Statistical Thermodynamics
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Transcript of Statistical Thermodynamics
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Statistical Statistical ThermodynamicsThermodynamics
Basic principles and applications
http://www.biochem.oulu.fi/Biocomputing/juffer/Teaching/Biocomputing/
From microscopic interactions tomacroscopic quantities.
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Thermodynamics (1)Thermodynamics (1)
First Law of Thermodynamics: The energy U of an isolated system is constant.
dwdqdU
Amount of heat supplied to the system.
Amount of work done on the system, e.g:
dTCdq V At constant V
dTCdq p At constant p
pdVdw Reversible pV-work
Mechanical work
qwU
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Thermodynamics (2)Thermodynamics (2)
)( charge of Change :
)(length of Change :
)( area surface of Change :
)( volumeof Change :
dqdq
dlfdl
dd
dVpdV
Types of mechanical work
potential ticElectrosta :
Length :
tensionSurface :
l
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Thermodynamics (3)Thermodynamics (3)
The temperature T of a system of N particles is a quantity related to the averaged kinetic energy of the disordered motion of the particle in the C-frame of reference.
2, 2
11ii
iavek vm
NET
Mass of particle iVelocity of particle i
Thermal equilibrium:Average kinetic energy is the same in all regions of the system
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Thermodynamics (4)Thermodynamics (4)
Second Law of Thermodynamics: The entropy S of an isolated system increases during any spontaneous change.
Lowentropy
gas
Highentropy
Spontaneous
Not observed
0S
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Thermodynamics (5)Thermodynamics (5)
Only irreversible processes generate entropy. The entropy change of reversible
process is exactly zero. Most basic irreversible process is
the generation of heat:
T
dqdS
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Thermodynamics (6)Thermodynamics (6)
System
Surroundings
System + Surroundings = Universe
0
universe
gssurroundinuniverse
dS
dSdSdS
For an equilibrium process
gssurroundindS
T
dq gssurroundin
dS
0universedS
Overall entropy productionThe universe seeks higher entropy
Heat:
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Thermodynamics (7)Thermodynamics (7)
0T
dqdSConcentrate on the system:
dq : heat supplied to the systemdS : Change of entropy of system due to internal processes.
0
0
0
dA
TdSdUT
dUdS
dqdU
At Constant Vno pV-work
dw=0
At constant pressureNo non-pV work
dp=0
0
0
0
dG
TdSdHT
dHdS
dHdq
Vdpdq
VdppdVpdVdq
VdppdVdUdH
pVUH
dwdqdU
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Thermodynamics (8)Thermodynamics (8)
energy free Gibbs
energy free Helmholtz
TSHG
TSUA
ST
H
T
G
STHG
universe theofentropy Total :
gssurroundin theofEntropy :
system theofEntropy :
T
GT
H
S
00 T
GG For a natural or
spontaneous change.
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Microscopic world versus macroscopic world
Classicalmechanics
Quantummechanics
Statisticalmechanics
Macroscopicquantities
Statistical Averaging
MicroscopicInteractionsPhase space
Wave function
Observables:Dynamic and
Thermodynamic quantities
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Concept of an ensembleConcept of an ensemble Members of the ensemble
represents a state of a system in accordance with external macroscopic parameters (e.g. N, V, T).
A state is either defined classically (a point in phase space) or quantum mechanically (quantum numbers).
Mechanical quantities M are averages over the ensemble: Pressure, volume, energy, ….. But not entropy S and free energies
(A, G).
1 2
N
i
iiMPM
Pi is probability of state i.
i
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A point in phase spaceA point in phase space
{p}
{q}
momenta
Coordinates
N particles
6N dimensionalspace
One-dimensionalharmonic oscillator
2N dimensionalphase spaceTrajectory
Up
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Quantum numbersQuantum numbers The state of the system is fully described by the
wave function. The state of the system is fully described by a
set of quantum numbers (n, m, l, …..). The total energy of the system is given from
Schrödinger equation. An observable such as the energy E corresponds to an operator such as H.
t,...,, 21 rr EH
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Probability of state Probability of state ii(quantum version)(quantum version)
i
iiMPM Average of quantum (discrete) states:Probability Pi depends on the type ofensemble.
iE
i eTVNQ
TVNP ,,
1),,(
Canonical ensemble (N,V,T)(E, p, … fluctuating) kT
1
k: Boltzmann’s constant
T: Temperature
E i
EieQ
i
Ei
Ei
i
i
e
eMM
Boltzmann factor
Partition function
P
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Free energy and entropy (1)Free energy and entropy (1)
QkTA ln
Partition function plays central in equilibrium statistical thermodynamics
Helmholtz free energy.
i
EieQ
QkT
QkTPPkS
VNiii ln
lnln
,
Entropy
i
Ei
Ei
TNie
ep
V
QkTp
,
ln Pressure
AUTS
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Free energy and entropy (2)Free energy and entropy (2)
Both entropy and free energy depend on the full phase space, or: One must sample all possible
states. Reason: Entropy:
QkTTSUA ln
Internal energyEnsemble average
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Free energy and entropy (3)Free energy and entropy (3)
Pi
States i
i
ii PPkS ln
S, A and G are very difficult to compute.
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The solute chemical potentialThe solute chemical potentialfrom thermodynamics (1)from thermodynamics (1)
Solute in a solvent
akT ln
m
ma
Standardchemical potential(reference state)
: Activity coefficientm : Molality (molarity)
mol/kg (mol/m3)m=1 mol/kg
Activity
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The solute chemical potentialThe solute chemical potentialfrom thermodynamics (2)from thermodynamics (2)
Solute in a solvent
lnln kTm
mkT
m
mkT ln Measures the deviation from
standard molality (molarity).
lnkTMeasures the deviation from ideality:
Interactions between solute molecules.1 if m0 At very low concentration (dilute).
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Equilibrium constant from Equilibrium constant from thermodynamics (1)thermodynamics (1)
ABBA Chemical equilibrium condition
ABBA
KkTG BAAB ln
m
mm
mm
m
aa
aK
BA
AB
BA
AB
BA
AB
m
mmkT
mm
mkTG
BA
AB
BA
AB
lnln
Experimentally: Measurement of concentrations provides Konly if it is assumed that =1 for all reactants.
K is dimensionless
0G
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Equilibrium constant from Equilibrium constant from thermodynamics (2)thermodynamics (2)
Solution
GasTransfer of solute from one phaseto another phase:
gassol AA
lnlnlnt kTfkTmp
pmkTG
fkTp
pkTgg lnln
Real gas
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Equilibrium constant from Equilibrium constant from thermodynamics (3)thermodynamics (3)
Previous formulation for G (or K) is not suitable for computational purposes. Instead what is required are expressions for: Standard chemical potential or some related
quantity. Activity coefficients, for low or moderate to high
concentrated solutions.