6.3228 6=3+3rot+0vib http://en.wikipedia.org/wiki/Heat_capacity#Heat_capacity http://www.physics.dcu.ie/~pvk/ThermalPhysics/SpecificHeat/index.htm 5

  6. Entropy: Thermodynamic Laws 1st Law heat absorbed by gas work done by gas 2nd Law: There exists an entropy function such that during any thermodynamic process with equality holding iff the process is reversible. 6

  7. Entropy: Adiabatic Expansion 1st Law adiabatic no heat transfer gas law kinetic 7

  8. Entropy: Reversible Processes Isothermal Adiabatic 8

  9. Entropy: Reversible Processes adiabatic isothermal 9

  10. Entropy: Free Ideal Gas Expansion A gas initially confined in a chamber with volume is released suddenly into a chamber with volume The gas does not push against anything movable, it does no work. Therefore the 1st law implies that the internal energy, and hence temperature, is constant. The ideal gas law implies that the pressure changes by the factor 1/a, hence the change in entropy is 10

  11. Thermodynamic Potentials Internal Energy Helmholz Free Energy Enthalpy Gibbs Free Energy A process is reversible if and only if equality holds, the equations are called Gibbs Equations http://en.wikipedia.org/wiki/Willard_Gibbs 11 http://en.wikipedia.org/wiki/Chemical_thermodynamics

  12. Equilibria and Reversibility For any process with constant row & column variable the corresponding variable in the table satisfies and then the system is in equilibrium if and only if 12

  13. Law of Mass Action [3] elucidated by C.M. Guldberg and P.Wage in 1860s For an arbitrary [chemical] transformation the reaction quotient Moreover, thermodynamics implies that 13

  14. Boltzmann’s Formula Take in the ideal gas law free expansion. additionalchamber originalchamber For each of the molecules, the number of its microstates is doubled (after the expansion it can be in either chamberwith equal probability), so the number W of microstates of the gas is multiplied by the factor hence the entropy increase is the increase of - the famous formula due to http://en.wikipedia.org/wiki/Ludwig_Boltzmann 14

  15. TUTORIAL 4 • Refine the method in vufoil 2 to explain how to derive • energy = 2nRTlog(2) by ‘mixing’ n-moles of each gas that are • initially contained in left and right haves of the container, in an • isothermal (at temperature T) and (exactly) reversible process. 2. Discuss the thermodynamics of reverse osmosis as applied to desalinate and/or purify water. 3. Derive the formula for the entropy change of N molecules of an ideal gas (as a function of pressure and volume) by computing the change of q/T over a path that consists of one isobaric path and one isochoric path. 4. Explain (i) how the Carnot cycle works, (ii) how to make separate salt into Cl and Na with heat using mass action. 5. Describe Boltzmann’s distribution, then how it explains the distribution of speeds of molecules in a gas. 15

  16. Boltzmann’s Formula Revisited Consider 2 systems that can exchange energy The number of states for the combined system equals Energy can flow between the systems but is conserved Entropy is maximized (Murphy’s Law) 16

  17. Entropy Formula Derived Consider system 1 that can be in states 1,2,3, …with probabilities What is its entropy? Consider N such systems. The law of large numbers  the number of systems in state j, so the number of states for the system of N systems is Boltzmann 1866 given by the multinomial theorem as Gibbs 1897 von Neumann 1927 Shannon 1948 Stirlings Approximation gives the entropy of 1 system 17

  18. Boltzmann’s Distribution Derived Consider system 1 that can be in states with energy interacting with an environment with energy temperature and entropy We wish to compute the probabilities system 1 is in a state with energy The entropy of the total system (system 1 + envir.) is which is maximized when where the Zustandsumme 18 or partition function

  19. Maxwell-Boltzmann Distribution Derived For continuously distributed energies, sums are replaced by integrals, therefore the MB distribution that describes the probability density for velocities of molecules in a gas is given by 19 http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution

  20. B. Distribution for Classical Harmonic Oscillator stiffness Recall that the energy of a CHO is whence Hence the expected internal energy of a CHO is so for an ideal gas for N molecules each vibrational mode contributes see p. 4 20

