MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.62 Lecture #19: Configurational Integral:Cluster Expansion Goal: For U( q ) ≠ 0, calculate Z to obtain corrections for non-ideal contributions to theequation of state. Z = ∫ ∫ d q3N e −U(q~)/kT ~ where U( q ) = Total Interaction Potential EnergySimplifications based on form of U( q ):1. Assume U( q ) is sum over pairs of atoms("pair potential") — pairwise additive interactions. 2. Assume U( q ) depends only on the distance between pairs of atoms, ri − rj . U(q) =∑ uij i< j ri − rj( ) rij = distance between atoms i and j sum over pairs, i< j prevents double counting uij ≡ pair interaction potential (Hard Sphere, Square Well,Sutherland, LJ, dipole-dipole, etc.), which is a function of r~i − r~ j . r~i ≡ position of ith atom Therefore — −∑uij kT Z = ∫ ∫ d q3N e i< j = ∫ ∫ d q3N ∏e−uij kT ~ ~ i< j The next step is a USEFUL TRICK … For distant particles uij = 0 ⇒ e− uij kT = 1 For convenience we define5.62 Spring 2008 Lecture #19, Page 2 e− uij kT ≡ (1 + fij ) ⇒ fij = e− uij kT − 1 giving fij = 0 when particles have no interaction. Z = ∫ ∫ d q3N ∏ (1+ fij ) evaluate as “cluster expansion” ~ i< j Z = ∫ ∫ d q3N (1+ f12 )(1+ f13 )(1+ f14 )…(1+ f1N )(1+ f23 )(1+ f24 )… ~ (1+ f2N )(1+ f34 )(1+ f35 )…(1+ f3N )… Z = ∫ ∫ d q3N ⎡⎣1+(f12 + f13 + f14 +…fN−1,N )+(f12f13 + f12f34 + f12f56 + f12f23 +…fN−2 N−1fN−1N )+ ~ (f12f34f56 +…+ f12f23f34 +… )+…⎤⎦ Define “Cluster Integrals” z1 = ∫ ∫ dq3N (1) = VN term involving no interactionsterms for independent two-particlez2 = ∫ ∫ dq3N (f12 + f13 + f14 + … + fN−1,N ) interactionsz3 = ∫ ∫ dq3N (f12f34 + f12f56 + … + f12f23 + …) terms for 2 simultaneous pair interactionsz4 = ∫ ∫ dq3N (f12f34f56 + … + f12f23f34 + …) terms for 3 simultaneous pair interactions revised 1/9/08 2:02 PM5.62 Spring 2008 Lecture #19, Page 3 Note that terms linked by one common-particle interaction, e.g. f12f23, are included in z3. But this causes no trouble because such integrals are trivially factorizable ∫∫∫ d3q d3q d3q f12f23 = ∫∫∫ d3q d3q f12d3q f23~1 ~2 ~3 ~cm12 ~12 ~23 = V ∫ d3q~12 f12 ∫ d3q~ 23 f23 We get VN–3 from the N–3 independent particles, and Vβ2 from the singly linked cluster. β is a two particle interaction integral. It has dimension of volume. See pages 263-266 and 277-291 of “Statistical Mechanics” by Mayer and Mayer (Wiley, 1940). A diagrammatic method for classifying interactions, a precursor to Feynmann diagrams, is presented. The fact that the 1,2 and 2,3 interaction integrals may be independently evaluated means that no approximations need to be made about excluding terms from the z2 and z3 integrals. There is, however, a class of term that cannot be factored; and this class first appears in the z4 integral when integrals over f12f23f13 type doubly linked terms must be evaluated. For our present purposes