The mathematical description of the motion of Atoms, Molecules & Other Particles. University of Rome La Sapienza - SAER - Mauro Valorani (2007)

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1 The mathematical description of the motion of Atoms, Molecules Other Particles

2 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 instantaneous position, 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 the interactions with other particles. This approach can be pursued only for system of small size given that in a mole of substance there are as many as 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 orders of 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 relaxation processes.

3 Relaxation Processes Main relaxation processes: 1 Translational relaxation: elastic collision among particles 2 Rotational Vibrational relaxation: excitation de-excitation of internal dof of polyatomic molecules 3 Electronic relaxation: formation and decay of excited states of atoms, molecules, ions, 4 Macroscopic chemical reactions elementary reactions among neutral particles 5 Macroscopic kinetic processes involving charged particles in plasma at low T Main issues in the mathematical description of a kinetic problem: 1 Identification of the kinetic mechanism 2 Identification of the rate equations, and the characteristic time scales associated with the relaxation process driving the kinetic mechanism The relaxation processes can be thus ranked according with their corresponding time scale:! 0 free mean path It might happen that after a time of the order of 1 Macroscopic chemical reactions are still far from their chemical equilibrium state 2 Translational Rotational Vibrational relaxation processes have regained a nearly thermal equilibrium state 3 All other slower processes remain nearly frozen In general, if the following inequalities hold:!! trasl! rot #!! vib #!! chem #!! elec # all fast processes can be considered in near-equilibrium all * processes are in non equilibrium all slow processes can be considered frozen gap! chem gap! fast!! *!! slow gap!! plasma gap

4 Statistical Mechanics One way to reduce the complexity of the mathematical description of the dynamics of an ensamble of particles is to introduce macroscopic properties on the basis of statistics over the ensamble. Fluctuations over the time scale of the order of 1 / N = 1 / 10 23! 10!12 s Let us suppose that a small perturbation is applied to a system in equilibrium. On time scale of the order! 0 r 0 / v! 2.5x10 #13 s very few collisions occur; the system can be described only by resorting to the Molecular Dynamics; are replaced by their average value. r 0 : free mean path v : translational speed On time scale of the order!! s 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!! s the system experienced very many collisions that have completely redistributed the energy of the perturbation; The system can be described by introducing the concept of local equilibrium, whose changes can be described by the thermodynamics of irreversible processes (Continuum description / Navier Stokes Equations. For!! 1s the system approaches an equilibrium condition described by the thermo-dynamics of reversible processes (thermo-static.

5 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 at the same time: 1 There exists no temperature gradients in the system and the temperature of the system is the same of the surrounding environment (Thermal equilibrium 2 There exists no pressure gradients in the system and the pressure of the system is the same of the surrounding environment (Mechanical equilibrium 3 There exists no composition variation in the system and the composition of the system is the same of the surrounding environment (Chemical equilibrium The state of the system under equilibrium conditions can be identified by two independent state variables, such as p and T. In equilibrium conditions, the chemical composition is a function of p and T, and thus it is a dependent state variable Y*=f(p,T

6 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 the small portion The state variables can take different values over adacent portions of the system, this yielding nonuniform 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 thermodynamics

7 Assumption of Local Equilibrium The assumption of local equilibrium enables to define a local value of the classic thermodynamic entropy. The assumption of local equilibrium holds when:!t #T #x!x T l!t T l T T!p p l p! 1 p!n N! 1 l v a! 1 a=speed of sound! 1 l=free mean path These conditions are violated in: rarefied gases across strong shock waves when the boundary conditions enforce changes to the system on a too fast scale

8 Atomic Molecular Weights Principles of Nonequilibrium Thermodynamics Entropy Production in Chemical Nonequilibrium

9 Unit of Amount of Substance Mixture of gases are made of different entities: atoms, molecules, ions, electrons Atomic/Molecular weights are relative masses obtained as the ratio of the weight of the unit of amount of substance (a mole of atoms or molecules with respect to the weight of a reference species, the isotope of Carbon with mass number 12 [Atomic/Molecular weights = Wx/Wref] By intl agreement, the atomic weight of the isotope of Carbon with mass number 12 has been fixed at Wref = kg = 12 g The unit of amount of substance (the mole is defined as the amount of substance of a system which contains as many elementary entities as there are atoms in Kg of unbound atoms of Carbon 12, at rest and in ground state One mole of substance contains Na= particles, with Na the Avogadro Number 1 Mole := particles 1 Mole 12 C := 12g of 12 C = W C g 1 Mole O = W O g = 16 g When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

10 Molecular Weight of a Compound S = E 1 N a1... E E ane W = Natoms a i! W i i=1 CH 3 O W CH 3O = 1 # W C + 3 # W H + 1# W O = 1 # # 1+ 1# 16 = 31

