Unformatted text preview: Summary of last lecture Mayer functions, cluster expansion, virial coefficients The Van der Waals equation Chemistry 161: Statistical Thermodynamics
Lecture 21 Thermodynamic Perturbation theory (TPT), correlation functions
Reading: McQuarrie Ch13.13 Key Concepts and Lessons:
a) b) c) d) High T expansion Correlation functions Relation of pair corr to TD functions Alternative  and better derivation of the WdW eq The Van der Waals equation:
We get therefore: p+a
2 = kT (1 + b ) 1 1 b Or in the canonical from: (p+ a )
2 1 b = kT This is famous van der Waals equation which was obtained here in a nonrigorous form as an extrapolation Of virial expansion from low densities to high densities. Thermodynamic Perturbation Theory Another useful idea is to treat some part of interaction potential as a `'small'' addition to the system with known property (not necessarily an ideal gas) an approach known as perturbative treatment. In other words we assume for interaction energy: U N (r1 , r2 ...rN ) = U (r , r ...rN ) + U (r , r ...rN )
0 N 1 2 1 N 1 2
where index `'0'' refers to an ideal system and index `'1'' denotes `'perturbation'' i.e. difference between ideal and real system. Our hope is that such perturbation is small (in some sense as we will outline below). (U ZN = e 0 N ( r1 , r2 ...rN )+U 1 ( r1 , r2 ...rN ) N ) dr dr ...dr
1 2 N Thermodynamic perturbation theory Dividing and multiplying by (U Z = e
0 N 0 N ( r1 , r2 ...rN ) ) dr dr ...dr
1 2 N We get: ZN = e
Or in short: ( 0 UN ( r1 , r2 ...rN ) ) dr dr ...dr
1 2 (U e 0 N ( r1 , r2 ...rN )+U 1 ( r1 , r2 ...rN ) N
0 N ) dr dr ...dr
1 2 2 N N (U e ( r1 , r2 ...rN ) ) dr dr ...dr
1 N ZN = Z 0 N exp ( U 1 N ) 0 where subscript index 0 means that we are averaging with the `'unperturbed'' Hamiltonian. For free energy we get obviously:
0 ZN A = ln N! N + ln exp ( U1 N ) 0 = A0 A1 Thermodynamic perturbation theory
Now next step is to expand the exponential in powers and view it a highT 1 expansion in powers of a small parameter :
= kT
2 U 1 + ....... N 0 exp ( U1 N ) 0 =1 U1 N
n 0 + 2 2 Similarly free energy can be presented as an expansion: A = 1 n =1 n! (
2 0 1 N 2 0 ) n 1 = U1 1 N
2 = = 3 ( ) (U )
U1 N
1 N 0 2 0 U1 N 3 U
0 3 ( ) U 1 N 0 +2 U 1 N 3 0 Thermodynamic perturbation theory We get first two terms of the expansion for free energy: A = A0 + 2 1 2kT +O ( )
2 From now on we assume that the interaction energy can be presented as a sum of pair potentials: U1 = N
And we get insightfully: i< j u1 rij ( )
= U 1 N 0 =
i< j u rij 1 ()
0 = N N 1 2 ( ) u (r )
1 12 N N 1 2Z
0 N ( ) 0 e 0 UN 1 u (r12 )dr1dr2 ..drN Correlation functions Now we introduce an extremely important concept of distribution functions which a joint probability to find particles at specific positions. The most general (and useless one) is the complete function: P N ( r , r2 ......rN ) = 1 e UN ZN a more useful concept is that of reduced ones. For example consider a joint probability density that we find particle 1 around 1 , particle 2 around ...2 particle n around r : n r r P ( r1 , r2 ......rn ) =
n e UN drn+1... drN ZN Now we note that particles are indistinguishable and therefore we should not care which particles we follow. To this end we get a better defined function which is a joint probability density to find any particle at r etc:
i r1 ( n) N! ( r1 , r2 ......rn ) = P n ( r1 , r2 ......rn ) N n ! Correlation functions The one particle correlation function is just local density it can be periodic (as in crystal) or constant as in liquid.: 1 V 1 ( ) N r1 dr1 = = V
n More generally we define correlation functions ( n) ( r1 , r2 ......rn ) = g ( r1 , r2 ......rn ) ( n) g characterizes nonindependence, correlation between particles. Of them the most important is pairwise correlation function g(2) because it can be determined experimentally from scattering. is the probability to observe a molecule around r provided that fisrt one is fixed at origin. It is normalized as follows: g(r) dr4 r dr = N
2 1 N Relation of thermodynamic functions to pairwise correlation function: Energy
First, we look at total energy
E= 3 NkT + kT 2 2 Ue ln Z N T
U =
N ,V 3 NkT + U 2 where U = dr1dr2 ...drN ZN Now we look at N(N1)/2 pairwise terms. Using particles 1 and 2 as an example we get: N(N 1) U= 2Z N 1 = 2 u ( r12 )
(2) N(N 1) e U u ( r12 ) dr1dr2 ...drN = 2 N2 u(r)g(r)4 r 2 dr; ( r12 ) dr1dr2 = 2V 0 u ( r12 ) e U dr3dr4 .....drN ZN dr1dr2 = the total energy is then: 3 E = + u(r)g(r, ,T )4 r 2 dr NkT 2 2kT 0 Relation of thermodynamic functions to pairwise correlation function: Pressure Correlation functions reflect how close molecules to each other and how strongly do they interact. Therefore they would reflect on pressure as we can see below.
