the mathematical description of the motion of atoms...
TRANSCRIPT
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The mathematical description of the motion of
Atoms, Molecules & Other Particles
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Particle Dynamics
Mixture of gases are made of different entities: atoms, molecules, ions, electrons.
In principle, the knowledge of the state of the system amounts at knowing the instantaneousposition, velocity and the number and nature of all particles in the system.
Molecular dynamics is a mathematical model describing the free motion of each particle and theinteractions with other particles. This approach can be pursued only for system of small sizegiven that in a mole of substance there are as many as 6.023 1023 particles to track and follow.
Let us suppose that this system is perturbed off a state of “equilibrium”; next, different particles,and different internal degree of freedom of complex particles, will recover their un-perturbed state(supposing that the equilibrium is “stable”) at different rates which might differ by several ordersof magnitudes.
This offers an opportunity to simplify the mathematical description of the system.
The processes driving the system back to its (stable) equilibrium state are called relaxationprocesses.
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Relaxation Processes
Main relaxation processes:1) Translational relaxation: elastic collision among particles2) Rotational & Vibrational relaxation: excitation & de-excitation of internal dof of polyatomic molecules3) Electronic relaxation: formation and decay of excited states of atoms, molecules, ions, …4) Macroscopic chemical reactions elementary reactions among neutral particles5) Macroscopic kinetic processes involving
charged particles in plasma at low TMain issues in the mathematical description of a kinetic problem:1) Identification of the kinetic mechanism2) Identification of the rate equations, and the characteristic time scales associated with the relaxation process driving
the kinetic mechanismThe relaxation processes can be thus ranked according with their corresponding time scale:
It might happen that after a time of the order of1) Macroscopic chemical reactions are still far from their chemical equilibrium state2) Translational Rotational & Vibrational relaxation processes have regained a nearly thermal equilibrium state3) All other slower processes remain nearly frozen
In general, if the following inequalities hold:
all fast processes can be considered in near-equilibriumall * processes are in non equilibriumall slow processes can be considered frozen
! 0
free mean path! " ! trasl " ! rot #
gap!! vib#
gap!! chem#
gap!! elec#
gap!! plasma
! chem
! fast ! !* ! ! slow
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Statistical Mechanics
One way to reduce the complexity of the mathematical description of the dynamics of an ensamble of particles is tointroduce macroscopic properties on the basis of statistics over the ensamble.
Fluctuations over the time scale of the order of are replaced by their average value.
Let us suppose that a small perturbation is applied to a system in equilibrium.
On time scale of the order very few collisions occur;the system can be described only by resorting to the Molecular Dynamics;
On time scale of the order the system experienced a number of collisions that redistribute the energy;The system can be described by introducing a distribution function f(t,x,v) and a set of equations describing their
evolution (Boltzmann Equations) [f is the probability of finding a particle with velocity v at (x,t) ].
On time scale of the order the system experienced very many collisions that have completely redistributed theenergy of the perturbation;
The system can be described by introducing the concept of local equilibrium, whose changes can be described by thethermodynamics of irreversible processes (Continuum description / Navier Stokes Equations).
For the system approaches an equilibrium condition described by the thermo-dynamics of reversible processes(thermo-static).
1 / N = 1 / 1023 ! 10!12 s
! 0 " r0 / v ! 2.5x10#13s
! ! 10"10 s
! ! 10"10 s
r0 : free mean pathv : translational speed
! ! 1s
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Thermodynamical Equilibrium
The system is described as a continuum.
A system is said to be in thermodynamic state of equilibrium if the following 3 conditions hold atthe same time:
1) There exists no temperature gradients in the system and the temperature of the system is thesame of the surrounding environment (Thermal equilibrium)
2) There exists no pressure gradients in the system and the pressure of the system is the same ofthe surrounding environment (Mechanical equilibrium)
3) There exists no composition variation in the system and the composition of the system is thesame of the surrounding environment (Chemical equilibrium)
The state of the system under equilibrium conditions can be identified by two independent statevariables, such as p and T.
In equilibrium conditions, the chemical composition is a function of p and T, and thus it is adependent state variable Y*=f(p,T)
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Thermodynamical Non Equilibrium
The system is described as a continuum.
It is assumed that a state of local equilibium holds.
Each portion of the system forms a macroscopic system surrounded by the surrounding
In each portion, we define thermodynamic state variables having a constant value over thesmall portion
The state variables can take different values over adjacent portions of the system, this yielding non-uniform spatio-temporal fields
Aim of the non-equilibrium thermodynamics is the quantitative prediction of the evolution of these fields
The assumption of local equilibrium enables to define a local value of entropy as in equilibrium thermo-dynamics
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Assumption of Local Equilibrium
The assumption of local equilibrium enables to define a local value of the classic thermodynamicentropy.
