USSC2001 Energy Lectures 4&5

Physical ChemistryChemical Thermodynamics

Bio-Organic Chemistry and Protein FoldingWayne M. Lawton

Department of MathematicsNational University of Singapore

2 Science Drive 2Singapore 117543

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Email matwml@nus.edu.sghttp://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2007/USC2001/

Tel (65) 6516-2749

Topics

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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.

Entropy – Boltzmann’s “Itsy Bitsy Teeny Weeny” pic.

Mass Action – it’s the law!

Derivation of Boltzmann’s Distribution

Derivation of Quantum Effects on Heat Capacity

Derivation of Mass Action for Ideal Gases

Entropy Driven Bioorganic Processes

Membrane MagicAssume slow motion, thermal equilibrium temp.

Argon+Helium Helium

F

LV

3

Helium-permeable membrane

T

piston

RV

RTnVp LLL RTnVp RRR

nnn VVV

LR nn V

VR

R

start nmoles of eachgas

2log2/2/

nRTdVV

nRTdVppw L

V

VL

L

V

V RL

RL VVV

Ideal Gases and Kinetic Theory

4

Ideal Gas Law nRTNkTpV Kinetic Theory nTCNkTcU vv vc dimensionless heat capacity at constant volume

for linear-molecular gases

HeAr,

22 , OH

2CO

vC molar heat capacity at constant volume

R = 8.314472(15) J / mole-K

RCRc pv

vibrotd ddd 23,2 effective degrees of freedom

0,0 vibrot dd for monotomic gases

0,2 vibrot dd

SH 2

1,2 vibrot dd0,3 vibrot dd

for linear-molecular gas

for nonlinear-mol. gas

Quantum Effects on Heat Capacity

5

“Theoretical”Gas

3=3+0rot+0vib

Experimental

Argon

Helium

Hydrogen

Carbon Dioxide

http://www.physics.dcu.ie/~pvk/ThermalPhysics/SpecificHeat/index.htm

http://en.wikipedia.org/wiki/Heat_capacity#Heat_capacity

4.9325

6.8857

Oxygen

Hydrogen Sulfide

5.0672

25 °C, 100 kPa

6.3228

3

3=3+0rot+0vib 3

5=3+2rot+0vib

5=3+2rot+0vib

7=3+2rot+2vib

6=3+3rot+0vib

Entropy: Thermodynamic Laws

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1st Law

q heat absorbed by gas

w-qU d

2nd Law: There exists an entropy function ),(SS Vp

such that during any thermodynamic process

dSTq /

w work done by gas pdV

with equality holding iff the process is reversible.

Entropy: Adiabatic Expansion

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1st Law

NkdTcdU v

pdVqdU pdVadiabatic

kinetic

gas law VNkTdV /

constantloglog VTcv

constant/11 vcpV

no heat transfer

Entropy: Reversible Processes

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Isothermal constant pV

Adiabatic

TqVpSVpS /),(),( 2233

0/),(),( 1122 TqVpSVpS

)/(log 233

2

1 VVkNpdVTV

V

0 q

Entropy: Reversible Processes

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P

V 11 ,VP

22 ,VP

33 ,VP

adiabatic

isothermal

vv cc VpVp /1122

/1111

3322 VpVp

vv cc VVppVV 1131323 )/()/(/

)log)1(log1

3

1

3( VV

vpp

v ccNk ),(),( 1133 VpSVpS

Entropy: Free Ideal Gas Expansion

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A gas initially confined in a chamber with volumeis released suddenly into a chamber with volume aV

V

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

anRakNS loglog

)log)1(log( Vcpc vvNk ),( VpS

Thermodynamic Potentials

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Internal Energy

Helmholz Free Energy

U

TSUA

Enthalpy pVUH

Gibbs Free Energy TSHG

pdVTdSpdVQdU

pdVSdTSdTTdSpdVQdA

VdpTdSVdpQdH

VdpSdTSdTTdSVdpQdG A process is reversible if and only if equality holds, the equations are called Gibbs Equations http://en.wikipedia.org/wiki/Willard_Gibbs

http://en.wikipedia.org/wiki/Chemical_thermodynamics

Equilibria and Reversibility

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For any process with constant row & column variable

and then the system is in equilibrium if and only if GAT

HUS

pVthe corresponding variable

in the table satisfies

0)variable( d

0)variable( d

Law of Mass Action

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[3] elucidated by C.M. Guldberg and P.Wage in 1860s

For an arbitrary [chemical] transformation

the reaction quotient ).(][][

][][TK

BA

DCba

dc

dDcCbBaA

Moreover, thermodynamics implies that

)(log)left()right( TKRTGG

Boltzmann’s Formula

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WkS log

Take in the ideal gas law free expansion. 2a

For each of the N molecules, the number of its

hence the entropy increase is the increase of

originalchamber

additionalchamber

microstates is doubled (after the expansion it can be in either chamber with equal probability), so the number W of microstates of the gas is multiplied by the factor

N2- the famous formula due to

http://en.wikipedia.org/wiki/Ludwig_Boltzmann

TUTORIAL 4

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4. Explain (i) how the Carnot cycle works, (ii) how to make separate salt into Cl and Na with heat using mass action.

1. Refine the method in vufoil 2 to explain how to derive energy = 2nRTlog(2) by ‘mixing’ n-moles of each gas that areinitially contained in left and right haves of the container, in

anisothermal (at temperature T) and (exactly) reversible process.