p = kT ln Z N V
N ,T Assume first that the liquid or gas is positioned in a cubic container. Then all coordinates span From 0 to V1/3: V 1/3 V 1/3 ZN =
0 0 ...e U dx1dx2 ..dxN dy1dy2 ..dyN dz1dz2 ...dzN Now we change variables so that all limits of integration run form 0 to 1 and volume dependence enters the coordinates. We need that to evaluate how volume change affects all intermolecular distances and therefore energy of their interactions: 1/ 3 ' k k x =V x 1 1 Z N = V N ...e U dx'1 dx'2 ..dx'N dy'1 dy'2 ..dy'N dz'1 dz'2 ...dz'N ;
0 0 U=
1,i, j,N u(rij ); rij = V 1/3rij' Relation of thermodynamic functions to pairwise correlation function: Pressure Therefore: ZN V V kT 1 1 = NV
N ,T N 1 0 0 ...e U dx '1 dx '2 ..dx ' N dy'1 dy'2 ..dy' N dz '1 dz '2 ...dz ' N N 1 1 ...e
0 0 U U dx '1 dx '2 ..dx ' N dy'1 dy'2 ..dy' N dz '1 dz '2 ...dz ' N V where du(rij ) drij rij du(rij ) U = = V drij dV 1<i < j < N 3V drij 1<i < j < N Going back to the original variables we get: ln Z N V p = kT =
N ,T 2 N 1 V 6VkT r12
V du ( r12 ) dr12 (2) ( r1,r2 ) dr1dr2 ; 6kT ru '(r)g(r)4 r 2 dr
0 Which is called `'pressure equation''  it establishes the relation between pair correlation and pressure; Also gives effectively the equation of state is we knew the pair correlation `'He who knows pair correlation functions owns the world'' (of Statistical
mechanics) (but they are not easy to calculate usually obtained in simulations) Back to TPT derivation of the vdW eq
Using our definition of correlation function we get for first term in perturbation expansion: 0 N Higher order terms in TPT require the knowledge of higher than second correlation functions Their derivation is may be complicated and require considerable approximations such as factorization (I.e. that higher order correlation functions can be presented as a product of pairwise ones  an essence of Kirkwood approximation (see McQuarrie, 13,7)) U1 0 = NN 1 2Z ( )e 0 UN 1 1 1 u (r12 )dr1dr2 ..drN = u (r12 ) 2 (2) () V (1) r12 dr1dr2 = u (r12 )g r12 dr12 2 2 () Now we assume that our unperturbed `'0'' system is a hard sphere fluid:
U1 N
0 = V 1 u (r)g HS (r)4 r 2 dr 2 0 2 We need to make an educated guess as to what is the pair correlation function in a hard sphere fluid? A first guess would reveal our true ignorance but it appears OK: g HS (r ) = 0 r < 1 r > U1 N 0 =2 2 V u1 (r)r 2 dr = aN u1 (r)r 2 dr where a = 2 TPT derivation of the vdW equation 0 We get finally: Z N = Z N e a N and more importantly for pressure: p = kT
0 ln Z N V N ,T a 2 p0 = kT kT a 2 kT Finally we have to assume smth about pressure in the hard sphere system. The assumption here is that of excluded volume that hard sphere system is like an ideal gas of noninteracting particles but each particle can move in effective volume Veff permitted by other particles. We have to avoid double counting and we get: Veff = V 2 N we get therefore:
0 ZN = V 3 /3 ( Nb
0 ln Z N V ) N ;
= b=2 3 /3 p0 = kT N ,T N 1 = 1 V Nb b
a 2 kT And finally the familiar vdW Equation: p = 1 kT b This equation is in a semiquantitative agreement with experiment but it gives many features of real liquids condensation, critical point etc etc. A better approximation uses less nave assumptions about g, a better separation into ideal and perturbation system etc. An example of such improved approach is the ChandlerWeeks Anderson (CWA) equation (Phys Rev Lett, v.25, p.149, 1970) Next lecture:
GasLiquid phase transitions, phase equilibria. Meanfield theories  ferromagnetism and DebyeHuckel. (Reif Ch 8.5; 8.6; 10.7) ...
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This note was uploaded on 05/04/2010 for the course CHEM 161 taught by Professor Shaklovich during the Spring '10 term at Harvard.
 Spring '10
 Shaklovich

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