Water and Aqueous Solutons 3. Lattce Model of a Flud Lattce Models Lattce models provde a mnmalst, or coarse-graned, framework for descrbng the translatonal, rotatonal, and conformatonal degrees of freedom of molecules, and are partcularly useful for problems n whch entropy of mxng, confguratonal entropy, or excluded volume are key varables. The lattce forms a bass for enumeratng dfferent confguratons of the system, or mcrostates. Each of these mcrostates may have a dfferent energy, whch s then used to calculate a partton functon. Q e E /k T (1) The thermodynamc quanttes then emerge from A k Tln Q S k P ln P U PE and other nternal varables (X) can be statstcally descrbed from N X PX P E 1 e E / kt We wll typcally work wth a macroscopc volume broken nto cells, typcally of a molecular sze, whch we can fll wth the fundamental buldng blocks n our problem (atoms, molecules, functonal groups) subject to certan constrants. In ths secton we wll concern ourselves wth the mxng of rgd partcles,.e., translatonal degrees of freedom. More generally, lattce models can nclude translatonal, rotatonal, and conformatonal degrees of freedom of molecules. Q Andre Tokmakoff 4/1/017
Ideal Lattce Gas The descrpton of a weakly nteractng flud, gas, soluton, or mxture s domnated by the translatonal entropy or entropy of mxng. In ths case, we are dealng wth how molecules occupy a volume, whch leads to a translatonal partton functon. We begn by defnng a lattce and the molecules that fll that lattce: Parameters: Total volume: V Cell volume: Number of stes: M = V/ Number of partcles: N (N M) Number of contacts each cell has wth adjacent cells: z We begn my assumng that all mcrostates (confguratons of occuped stes n the volume) are equally probable,.e., E = constant. Ths s the mcrocanoncal ensemble, so the entropy of the flud s gven by oltzmann s equaton S k ln () where Ω s the number of mcrostates avalable to the system. If M s not equal to N, then the permutatons for puttng N ndstngushable partcles nto M stes s gven by the bnomal dstrbuton: M! Vacances (3) N!( M N)! are ndstngushable Partcles are ndstngushable Also, on cubc lattce, we have 6 contacts that each cell makes wth ts neghbors. The contact number s z, whch wll vary for D (z = 4) and 3D (z = 6) problems. How do we choose the sze of v? It has to be consdered on a case-by-case bass. The objectve of these models s to treat the cell as the volume that a partcle excludes to occupaton by other partcles. Ths need not correspond to an actual molecular dmenson n the atomc sense. In the case of the tradtonal dervaton of the translatonal partton functon for an deal gas, v s equvalent to the quantzaton volume h mk T 3/. 3 From Ω we can obtan the entropy of mxng from S k ln wth the help of Sterlng s approxmaton ln( M!) M ln( M ) M : S k M ln M Nln N ( M N)ln( M N) (4) Mk xln x(1 x)ln(1 x) In the last lne we ntroduced a partcle fll factor
x N / M whch quantfes the fracton of cells that are occuped by partcles, and s also known as the mole fracton or the packng rato. Snce x 1, the entropy of mxng s always postve. For the case of a dlute soluton or gas, N M, and (1 x) 1, so Sdlute Nk ln x or nrln x We can derve the deal gas law p Nk T / V from ths result by makng use of the thermodynamc dentty p T S V N. 3
nary Flud Entropy of Mxng The thermodynamcs of the mxng process s mportant to phase equlbra, hydrophobcty, solublty, and related solvaton problems. The process of mxng two pure substances A and s shown below. We defne the composton of the system through the number of A and partcles: NA and N and the total number of partcles N = NA + N, whch also equals the number of cells. We begn wth two contaners of the homogeneous pure fluds and mx them together, keepng the total number of cells constant. In the case of the pure fluds before mxng, all cells of the contaner are ntally flled, so there s only one accessble mcrostate, Ωpure = 1, and S pure k ln1 0 When the two contaners are mxed, the number of possble mcrostates are gven by the bnomal dstrbuton: mx N! N A!N!. N! A 1 1 mx N A!N! If these partcles have no nteractons, each mcrostate s equally probable, and smlar to eq. (4) we obtan the entropy of the mxture as S mx Nk x A ln x A x ln x For the mxture, we defne the mole fractons for the two components: xa NA / N and x N / N. As before, snce x A and x 1, the entropy for the mxture s always postve. The entropy of mxng s then calculated from S mx S mx (S pure A S pure ). Snce the entropy of the pure substances n ths model s zero, S mx S mx. A plot of ths functon as a functon of mole fractons llustrates that the maxmum entropy mxture has xa = x = 0.5. (5) 4
