Transcript Entropy - Department of Mathematics

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 [email protected]
http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2007/USC2001/
Tel (65) 6516-2749
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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
Membrane Magic
Assume slow motion, thermal equilibrium temp. T
Helium-permeable membrane
V  VL  VR
start n
moles
piston
of each
F
gas
VL
VR
pLVL  nL RT Argon+Helium Helium p RVR  nR RT
nL  n 
V VR
V
n
 nRT
w    pL  pR  dVL   
V /2
V /2
 VL
V
V
nR 
VR
V
n

 dVL  nRT log 2
3

Ideal Gases and Kinetic Theory
Ideal Gas Law pV  NkT  nRT
Kinetic Theory U  c NkT  C nT
cv d
v
v
dimensionless heat capacity at constant volume
 2 , d  3  d rot  2d vib effective degrees of freedom
C v  molar heat capacity at constant volume
 cv R  C p  R R = 8.314472(15) J / mole-K
Ar, He
d rot  0, d vib  0 for monotomic gases
d rot  2, d vib  0 for linear-molecular gases H 2 , O2
d rot  2, d vib  1 for linear-molecular gas CO2
d rot  3, d vib  0 for nonlinear-mol. gas H 2 S
4
Quantum Effects on Heat Capacity
25 °C, 100 kPa
Gas
“Theoretical”
Argon
3=3+0rot+0vib
3
Helium
3=3+0rot+0vib
3
Hydrogen
5=3+2rot+0vib
4.9325
Oxygen
5=3+2rot+0vib
Carbon Dioxide
Hydrogen Sulfide
7=3+2rot+2vib
5.0672
6.8857
6.3228
6=3+3rot+0vib
Experimental
http://en.wikipedia.org/wiki/Heat_capacity#Heat_capacity
http://www.physics.dcu.ie/~pvk/ThermalPhysics/SpecificHeat/index.htm
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Entropy: Thermodynamic Laws
1st Law
dU  q - w
q  heat absorbed by gas
w  work done by gas  pdV
2nd Law: There exists an entropy function S  S ( p,V )
such that during any thermodynamic process
q / T  dS
with equality holding iff the process is reversible.
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Entropy: Adiabatic Expansion
dU  q  pdV
adiabatic


pdV
no heat transfer
gas law
 NkTdV / V
kinetic dU  c NkdT
v
 cv log T  log V  constant
11 / cv
pV
 constant 7
1st Law
Entropy: Reversible Processes
Isothermal

pV  constant
S ( p3 , V3 )  S ( p2 , V2 )   q / T
1 V3
T V
2
 pdV  N k log (V3 / V2 )
Adiabatic
 q0
S ( p2 ,V2 )  S ( p1,V1 )   q / T  0
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P
Entropy: Reversible Processes
adiabatic
P2 ,V2 
P1 ,V1 
p1V1
isothermal
11 / cv
 p2V2
11 / cv
p 2V2  p3V3
P3 ,V3 
 V3 / V2  ( p3 / p1 ) (V3 / V1 )
cv
1 cv
V
S ( p3 , V3 )  S ( p1 , V1 ) 
p3
V3
Nk (cv log p1  (1  cv ) log V1 )
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Entropy: Free Ideal Gas Expansion
S ( p, V ) 
Nk (cv log p  (1  cv ) log V )
A gas initially confined in a chamber with volume V
is released suddenly into a chamber with volume aV
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
 S  N k log a  nR log a
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Thermodynamic Potentials
Internal Energy U
dU  Q  pdV  TdS  pdV
Helmholz Free Energy A  U  TS
dA  Q  pdV  TdS  SdT   SdT  pdV
Enthalpy H  U  pV
dH  Q  Vdp  TdS  Vdp
Gibbs Free Energy G  H  TS
dG  Q  Vdp  TdS  SdT   SdT  Vdp
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
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Equilibria and Reversibility
For any process with constant row & column variable
the corresponding variable
in the table satisfies
d ( variable )  0
V
S U
T A
p
H
G
and then the system is in equilibrium if and only if
d ( variable )  0
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Law of Mass Action
[3] elucidated by C.M. Guldberg and P.Wage in 1860s
For an arbitrary [chemical] transformation
aA  bB  cC  dD
c
the reaction quotient
d
[C ] [ D]

K
(
T
).
a
b
[ A] [ B]
Moreover, thermodynamics implies that
G (right )  G (left )   RT log K (T )
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Take
a2
Boltzmann’s Formula
in the ideal gas law free expansion.
original
chamber
additional
chamber
For each of the N molecules, the number of its
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
2
N
hence the entropy increase is the increase of
S  k log W
- the famous formula due to
http://en.wikipedia.org/wiki/Ludwig_Boltzmann
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TUTORIAL 4
1. 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.
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Boltzmann’s Formula Revisited
Consider 2 systems that can exchange energy
S1  k log W1
W1  W1 ( E1 )
S2  k logW2
W2  W2 ( E2 )
The number of states for the combined system equals
W  W1 W2  S  k log W  S1  S2
Energy can flow between the systems but is conserved
E  E1  E2
Entropy is maximized (Murphy’s Law)
S1
S 2
1
T1 