5. Describe Boltzmann’s distribution, then how it explains the distribution of speeds of molecules in a gas.

2. Discuss the thermodynamics of reverse osmosis as appliedto 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.

Boltzmann’s Formula Revisited

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11 logWkS

Consider 2 systems that can exchange energy

The number of states for the combined system equals

Entropy is maximized (Murphy’s Law)

22 logWkS

2121 log SSWkSWWW Energy can flow between the systems but is conserved

)( 111 EWW

21 EEE

mequilibriuthermal12

2111

21

TE

S

E

ST

EE

)( 222 EWW

Entropy Formula Derived

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Consider system 1 that can be in states 1,2,3, …with

jj NpN

probabilities ,...,, 321 ppp What is its entropy?

Consider N such systems. The law of large numbers

1

!/!jN

jN NNW

so the number of states for the system of N systems is

given by the multinomial theorem as

the number of systems in state j,

Stirlings Approximation gives the entropy of 1 system

jj jNN ppkWNkppS logloglim...),( 121

Boltzmann 1866

Gibbs 1897

Shannon 1948

von Neumann 1927

Boltzmann’s Distribution Derived

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,...,, 321 EEEConsider system 1 that can be in states with energy

interacting with an environment with

energy

,...,, 321 EEE

TtemperatureE )(ESS We wish to compute the probabilities ,...,, 321 pppsystem 1 is in a state with energy

The entropy of the total system (system 1 + envir.) is

j

jjjj

jjj pkTEESppkEESp log/)(log)(

which is maximized when )(// TZep kTEj

j

and entropy

where the Zustandsumme j

kTE jeTZ /)(or partition function

Maxwell-Boltzmann Distribution Derived

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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

)(/),,( 2/)( 222

TZevvvp kTvvvmzyx

zyx 2/3

)(

2/)( )/2()(2

,,

222

mkTdvdvdveTZ zyx

Rvvv

kTvvvm

zyx

yyx

http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution

kTmvevkTmvp 2/22/3 2

4)2/()( 222zyx vvvv

B. Distribution for Classical Harmonic Oscillator

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Recall that the energy of a CHO is

22

21

21),( xxmxxE

whence )(/),( 2/)( 22

TZexxp kTxxm mTkxdxdeTZ

Rxx

kTxxm /2)(2

22

),(

2/)(

stiffness

Hence the expected internal energy of a CHO is

TkxdxdxxpxxEExx

),(),(),(

so for an ideal gas

1vib.modemode.vib dNkTcENU v

for N molecules each vibrational mode contributessee p. 4

B. Distribution for Quantum Harmonic Oscillator

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The energy levels occur only in discrete quanta

0),( 21 nnEn

Case 1 gives the classical result, Case 2 ‘freezes’ out the vibration

whence

1/2/

0

/ ]1[)( kTkTkTE eeeTZ n

so 0

/1)( nkTE EeTZE n

1/ ]1[ kTe 2/ kTEkT0/ kTeEkT

Case 1

Case 2

http://www.fordham.edu/academics/programs_at_fordham_/chemistry/courses/fall_2008/physical_chemistry_i/lectures/equipartition_6542.asp

Thermodynamic Potentials and the Partition Function

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jj

jkTE

j ppSTZep n log)(//

)(log/)](log/[ TZkTETZkTEp nj

j )(log TZTSUA

NTV

ZkTp

,

Other Relationships

NVT

ZkTU

,

2

VTn

ZRT

,

Molar

Chemical

Potential

Chemical Potentials

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j jjdnpdVTdSdU

j jjdnpdVSdTdA

Gibbs Equations for interchanging particles are:

j jjdnVdpTdSdH

j jjdnVdpSdTdG

jiji npTjnVSjj n

G

n

U

,,,,

Material and Reaction Equilibria

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A system is in material equilibrium if and only if

j jjdn 0

In constant T, p is in equilibrium if and only if

A reaction represented by

nnmmmm AvAvAvAv 1111 )()(

that has gone to extent ij vdn ,

j jjvd

dG0

Reaction Equilibrium for an Ideal Gas Mixture

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Then )/log()( PPRTT iii

So

where

denotes the partial pressure of the i-th gas.iP denote quantities at 1 atmosphere pressure

and

i

viii iii i

iPPRTTvv )/log()(0

i

vi

iPPRTTG )/(log)(

Hey - ain’t this Mass Action ?

)(log TKRT

Haber Process

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322

23

]/)H([]/)N([

]/)HN([)(

PPPP

PPTK

eqeq

eq

322 HN2H3N

http://en.wikipedia.org/wiki/Fritz_Haber

http://en.wikipedia.org/wiki/Haber_process

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

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Entropy in Bioorganic ChemistryThe 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 quantumchemical ….

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TUTORIAL 5

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4. Discuss the thermodynamics of the Haber process.

1. Learn Stirling’s Approximation is and use it to derive the entropy formula on vufoil 17.

5. Discuss the role of entropy in several metabolic processes, use the following and other websites

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 ?

jp

vc

http://www.proteinscience.org/cgi/content/abstract/5/3/507

http://en.wikipedia.org/wiki/Entropy_and_life

http://en.wikipedia.org/wiki/Diatomic

References

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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.