The assumption of local equilibrium holds when:
!T " #T#x
!x " $T l % !TT
"l $TT!1 l=free mean path
!pp
"l $pp!1
!N j
N j
!1
l $va!1 a=speed of sound
These conditions are violated in:rarefied gases across strong shock waveswhen the boundary conditions enforce changes to the system on a too “fast” scale
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Atomic & Molecular WeightsPrinciples of Nonequilibrium Thermodynamics
Entropy Production in Chemical Nonequilibrium
http://en.wikipedia.org/wiki/Category:Physical_chemistryhttp://en.wikipedia.org/wiki/Category:Thermodynamics
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Unit of Amount of Substance
Mixture of gases are made of different entities: atoms, molecules, ions, electrons
The unit of amount of substance (the mole) is defined as the amount of substance of a systemwhich contains as many elementary entities as there are atoms in 0.012 Kg of unbound atoms ofCarbon 12, at rest and in ground state
By intl agreement, the atomic weight of the isotope of Carbon with mass number 12 hasbeen fixed at Wref = 0.012 kg = 12 g
Atomic/Molecular weights are relative masses obtained as the ratio of the weight of the unit ofamount of substance (a mole) of atoms or molecules with respect to the weight of a referencespecies, the isotope of Carbon with mass number 12 [Atomic/Molecular weights = Wx/Wref]
One mole of substance contains Na=6.023 1023 particles, with Na the Avogadro Number
1 Mole := 6.023 1023 particles1 Mole 12C := 12g of 12C = WC g1 Mole O = WO g = 16 g
When the mole is used, the elementary entities mustbe specified and may be atoms, molecules, ions,electrons, other particles, or specified groups of suchparticles.
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Molecular Weight of a Compound
S j = Ea1j
1 ... EaNEjNE
W j = aij
i=1
Natoms
! W i
CH 3O"WCH3O = 1#WC + 3#W H +1#WO
= 1#12 + 3#1+1#16 = 31
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Extensive and Intensive Properties
Extensive Property: ! = ! jj=1,N"
Intensive Property: ! j = ! j=1,N
!1
!4
!3
!2
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Composition of a Gaseous Mixture
Number of Particles Npart ,i [part]
Mole Ni [mol] ! N = Nii=1
Nspecies
"
Mass Mi [Kg] ! M = Mii=1
Nspecies
"
Number Density ni =Npart ,i
V [ part
m3 ]
n[ ] = ni[ ]i=1
Nspecies
!
Molar Concentration Ni[ ] = Ni
V [mol
m3 ]
N[ ] = Ni[ ]i=1
Nspecies
!
Partial Density "i =Mi
V [Kg
m3 ]
" = "ii=1
Nspecies
!
Extensive Variables:
Intensive Variables:
Number of Particles is extensive ! Npart ,i = (Npart ,i )j
j=1
4
" ; Npart = (Npart )j
j=1
4
"
Mole is extensive ! Ni = (Ni )j
j=1
4
" ; N = (N ) jj=1
4
"
Mass is extensive ! Mi = (Mi )j
j=1
4
" ; M = (M ) jj=1
4
"
Number Densities are intensive ! npart ,i( )V= Npart ,i
V=Npart ,i / 4V / 4
= npart ,i( )V/4
Molar Concentrations are intensive ! Ni[ ]( )V= Ni
V=Ni / 4V / 4
= Ni[ ]( )V/4
Densities are intensive ! "i( )V= Mi
V=Mi / 4V / 4
= "i( )V/4
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Fuel/Oxidant Parameters
Suppose the mixture be formed by f Kg/s of fuel and 1-f Kg/s of oxidizer.Next:Any intensive property: !mix = f!F + (1" f )!OX
Def.: Mixture Fraction Z #!mix "!OX
!F "!OX
Def.: Mixture Ratio of#
1" ff
Def.: Equivalence Ratio $ #
fo
%&'
()*
fo
%&'
()* stechio
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Mass and Molar Fractions
Mi =WiNi
M =WN
Yi =Mi
M=WiNi
WN=Wi
WXi
Molar Fraction Xi =Ni
N=
Ni[ ]N[ ] [!]
Mass Fraction Yi =Mi
M= "i"
[!]
Definition
Transformation Rules
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Molecular Weight of a Mixture
W =1N
WiNii! Kg
K-molei
K-molei
K-mole"
#$
%
&'
R =1M
RiMii!
R =(W
Ri =(Wi
R =!W
=1M
RiMii" = !