11 Extensive and Intensive Properties! 4! 3! 1! 2 Extensive Property:! =! =1,N Intensive Property:! =! =1,N

12 Composition of a Gaseous Mixture Extensive Variables: Number of Particles N part,i [ part] Nspecies Mole N i [mol]! N = N i Mass M i [Kg]! M = M i Number Density Molar Concentration Partial Density n i = N part,i V Nspecies! [ ] [ n] = n i i=1 [ N i ] = N i V Nspecies! [ N ] = N i i = M i V i=1 Nspecies! = i i=1 [ part m 3 ] [ ] [ mol m 3 ] [ Kg m 3 ] i=1 Nspecies i=1 Intensive Variables: Number of Particles is extensive! N part,i = (N part,i ; N part = (N part Mole is extensive! N i = (N i ; N = (N Mass is extensive! M i = (M i ; M = (M Number Densities are intensive Molar Concentrations are intensive! Densities are intensive 4 =1 4 =1 4 =1! ( n part,i V = N part,i V = N part,i / 4 V / 4 4 =1 4 =1 ([ N i ] V = N i V = N i / 4 V / 4 = ([ N i ] V/4! i ( V = M i V = M i / 4 V / 4 = i ( V/4 4 =1 = ( n part,i V/4

13 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: Def.: Mixture Fraction Def.: Mixture Ratio! mix = f! F + (1 f! OX Def.: Equivalence Ratio # Z #! mix! OX! F! OX o f # 1 f f ' f ' o f o ( * ( * stechio

14 Mass and Molar Fractions Definition Molar Fraction Mass Fraction Transformation Rules M i = W i N i M = WN X i = N i N = N i N Y i = M i M = i Y i = M i M =W i N i WN =W i W X i [ ] [ ] [!] [!]

15 Molecular Weight of a Mixture W = 1 N R = 1 M R = ( W R i = ( W i! i! i W i N i R im i # Kg K-mole i K-mole i K-mole R =! W = 1 M W = # i ' R M i i =! 1 M i W i M i 1 M i W i M ' ( 1 i W =! W i X i = # i! i Y i W i ' (1

16 Principles of Non-Equilibrium Thermodynamics

17 Principles of Nonequilibrium Thermodynamics Assumptions: System in local thermal and mechanical equilibrium System in local chemical nonequilibrium State variables: Pressure (intensive p Volume (extensive V Moles (extensive N i

18 Zero-th Principle of Nonequilibrium Thermodynamics There exists an intensive variable, the temperature T, which is a state function obeying to the equation of state: T = T (p,v, N T is the same for systems A, B, C if they are in thermodynamic equilibium, that is: If and then T A = T C T B = T C T A = T B The Zero-th Principle ensures the existence of a Thermal Equation of State of the form: p = p(t,v, N

19 First Principle of Nonequilibrium Thermodynamics There exists an extensive variable, the internal energy U, which is a state function U=U(p,V,N, and whose variation can be computed as: Open system du =!Q!W + Nspecies # µ dn =1 Closed system (dn =0 du =!Q!W

20 First Principle: Sensible Energy The internal energy U is a measure of the energy stored in the internal degrees of freedom of the molecule: Translational energy Rotational energy Vibrational energy These contribution can be computed by the Partition Functions obtained via Statistical Mechanics Under condition of thermal equilibrium, these contributions (the Partition Functions depends only on temperature, and for this reason, they are a measure of the sensible energy of the molecule

21 First Principle/Sensible Energy Partition Functions For separable degree of freedoms the Partition Function reads: Translational PF Rotational PF Q rotation = 8! 2 I k B T # h 2 ' Vibrational PF (harmonic oscillator # Q vibration = 1! e! Q i = Q i,1 Q i,2 Q i,3... Q traslation = 2! m k T B # h K B T ( ' 1 2!1 h ' V 3 I=Moment of Inertia of Molecule =Natural Frequency of Oscillation

22 First Principle: Energy of Formation Since all processes at the molecular level are conservative, then the internal energy of a gas is an invariant and only variation of internal energy are of interest; Thus, a reference level is required to define defined U, named energy of formation : 1 The energy of formation of pure substances at standard conditions is 0 by definition; 2 The energy of formation of a compound is found by adding the bounding energy needed to form it from pure substances.

