This is the author’s final, peer-reviewed manuscript as accepted for publication. The publisher-formatted version may be available through the publisher’s web site or your institution’s library. Kirkwood–Buff integrals for ideal solutions Elizabeth A. Ploetz, Nikolaos Bentenitis, Paul E. Smith. How to cite this manuscript If you make reference to this version of the manuscript, use the following information: Ploetz, E.A., Bentenitis, N., & Smith, P.E. (2010). Kirkwood–Buff integrals for ideal solutions. Retrieved from http://krex.ksu.edu Published Version Information Citation: Ploetz, E.A., Bentenitis, N., & Smith, P.E. (2010). Kirkwood–Buff integrals for ideal solutions. The journal of chemical physics, 132(16), 9. Copyright: © 2010 American Institute of Physics. Digital Object Identifier (DOI): doi:10.1063/1.3398466 Publisher’s Link: http://jcp.aip.org/resource/1/jcpsa6/v132/i16/p164501_s1 This item was retrieved from the K-State Research Exchange (K-REx), the institutional repository of Kansas State University. K-REx is available at http://krex.ksu.edu THE JOURNAL OF CHEMICAL PHYSICS 132, 164501 ͑2010͒ Kirkwood–Buff integrals for ideal solutions ͒ Elizabeth A. Ploetz,1 Nikolaos Bentenitis,2 and Paul E. Smith1,a 1Department of Chemistry, Kansas State University, Manhattan, Kansas 66506, USA 2Department of Chemistry and Biochemistry, Southwestern University, Georgetown, Texas 78626, USA ͑Received 20 May 2009; accepted 29 March 2010; published online 22 April 2010͒ The Kirkwood–Buff ͑KB͒ theory of solutions is a rigorous theory of solution mixtures which relates the molecular distributions between the solution components to the thermodynamic properties of the mixture. Ideal solutions represent a useful reference for understanding the properties of real solutions. Here, we derive expressions for the KB integrals, the central components of KB theory, in ideal solutions of any number of components corresponding to the three main concentration scales. The results are illustrated by use of molecular dynamics simulations for two binary solutions mixtures, benzene with toluene, and methanethiol with dimethylsulfide, which closely approach ideal behavior, and a binary mixture of benzene and methanol which is nonideal. Simulations of a quaternary mixture containing benzene, toluene, methanethiol, and dimethylsulfide suggest this system displays ideal behavior and that ideal behavior is not limited to mixtures containing a small number of components. © 2010 American Institute of Physics. ͓doi:10.1063/1.3398466͔ I. INTRODUCTION solutions. In SI solutions the KBIs are neither zero nor inde- pendent of composition. The KBIs for these solutions can be The Kirkwood–Buff ͑KB͒ theory of solutions has pro- ͑␬ ͒ expressed in terms of the isothermal compressibility T and vided a wealth of data and insight into the properties of so- ͑ ͒ 1–13 molar volumes Vi of the pure components at the same T lution mixtures. KB theory can be applied to any stable and P. Expressions for binary, ternary, and quaternary solu- mixture containing any number of components. In particular, tions are available.2,20,21 Ben-Naim has also shown that SI the theory provides exact relationships from the molecular ⌬ solutions including up to four components satisfy Gij=Gii distributions between the various species in solution to the 22 +Gjj−2Gij=0 for all i, j pairs. Our recent analysis of the corresponding thermodynamics of the mixture. The central results obtained from the KB theory of solutions for n=1 to quantities of interest in KB theory are the KB integrals ͑ ͒ 1 4 components proposed the following general expression for KBIs , the KBIs:21 ϱ ␲͵ ͓ ␮VT͑ ͒ ͔ 2 ͑ ͒ n Gij = Gji =4 gij r −1 r dr, 1 SI ␬ ␳ 2 ͑ ͒ 0 Gij = RT T − Vi − Vj + Sn, Sn = ͚ kVk , 2 k=1 where g is the corresponding radial distribution function ij R ␳ N /V ͑rdf͒ between species i and j, and r is their intermolecular where is the gas constant and i = i are the number separation. In applying KB theory to understand solutions, it densities of each species. The above expression was postu- i j n is often useful to compare the properties of a real solution lated to be valid for any and combination in any com- with those of a corresponding ideal solution.2,6,14,15 In par- ponent SI solution. This can be shown to be true for n 21 ticular, deviations from ideal solution behavior provide indi- =1–4 components. Here we provide rigorous proof that cations of specific associations or affinities between the so- this expression is valid for any number of components at 16,17 T P lution components which affect the thermodynamics. constant and . Furthermore, we also determine the corre- Furthermore, assuming