Ideal systems be they gasses, solutons, or any varety of molecular ensembles are characterzed by no nteractons between partcles. Under these condtons, the free energy of mxng s purely entropc wth ΔAmx = TΔSmx. Intermolecular Interactons To look at real (non-deal) systems, we now add nteractons between partcles by assgnng an nteracton energy ω between two cells whch are n contact. The nteracton energy can be postve (destablzng) or negatve (favorable). Wth the addton of ntermolecular nteractons, each mcrostate wll have a dstnct energy, the canoncal partton functon can be obtaned from eq. (1), and other thermodynamc propertes follow. In the case of a mxture, we assgn separate nteracton energes for each adjonng A-A, -, or A- par n a gven mcrostate: AA,,. How do we calculate the energy of a mcrostate? m s the total number of molecular contacts n the volume, and these can be dvded nto A-A, -, or A- contacts: m maa m m Whle m s constant, the counts of specfc contacts mj vary by mcrostate. Then the energy of the mxture for the sngle th mcrostate can be wrtten as E mx m AA AA m m (6) and the nternal energy comes from an ensemble average of ths quantty. An exact calculaton of the nternal energy from the partton functon would requre a sum over all possble confguratons wth ther ndvdual contact numbers. Instead, we can use a smpler, approxmate approach whch uses a strategy that starts by expressng each term n eq. (6) n terms of m. We know: Then we have m AA (Total contacts for A) (Contactsof A wth ) zn A m zn m m (8) (7) 5
E mx z N AA A z N U pure A U pure m m AA AA Here n the second step, we recognze that the frst two terms are just the energy of the two pure lquds before mxng. These are calculated by takng the number of cells n the pure lqud (N) tmes the number of contacts per cell (z) and then dvde by two, so you do not double count the contacts. U pure, z N (10) Equaton (9) descrbes the energy of a mcrostate n terms of m. To smplfy our calculaton of Umx, we make a mean feld approxmaton, whch replaces m wth ts statstcal average m: probablty of contact ste beng m # of contact stes for A N ( NAz) zxaxn N Then for the energy for the mxed state Umx Emx, we obtan: U mx U pure A U pure x A x Nk T (1) Here we have ntroduced the untless exchange parameter, z AA z (13) kt kt whch measures the relatve change of ntermolecular nteracton for one cell n the lattce swtchng from an A-A and - contact to A- contacts. Ths average change of nteracton energes s expressed n unts of kt. Dvdng by z gves the average nteracton energy per contact. 0 unfavorable A nteracton 0 favorable A nteracton We can now determne the change n nternal energy on mxng: (9) (11) U mx U mx U pure A U pure (14) x A x Nk T Note ΔUmx as a functon of composton has ts maxmum value for a mxture wth xa=0.5. Note that n the mean feld approxmaton, the canoncal partton functon s 6
N! NA N Q qa q exp Umx / kt N! N! A We kept the nternal molecular partton functons here for completeness, but for the smple partcles n ths model q q 1. A Free Energy of Mxng 1 Usng eqs. (5) and (14), we can now obtan the free energy of mxng: A mx U mx T S mx Nk T x A x x A ln x A x ln x Ths functon s plotted below as a functon of mole fracton for dfferent values of the exchange parameter. When there are no ntermolecular nteractons (χ = 0), the mxng s spontaneous for any mole fracton and purely entropc. Any strongly favorable A- nteracton (χ < 0) only serves to decrease the free energy further for all mole fractons. As χ ncreases, we see the free energy for mxng rse, wth the bggest changes for the 50/50 mxture. To descrbe the consequences, let s look at the curve for 3, for whch certan compostons are mscble (ΔAmx < 0) and others mmscble (ΔAmx > 0). Consder what would happen f we prepare a 50/50 mxture of ths soluton. The free energy of mxng s postve at the equlbrum composton of the xa = 0.5 homogeneous mxture, ndcatng that the two components are mmscble. However, there are other mxture compostons that do have a negatve free energy of mxng. Under these condtons the soluton can separate nto two 1. J. H. Hldebrand and R. L. Scott, Regular Solutons. (Prentce-Hall, Englewood Clffs, N.J., 196). 7
phases n such a way that ΔAmx s mnmzed. Ths occurs at mole fractons of x A 0.07 & 0.93, whch shows us that one phase wll be characterzed by xa x and the other wth x A x. If we prepare an unequal mxture wth postve ΔAmx, for example xa = 0.3, the system wll stll spontaneously phase separate although mass conservaton wll dctate that the total mass of the fracton wth xa 0.07 wll be greater than the mass of the fracton at xa 0.93. As ncreases beyond 3, the mole fracton of the lesser component decreases as expected for the hydrophobc effect. Consder f A = water and = ol. and are small and negatve, s large and negatve, and 1. Crtcal ehavor AA Note that 50/50 mxtures wth.8 have a negatve free energy of mxng to create a sngle homogeneous phase, yet, the system can stll lower the free energy further by phase separatng. As seen n the fgure, marks a crossover from one phase mxtures to two phase mxtures, whch s the sgnature of a crtcal pont. We can fnd the condtons for phase equlbra by locatng the free energy mnma as a functon of χ, whch leads to the phase dagrams as a functon of χ and T below. The crtcal temperature for crossover from one- to two-phase behavor s T0, and Δω s the average dfferental change n nteracton energy defned n eq. (13). 8
Readngs 1. K. Dll and S. romberg, Molecular Drvng Forces: Statstcal Thermodynamcs n ology, Chemstry, Physcs, and Nanoscence. (Taylor & Francs Group, New York, 010).. W. W. Graessley, Polymerc Lquds and Networks: Structure and Propertes. (Garland Scence, New York, 004), Ch. 3. 9