1Wi
Mi
Mi"
W =1Wi
Mi
Mi"#$%
&'(
)1
W = WiXii! = Yi
Wii!"#$
%&'
(1
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Principles of Non-Equilibrium Thermodynamics
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Principles of Nonequilibrium Thermodynamics
Assumptions: System in local thermal and mechanical equilibriumSystem in local chemical nonequilibrium
State variables: Pressure (intensive) pVolume (extensive) VMoles (extensive) Ni
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Zero-th Principle of Nonequilibrium Thermodynamics
There exists an intensive variable, the temperature T, which is a state functionobeying to the equation of state:
T = T (p,V ,N j )
T is the same for systems A, B, C if they are in thermodynamic equilibium, that is:
If TA = TCand TB = TCthen TA = TB
The Zero-th Principle ensures the existence of a Thermal Equation of State of theform:
p = p(T ,V ,N j )
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First Principle of Nonequilibrium Thermodynamics
There exists an extensive variable, the internal energy U, which is a statefunction U=U(p,V,Nj), and whose variation can be computed as:
dU = !Q " !W + µ jj=1
Nspecies
# dN jOpen system
Closed system (dNj=0) dU = !Q " !W
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First Principle: Sensible Energy
The internal energy U is a measure of the energy stored in the internal degreesof freedom of the molecule:
Translational energyRotational energyVibrational energy
These contribution can be computed by the Partition Functions obtained viaStatistical Mechanics
Under condition of thermal equilibrium, these contributions (the PartitionFunctions) depends only on temperature, and for this reason, they are a measureof the sensible energy of the molecule
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First Principle/Sensible Energy & Partition Functions
• For separable degree of freedoms the Partition Function reads:
• Translational PF
• Rotational PF
• Vibrational PF (harmonic oscillator)
Qi = Qi,1Qi,2Qi,3...
Qtraslation =2! m kBT
h2"#$
%&'
12V
13
Qrotation =8! 2I kBT
h2
"#$
%&'
12
I=Moment of Inertia of Molecule
Qvibration = 1! e!h "KBT
#
$%
&
'(
!1
"=Natural Frequency of Oscillation
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First Principle: Energy of Formation
Since all processes at the molecular level are conservative, then the internalenergy of a gas is an invariant and only variation of internal energy are ofinterest;
Thus, a reference level is required to define defined U, named energy offormation :
1) The energy of formation of pure substances at standard conditions is 0 bydefinition;
2) The energy of formation of a compound is found by adding the boundingenergy needed to form it from pure substances.
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Second Principle of Nonequilibrium Thermodynamics
There exists:1) an absolute scale of the temperature, and2) an extensive variable, the entropy S, which is a state function S=S(p,V,Nj),whose variation for a closed system can be computed as:
TdS ! "Qwhere the sign := holds for changes due to reversible processes> holds for changes due to irreversible processes
For a systems with both external and internal irreversible processes:
dS = dextS + dintS
dS = dextS interaction with the outside of the systemdS = dintS due to irreversible processes inside the system
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Third Principle of Nonequilibrium Thermodynamics
The entropy of a perfect crystal is zero at T=0K
NB: the Third Principle identifies the reference value for the entropy
NB: a substance at 0K in a state different from that of a perfect crystal has anentropy different from zero
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Fundamental Eqns of Chemical Thermodynamics
dU = !Q! "!W
"+ µ jdN j
j# = TdS! " pdV
"+ µ jdN j
j#
Since U is a state function, its variation does not depend on the process, and thus we cancompute the heat and work exchanged with outside as if the processes be reversible:
!Q = TdS !W = pdVfrom which the First Principle can be written as:
which is the differential definition of the caloric equation of state: U=U(S,V,Nj)
dU = TdS ! pdV + µ jdN jj" =
#U#S
$%&
'()V ,Nj
dS + #U#V
$%&
'() S ,Nj
dV +#U#N j
$
%&'
() S ,V
dN jj"
T =#U#S
$%&
'()V ,Nj
p = !#U#V
$%&
'() S ,Nj
µ j =#U#N j
$
%&'
() S ,V
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Closed Systems in Chemical Nonequilibrium
If the chemical state in a closed system changes because of chemical reactions,one can apply the first principle to the (open) subsystems comprising a singlespecies:
dU j = !Qj " !Wj + µ jdN j
!Qj = TdSj !Wj = pjdV
dU j = TdSj " pjdV + µ jdN j
d U jj!
"
#$%
&'= Td Sj
j!
"
#$%
&'( dV pj
j!
"
#$%
&'+ µ jdN j
j!
dU = TdS ( pdV + µ jdN jj!
which ensures the existence of a Caloric Equationof State of the form:
U =U(S,V ,N j )
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Fundamental Eqns of Chemical Thermodynamics
dU = TdS ! pdV + µ jdN jj"
From:
derives:
which is the differential definition of the equation of state: S=S(U,V,Nj)
dS = 1TdU +
pTdV !