 T21  thermal equilibriu m
E E1 E E2
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Entropy Formula Derived
Consider system 1 that can be in states 1,2,3, …with
probabilities p1, p2 , p3 ,... What is its entropy?
Consider N such systems. The law of large numbers
 the number of systems in state j, N j  Np j
so the number of states for the system of N systems is
given by the multinomial theorem as
WN  N! /  N j !
N j 1
Boltzmann 1866
Gibbs 1897
von Neumann 1927
Shannon 1948
Stirlings Approximation gives the entropy of 1 system
S ( p1 , p2 ...)  k lim N  N log WN  k  j p j log p j
1
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Boltzmann’s Distribution Derived
Consider system 1 that can be in states with energy
E , E , E ,... interacting with an environment with
1
2
3
energy E temperature T and entropy S  S (E )
We wish to compute the probabilities p1 , p2 , p3 ,...
system 1 is in a state with energy E1 , E2 , E3 ,...
The entropy of the total system (system 1 + envir.) is
 p S ( E  E )  k log p    p S ( E )  E
j
j
j
j
j
j
j
/ T  k log p j

 E j / kT
which is maximized when p j  e
where the Zustandsumme Z (T ) 
or partition function
/ Z (T )
 E j / kT
e
j
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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
p(vx , vy , vz )  e
Z (T ) 
 e
 m ( vx2 v2y vz2 ) / 2 kT
 m ( vx2  v 2y  v 2y ) / 2 kT
/ Z (T )
dvx dv y dvz  (2 kT / m )
( v x ,v y , v z )R 2
3/ 2
 p(v)  (m / 2 kT ) 4 v e
3/ 2
v  v v v
2
x
2
y
2
z
http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution
2  mv2 / 2 kT
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B. Distribution for Classical Harmonic Oscillator
stiffness
Recall that the energy of a CHO is
E ( x, x )  mx  x
 ( mx 2  x 2 ) / 2 kT
whence p( x, x
)  e
/ Z (T )

Z (T )   e
dxdx  2 k T / m
1
2
2
1

2
2
 ( mx 2  x 2 ) / 2 kT
( x , x )R 2
Hence the expected internal energy of a CHO is
 E    E ( x, x ) p( x, x)dxdx  k T so for an ideal gas
( x , x )
for N molecules each vibrational mode contributes
U vib. mode  N  E   cv NkT  d vib.mode  1 see p. 4
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B. Distribution for Quantum Harmonic Oscillator
The energy levels occur only in discrete quanta
1
En   (n  ), n  0 whence
2
  E / kT
n
Z (T )   0 e
so
 E   Z (T )
1
  [e
Case 1
Case 2
e
  / 2 kT
e
  E / kT
n
0
 / kT
 1]
[1  e
  / kT 1
]
En
1
kT     E   kT   / 2
  / kT
kT     E    e
0
Case 1 gives the classical result, Case 2 ‘freezes’ out the vibration
http://www.fordham.edu/academics/programs_at_fordham_/chemistry/courses/fa
ll_2008/physical_chemistry_i/lectures/equipartition_6542.asp
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Thermodynamic Potentials and the Partition Function
pj  e
 En / kT
/ Z (T )  S   p j log p j
j
  p j [ En / kT  log Z (T )]   E  / kT  log Z (T )
j
 A  U  TS  log Z (T )
 Z 
p  kT  
 V T , N
Molar
Chemical
Potential
Other Relationships
 Z 
U  kT  
 T V , N
2
 Z 
   RT  
 n T ,V
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Chemical Potentials
Gibbs Equations for interchanging particles are:
dU  TdS  pdV   j  j dn j
dA   SdT  pdV   j  j dn j
dH  TdS  Vdp   j  j dn j
dG   SdT  Vdp   j  j dn j
 U
 j  

n
 j

 G 


 

 n 
 S ,V ,ni j
 j T , p ,ni j
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Material and Reaction Equilibria
A system is in material equilibrium if and only if


dn

0
j
j
j
A reaction represented by
(v1 ) A1    (vm ) Am  vm1 Am1    vn An
that has gone to extent  ,
dn j  vi
In constant T, p is in equilibrium if and only if
dG
  j  jv j  0
d
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Reaction Equilibrium for an Ideal Gas Mixture
i   (T )  RT log( Pi / P )
where  denote quantities at 1 atmosphere pressure

i
Then
and
Pi
So 0 

denotes the partial pressure of the i-th gas.
 v    v  (T )  RT  log( P / P )
i i
i
i i

i
 vi
i
i
 G (T )   RT log i ( Pi / P )

 vi
  RT log K (T )
Hey - ain’t this Mass Action ?
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Haber Process
http://en.wikipedia.org/wiki/Haber_process
http://en.wikipedia.org/wiki/Fritz_Haber
N 2  3H 2  2 NH3
 2
K (T ) 
[ P( N H 3 )eq / P ]

 3
[ P( N 2 ) eq / P ][ P(H 2 ) eq / P ]
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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/XX4012001N34T6
86.pdf - Similar pages
by CM Visser - 1978 - Cited by 5 – Related
articles - All 2 versions
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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 ….
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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 p j 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 cv ?
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
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.
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