ideal behavior for some components sponding expressions for the KBIs which result in ideal be- may be the only possible approach to describe solutions con- havior on the molality and molarity concentration scales. The taining a large number of components, where experimental results are then illustrated using data obtained from computer data are usually very rare.18,19 Consequently, it is important simulations. to understand the nature of ideal solutions as formulated by KB theory. Ideality in solution mixtures at constant pressure ͑P͒ and II. THEORY ͑ ͒ temperature T can be expressed using a variety of concen- The chemical potential ͑␮͒ of a solute in solution can be tration scales. The most common is the mole fraction con- expressed in a variety of ways using a series of different centration scale where the corresponding activity coefficients concentration scales. There are three major expressions re- are unity for all species at all solution compositions. These lated to whether the species concentrations are represented in ͑ ͒ mixtures are also known as symmetric ideal SI or perfect terms of mole fractions ͑x͒, molalities ͑m͒, or molarities ͑c͒. It is then typical to write a͒ Author to whom correspondence should be addressed. Tel.: 785-532-5109. ␮ ␮o,x ␮o,x ␮ex FAX: 785-532-6666. Electronic mail: [email protected]. i = i + RT ln fixi = i + i + RT ln xi, 0021-9606/2010/132͑16͒/164501/9/$30.00132, 164501-1 © 2010 American Institute of Physics Downloaded 04 Aug 2011 to 129.130.37.167. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 164501-2 Ploetz, Bentenitis, and Smith J. Chem. Phys. 132, 164501 ͑2010͒ ␮ץ , ␮ = ␮o,m + RT ln ␥ m i i i i ␤ͩ i ͪ = ␮ , ij ץ ln mj T,P,m ␮ ␮o,c kj i = i + RT ln yici, ␮ ␮ץ i ij 3␳ ͑ ͒ ␤ͩ ͪ = , ͑7͒⌳ ء␮ ␮ ln ␳ ␦ − ␾ ץ i = i + RT ln i i, 3 j T,P,Nkj ij j ͑ ␥ ͒ where a set of activity coefficients f , ,y are used to quan- which are general for any number of components. Alterna- tify deviations from ideal behavior on the mole fraction, mo- tively, if one starts from Eq. ͑4͒ and then takes derivatives lality, and molarity concentration scales, respectively. The with respect to pressure with all Nj and T constant one finds final equation is the statistical mechanical expression for the that chemical potential in terms of the pseudochemical potential n thermal de Broglie wavelength ͑⌳͒, and number ,͒ء␮͑ ␬ ͑␦ ͒ ͑ ͒ density.2 The standard chemical potentials ͑␮o͒ can refer to RT T = ͚ ij + Nij Vj 8 j=1 an infinitely dilute solute or the pure solute at the same T and P. They are constants and independent of the concentration for any i component. of i, whereas the pseudochemical potential is composition The above equations relate the KBIs to properties of the dependent. The following discussion is focused on changes solution. It is easier, albeit less general, if we select a par- in the chemical potentials with composition, and therefore ticular species to continue. We choose i=1 and k1, and ͑ ͒ ͑ ͒ ͑ ␳ ͒ the choice of standard state is largely irrelevant. We also note then write Eqs. 5 and 8 after multiplication by 1 in an RT d ln y when T is constant. It is generally nϫn matrix form so that= ءthat d␮ known that the activity coefficients decrease when the solute ␳ V ␳ V ␳ V ␳ V 1+N ␳ RT␬ displays significant solute self-association, whereas the activ- 1 1 1 2 1 3 ¯ 1 n 11 1 T ␮ ␮ ␮ ␮ ␾ ity coefficients increase when the solute is significantly sol- 12 22 32 n2 N12 − 2 ␮ ␮ ␮ ␮ ␾ vated by the other species in solution. 13 23 33 n3 N13 = − 3 . The general approach used here to determine the re- ΄ ΅΄ ΅ ΄ ΅ ]]] ] quired KBIs involves generating a series of expressions in- ␮ ␮ ␮ ␮ N − ␾ volving derivatives of the chemical potentials which are 1n 2n 3n ¯ nn 1n n valid for real solutions, followed by application of the appro- ͑9͒ priate conditions corresponding to ideality for the various Hence, we have a set of simultaneous equations which can concentration scales. Let us consider the species number be solved quite easily for the required KBIs to give densities in the grand canonical ensemble to be functions of ␳ ␬ temperature and all the chemical potentials. One can then 1+N11 1RT T 23 ␾ write N12 − 2 n −1 ␾ ͑ ͒ N13 = Mn − 3 , 10 ␳ ␤ ͑␦ ͒ ␮ ͑ ͒ d ln i = ͚ ij + Nij d j, 4 ΄ ΅ ΄ ΅ ] ] j=1 ␾ N1n − n which is valid for changes in the number density of any where M is the matrix from Eq. ͑9͒. This is the approach we component in any multicomponent system and any ͑thermo- n used recently for outlining a general KB inversion dynamically reasonable͒ ensemble with T constant. Here, procedure.21 After the appropriate chemical potential deriva- N =␳ G , ␤=1/RT, and ␦ is the Kroenecker delta function.