23 Second Principle of Nonequilibrium Thermodynamics There exists: 1 an absolute scale of the temperature, and 2 an extensive variable, the entropy S, which is a state function S=S(p,V,N, whose variation for a closed system can be computed as: TdS! Q where 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 = d ext S ds = d int S ds = d ext S + d int S interaction with the outside of the system due to irreversible processes inside the system

24 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 an entropy different from zero

25 Fundamental Eqns of Chemical Thermodynamics Since U is a state function, its variation does not depend on the process, and thus we can compute the heat and work exchanged with outside as if the processes be reversible:!q = TdS!W = pdv from which the First Principle can be written as: du =!Q!!W + # µ dn = TdS! pdv + # µ dn which is the differential definition of the caloric equation of state: U=U(S,V,N du = TdS! pdv + µ dn = #U ' #S ( T = #U ' #S ( V,N p =! #U ' #V ( S,N V,N ds + #U ' #V ( S,N µ = #U ' #N ( dv + S,V #U #N ' ( S,V dn

26 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 single species: du =!Q!W + µ dn!q = TdS!W = p dv d! U ' # = Td # du = TdS p dv + µ dn! du = TdS ( pdv + S! ' ( dv # µ dn! p ' +! µ dn which ensures the existence of a Caloric Equation of State of the form: U = U(S,V, N

27 Fundamental Eqns of Chemical Thermodynamics From: du = TdS! pdv + derives: ds = 1 T du + p T dv! 1 T µ dn which is the differential definition of the equation of state: S=S(U,V,N µ dn

28 Thermodynamics Potentials: Enthalpy Introducing the Enthalpy: H = U + pv After differentiation: dh = du + pdv + Vdp dh = TdS! pdv + µ dn + pdv + Vdp dh = TdS + Vdp + µ dn which is the differential definition of the equation of state: H=H(S,p,N # dh = TdS + Vdp +! µ dn = H S # T = H S ' ( p,n # V = H p ' ( S,N ' ( p,n # ds + H p ' ( S,N # µ = H N ( ' dp + p,s! # H N ( ' p,s dn

29 Thermodynamics Potentials: Helmoltz Free Energy Introducing the Helmoltz Energy: F = U! TS After differentiation: df = du! TdS! SdT df = TdS! pdv + µ dn! TdS! SdT df =!SdT! pdv + µ dn which is the differential definition of the equation of state: F=F(T,V,N df =!SdT! pdv + µ dn = #F ' #T ( S =! #F ' #T ( V,N p =! #F ' #V ( V,N V,N dt + #F ' #V ( V,N µ = #F ' #N ( dv + V,T #F ' #N ( V,T dn

30 Thermodynamics Potentials: Gibbs Free Energy Introducing the Gibbs Energy: After differentiation: G = H! TS dg = dh! TdS! SdT dg = TdS + Vdp + µ dn! TdS! SdT dg =!SdT + Vdp + µ dn which is the differential definition of the equation of state: G=G(T,p,N dg =!SdT + Vdp + µ dn = #G ' #T ( p,n!s = #G ' #T ( p,n V = #G ' #p ( T,N dt + #G ' #p ( T,N µ = #G ' #N ( p,t dp + #G #N ' ( p,t dn

31 Gibbs Free Energy Integration at const T, p and µ of: dg =!SdT + Vdp + yields the definition of G: G! G 0 = N µ dn # µ dn = # µ N 0 The Gibbs Free Energy per unit mole can be also referred to as Chemical Potential: µ =!G #!N ' p,t = H ( TS

32 Entropy Production due to Chemical Nonequilibrium From the first principle From the second principle Open system Closed system ds = 1 T du + p T dv! 1 T ds = d ext S + d int S d ext S + d int S = 1 T du + p T dv! 1 T d ext S = 1 T du + p T dv Internal irreversible processes d int S =! 1 T µ dn µ dn µ dn This is the most significant result in the study of systems in chemical nonequilibrium

33 Reversible Chemical Processes Internal reversible processes Frozen processes Equilibium processes d int S =! 1 T dn = 0! N = const µ dn = 0! µ dn = 0 dn # 0

34 Condition of Chemical Equilibrium (1! µ dn = 0 dn # 0 It can be proved that S int has a maximum at equilibrium For a system at constant T and p:! dg =!SdT + Vdp + µ dn = dt =dp=0 µ dn =!ds int Thus, G has a minimum at equilibrium ds int! 0 dg! 0

35 Composition at off equilibrium At chemical equilibrium, the composition is a dependent state variable N! N * Min p,t =const (! G p,t, N ( p,t G p,t, N # N * = N ( p,t Off chemical equilibrium, the composition is an independent state variable * ( > G( p,t, N ( p,t

36 Stoichiometry Kinetics Off chemical equilibrium, there exists a spontaneous tendency to change the system composition so as to reach the chemical equilibrium state: * ( > G( p,t, N ( p,t G p,t, N N Irreversible Process *!!!!!! N ( p,t The irreversible process evolves following PATHS constraint by the STOICHIOMETRY (atomic mass conservation, and at a pace defined by the CHEMICAL KINETICS

Chapter 3. Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc.

Chapter 3. Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc. Chapter 3 Property Relations The essence of macroscopic thermodynamics Dependence of U, H, S, G, and F on T, P, V, etc. Concepts Energy functions F and G Chemical potential, µ Partial Molar properties