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.
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
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
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
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
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
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:
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
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.
Microscopic world versus macroscopic world
Classicalmechanics
Quantummechanics
Statisticalmechanics
Macroscopicquantities
Statistical Averaging
MicroscopicInteractionsPhase space
Wave function
Observables:Dynamic and
Thermodynamic quantities
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
A point in phase spaceA point in phase space
{p}
{q}
momenta
Coordinates
N particles
6N dimensionalspace
One-dimensionalharmonic oscillator
2N dimensionalphase spaceTrajectory
Up
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
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
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
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
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.
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
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).
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
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
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.
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