it is sufficient to remark that the neglected terms in the z4 (triples) and higher terms result in a dependence of Z on powers of the density higher than 1, Z =[Veβ1ρeβ2ρ2 …]N where ρ = N/V. The assumption of independent binary collisions (Boltzmann’s stosszahl ansatz) is valid when the mean time between collisions is long relative to the time of one collision. When the inter-particle interaction is 1/r (Coulomb), the interaction length is infinite and there is no such thing as independent binary collisions. When the density is very high, the mean time between collisions can become comparable to the duration of a collision. revised 1/9/08 2:02 PM5.62 Spring 2008 Lecture #19, Page 4 So, Z = z1 + z2 + z3 + z4 + … try to simplify these integrals — z2 = ∫ d q3N (f12 + f13 + f14 +…+ fN−1,N )~ kT −1)+(e−u13 kT −1)+(e−u14= ∫ d q~ 3N [(e−u12 kT −1)+… ]  Each term (e− uij kT − 1) has the same functional form, so z2 is just equal to (# of terms) × (configurationally averaged value of any term). The # of terms is just the number of ways of choosing pairs from N molecules, which is N(N − 1) .2 N N(N 1) −⎡ ⎤[3 3 3d d d∫ ∫z r r r= ⎢ ⎥…21 22~ ~ ~ ⎣ ⎦N(N 1) − 3 3d dr rN(N 1) − ( )N 2 3 3V d d f−∫ ∫ r r121 2 22~ ~ ~  =  3 3d f d∫ ∫ ∫ ∫ r r121 2 32N~ ~ ~ ~,rr= 1~relative coordinates −r r r= 12 1 2 Change from laboratory coordinates r~1 and r2 to center of mass and relative coordinates, since~ f12 depends only on r1 − r2 .   r12 when m1 ≠ m2) e−u12 kT − 1( )] f12 (0,0)  r1  r2  RCM (NOTE: CM is not at midpoint of (r~1 + r~2 )[m1 = m2 ]Coordinate transformation r~12 = r~1 – r~2 R CM = 12 N(N −1) VN−2 ∫ d3R∫ d3r(e−u12 kT −1)~z2 = 2 CM ~12 N(N −1) VN−1 ∫ d3r(e−u12 get one factor of V from d3RCM integralkT −1)= 2 ~12 revised 1/9/08 2:02 PM5.62 Spring 2008 Lecture #19, Page 5 Define β ≡ ∫ d3r~12 (e−u12 kT −1) Cartesian coordinates ≡ 4π∫ dr r2 (e−u(r ) kT −1) Spherical coordinates (integrated over θ, φ) So N(N − 1) VN−1β = N(N − 1) VN ⎛β⎞z2 = 2 2 ⎝⎜ V⎠⎟ for large N, z2 ≈ N22VNV⎟ ⎞⎠β⎛⎜⎝Look at z3 z3 = ∫  ∫ dq3N (f12f34 + f12f56 +…+ f12f23 +… )N(N − 1)(N − 2)(N − 3) = 2 ⋅ 2 ⋅ 2 ∫∫∫∫ d3r~1d3r~2 d3r~3d3r~4 f12f34 ∫∫ d3r~5 d3r~N 2 ⋅ 2 ⋅ 2 comes from 1,2 pair times 3,4 pair times 1,2 and 3,4 order N large, so ≈ N4 = N234 VN− 4 ∫ ∫ d3r~1d3r~2 f12 ∫ ∫ d3r~3d3r~4 f34 = N234 VN− 2β2 = N234 VN ⎛⎝⎜ β⎞ 2 V⎠⎟ IN GENERAL Z = z1 + z2 + z3 + z4 + … ⎞⎟⎠ 2 2 β⎛⎜⎝ Z = VN + VN N2 1+ N+ VN N4 23 ⎞⎟⎠ ⎛⎜⎝ ⎛⎜⎝ β β + … V V 2 2⎞⎟⎠+ … N 1 ⎛⎜⎝ βN2 ⎞⎟⎠ N ⎛⎜⎝ ⎞⎟⎠ ⎛⎜⎝ ⎞⎟⎠ = VN + 2 V 2 V x2 2! + … βN2 N = VN exp ⎡ ⎣⎢ ⎤⎞⎟⎠⎦⎥ ⎛⎜⎝ since ex = 1+ x +V Z =[VeβN 2V ]N where β = β(T) = 4π∫0 ∞⎡⎣e−u(r ) kT − 1⎤⎦ r2dr So we have summed over all orders of interaction. Everything is expressed in terms ofone pair-interaction potential. So each simplified form of u(r) gives an explicit way tocompute the contribution to Z! revised 1/9/08 2:02