  21. B. Distribution for Quantum Harmonic Oscillator The energy levels occur only in discrete quanta whence so Case 1 Case 2 Case 1 gives the classical result, Case 2 ‘freezes’ out the vibration http://www.fordham.edu/academics/programs_at_fordham_/chemistry/courses/fall_2008/physical_chemistry_i/lectures/equipartition_6542.asp 21

  22. Thermodynamic Potentials and the Partition Function Other Relationships MolarChemical Potential 22

  23. Chemical Potentials Gibbs Equations for interchanging particles are: 23

  24. Material and Reaction Equilibria A system is in material equilibrium if and only if A reaction represented by that has gone to extent In constant T, p is in equilibrium if and only if 24

  25. Reaction Equilibrium for an Ideal Gas Mixture Then where denote quantities at 1 atmosphere pressure and denotes the partial pressure of the i-th gas. So Hey - ain’t this Mass Action ? 25

  26. Haber Process http://en.wikipedia.org/wiki/Haber_process http://en.wikipedia.org/wiki/Fritz_Haber 26

  27. Entropy in Bioorganic Chemistry • Bioorganic Chemistry and the Origin of Life • A challenging theme in bioorganic chemistry is the unification of .... that in every spontaneous process the entropy increases, or, put otherwise, ...www.springerlink.com/index/XX4012001N34T686.pdf - Similar pagesby CM Visser - 1978 - Cited by 5 – Related articles - All 2 versions 27

  28. Entropy in Bioorganic Chemistry The Bioorganic Chemistry Laboratory led by Prof. Qingxiang Guo works on the molecular recognition, electron transfer reactions in supramolecular systems and green chemistry. The research projects are supported by the Ministry of Science and Technology (MOST), the CAS, the Ministry of Education and the National Science Foundation of China (NSFC). Employing experimental and theoretical methods, such as artificial neural networks and genetic algorithm, researchers in the lab predicted the driving forces and composition of driving forces for the molecular recognition of cyclodextrins. The binding constants for the inclusion complexation of cyclodextrins with substartes calculated were closed to the experimental data (J. Phys. Chem. B, 1999). Enthalpy-entropy compensation effect was observed widely existent in the chemical and biological process. They studied the enthalpy-entropy compensation in protein unfolding and molecular recognition of cyclodextrin and suggested a new model for enthalpy-entropy compensation with a huge amount of experimental parameters and theoretical analysis (Chem. Rev. 2002). They designed and synthesized some electron-accepting receptors with cyclodextrin as the framework. Supramolecular systems of the receptor with electron-donating substrates, such as naphthalene derivatives was formed by the host-guest interaction. The high efficient photoinduced elctron transfer reaction in the supramolecular system was observed in the lab (J. Org. Chem. 2002). In order to increase the efficiency and selectivity and reduce the generation of waste in organic synthetic reactions, they studied the organic reactions in solventless or in environmentally benign solvent, e.g. water and supercritical fluids. Recently, a novel coupling reaction of carbonyl compounds in the presence of alkali metals without solvent was developed. Based on the product analysis, the ESR evidence and quantum chemical …. 28

  29. TUTORIAL 5 1. Learn Stirling’s Approximation is and use it to derive the entropy formula on vufoil 17. 2. Learn the Method of Lagrange Multipliers and use it to derive the formula for on vufoil 18. 3. What are typical values of for rotational and vibrational energies of diatomic molecules, how do they compare with kT at room temperature, and how do they effect ? http://en.wikipedia.org/wiki/Diatomic 4. Discuss the thermodynamics of the Haber process. 5. Discuss the role of entropy in several metabolic processes, use the following and other websites http://en.wikipedia.org/wiki/Entropy_and_life http://www.proteinscience.org/cgi/content/abstract/5/3/507 29

  30. References 1. Atkins, P.W., Physical Chemistry, Oxford, 1982. 2. Levine, I.N., Physical Chemistry, McGraw, 1983. 3. Munowitz,M.,Principles of Chemistry,Norton,2000. 4. Petz, D.,Entropy, von Neumann and the von Neumann entropy, http://arxiv.org/PS_cache/math-ph/pdf/0102/0102013v1.pdf. 5. Branden, C. and Tooze, J., Introduction to Protein Structure, Garland, 1991. 6. Huang, K., Lectures on Statistical Physics and Protein Folding, World Scientific, 2005. 7. Schrodinger, E., What is Life with Mind and Matter and Autobiographical Sketches, 1944. 30

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