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
1
Email [email protected]://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2007/USC2001/
Tel (65) 6516-2749
Topics
2
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
6
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
7
1st Law
NkdTcdU v
pdVqdU pdVadiabatic
kinetic
gas law VNkTdV /
constantloglog VTcv
constant/11 vcpV
no heat transfer
Entropy: Reversible Processes
8
Isothermal constant pV
Adiabatic
TqVpSVpS /),(),( 2233
0/),(),( 1122 TqVpSVpS
)/(log 233
2
1 VVkNpdVTV
V
0 q
Entropy: Reversible Processes
9
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
10
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
11
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
12
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
13
[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
14
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
15
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
16
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
17
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
18
,...,, 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
19
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
20
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
21
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
22
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
23
j jjdnpdVTdSdU
j jjdnpdVSdTdA
Gibbs Equations for interchanging particles are:
j jjdnVdpTdSdH
j jjdnVdpSdTdG
jiji npTjnVSjj n
G
n
U
,,,,
Material and Reaction Equilibria
24
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
25
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
26
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
27
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 ….
28
TUTORIAL 5
29
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
30
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.