1T
µ jdN jj"
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Thermodynamics Potentials: Enthalpy
dH = dU + pdV +Vdp
dH = TdS ! pdV + µ jdN jj" + pdV +Vdp
dH = TdS +Vdp + µ jdN jj"
Introducing the Enthalpy:
After differentiation:
which is the differential definition of the equation of state: H=H(S,p,Nj)
H =U + pV
dH = TdS +Vdp + µ jdN jj! =
"H"S
#$%
&'( p,Nj
dS +"H"p
#$%
&'( S ,Nj
dp +"H"N j
#
$%&
'( p,S
dN jj!
T ="H"S
#$%
&'( p,Nj
V ="H"p
#$%
&'( S ,Nj
µ j ="H"N j
#
$%&
'( p,S
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Thermodynamics Potentials: Helmoltz Free Energy
dF = dU ! TdS ! SdT
dF = TdS ! pdV + µ jdN jj" ! TdS ! SdT
dF = !SdT ! pdV + µ jdN jj"
Introducing the Helmoltz Energy:
After differentiation:
which is the differential definition of the equation of state: F=F(T,V,Nj)
F =U ! TS
dF = !SdT ! pdV + µ jdN jj" =
#F#T
$%&
'()V ,Nj
dT +#F#V
$%&
'()V ,Nj
dV +#F#N j
$
%&'
()V ,T
dN jj"
S = !#F#T
$%&
'()V ,Nj
p = !#F#V
$%&
'()V ,Nj
µ j =#F#N j
$
%&'
()V ,T
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Thermodynamics Potentials: Gibbs Free Energy
dG = dH ! TdS ! SdT
dG = TdS +Vdp + µ jdN jj" ! TdS ! SdT
dG = !SdT +Vdp + µ jdN jj"
Introducing the Gibbs Energy:
After differentiation:
which is the differential definition of the equation of state: G=G(T,p,Nj)
G = H ! TS
dG = !SdT +Vdp + µ jdN jj" =
#G#T
$%&
'() p,Nj
dT +#G#p
$%&
'() T ,Nj
dp + #G#N j
$
%&'
() p,T
dN jj"
!S = #G#T
$%&
'() p,Nj
V =#G#p
$%&
'() T ,Nj
µ j =#G#N j
$
%&'
() p,T
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Gibbs Free Energy
dG = !SdT +Vdp + µ jdN jj"
Integration at const T, p and µj of:
yields the definition of G:
G !G0 = µ jdN j0
N j
"j# = µ jN j
j#
µ j =!G!N j
"
#$%
&' p,T
= H j ( TSj
The Gibbs Free Energy per unit mole can be also referred to as Chemical Potential:
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Entropy Production due to Chemical Nonequilibrium
dS = dextS + dintS
dS = 1TdU +
pTdV !
1T
µ jdN jj"From the first principle
From the second principle
Open system dextS + dintS =1TdU +
pTdV !
1T
µ jdN jj"
Closed system dextS =1TdU +
pTdV
Internal irreversible processes dintS = !1T
µ jdN jj"
This is the most significant result in the study of systems in chemical nonequilibrium
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Reversible Chemical Processes
Internal reversible processes dintS = !1T
µ jdN jj" = 0
Frozen processes dN j = 0 ! N j = const
Equilibium processes µ jdN jj! = 0 " dN j # 0
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Condition of Chemical Equilibrium (1)
µ jdN jj! = 0" dN j # 0
It can be proved that Sint has a maximum at equilibrium
dG = !SdT +Vdp + µ jdN j
j" =
dT =dp=0!µ jdN j
j" = !dSint
Thus, G has a minimum at equilibrium
For a system at constant T and p:
dSint ! 0
dG ! 0
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Composition at & off equilibrium
N j ! N j* p,T( )"G p,T ,N j( ) > G p,T ,N j
* p,T( )( )
Minp,T =const
G p,T ,N j( )!" #$ % N j = N j* p,T( )
At chemical equilibrium, the composition is a dependent state variable
Off chemical equilibrium, the composition is an independent state variable
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Stoichiometry & Kinetics
G p,T ,N j( ) > G p,T ,N j* p,T( )( )
N jIrreversible Process! "!!!!! N j
* p,T( )
Off chemical equilibrium, there exists a spontaneous tendency to change the systemcomposition so as to reach the chemical equilibrium state:
The irreversible process evolves following PATHS constraint by the STOICHIOMETRY(atomic mass conservation), and at a pace defined by the CHEMICAL KINETICS