Are there any other ways of estimating fusion enthalpies and melting temperatures? Mobile Order and Disorder Theory

ln x = -[(  fus H/R)(1/T-1/T fus )+  (  trans H/R)(1/T-1/T trans )] – ln , where x is the observed solubility in mole fraction,  fus H is the enthalpy of fusion, T fus is the melting point of the compound of interest, and R and T refer to the gas constant and temperature of measurement, respectively and  represents the activity coefficient. Mobile Order and Disorder Theory (MOD) ln  B = A + B + D + F + O + OH(1) where the solubility  B on a volume fraction scale (  B = x B V B /(x B V B -(1-x B )V S ) of a solute B in a solvent S is evaluated by a series of terms.

ln  B = A + B + D + F + O + OH (2) A = - (  fus H/R)(1/T-1/T fus ) -  (  trans H/R)(1/T-1/T trans ) (3) B = 0.5  S (V B /V S – 1) ln(  B +  S V B /V S ) (4) D = -  S 2 V B (  B -  S ) 2 /[RT(1.0 + max(K OH, K O )  S /V S )] (5) F = - r S  S V B /V S +  OHi  S (r S +b i ) (6) O =  Oi ln[(1 + K Oi (  S /V S - Oi  B /V B )] (7) OH =  OHi [ln(1 + K OHi  S /V S + K BBi  B /V B ) – ln(1 +K BBi V B )] (8) The terms represent different factors that can influence the solubility of each respective compound, including such factors as hydrogen bonding, nonspecific cohesion forces, entropic factors, and others.

 S is the volume fraction of the solvent, S; calculated from 1-  B  B is the volume fraction of the solute; calculated from solubility V B is the molar volumes of the solute V S is the molar volumes of the solute V B & V S can be estimated by group additivity  B &  S are modified cohesion parameters;  S values are tabulated for most common solvents;  B is an unknown. K O, K OH, & K BB refer to stability constants that describe the strength of association between solute-solvent and solute-solute molecules respectively resulting from hydrogen bonding r S & b are structuration factors associated with amphiphilic solvents

Table. Standard group interaction stability constants and related parameters at 298 K a Term Value Comment r S 0for non-associated solvents (all hydrocarbons, esters, ketones nitriles) r S 1for strongly associated solvents forming single hydrogen bonds (alcohols) r S 2for water and diols (molecules involved in double hydrogen bonded chains b0for non-aqueous solvents K OH 40solute donor : –OH; solvent acceptor: -C  N; -NO 2 K OH 200solute donor :–OH; solvent acceptor: aromatic ring;CH 2 Cl 2 K OH 230solute donor : secondary amine; solvent acceptor: -OH K OH 300solute donor : –OH; solvent acceptor: CHCl 3 K OH 1000solute donor : secondary amide; solvent acceptor: -OH K OH 1500solute donor: aromatic or conjugated amine; solvent acceptor: -OH K OH 2000solute donor: –OH; solvent acceptor: ketone

K OH 2500solute donor: –OH; solvent acceptor: ester; ether K OH 5000solute donor: –OH; solvent acceptor: -OH K O 110solute acceptor: ester. ether; HN-N= ; solvent donor: -OH K O 170solute acceptor: ketone; solvent donor: -OH K O 300solute acceptor: tertiary amine; solvent donor: -OH K O 600solute acceptor: tertiary amide; solvent donor: -OH K BB 0solute acceptor: secondary amine; solvent donor: secondary amine K BB 1000solute acceptor: secondary amide; solvent donor : secondary amide K BB 1500solute acceptor: aromatic or conjugated amine; solvent donor: aromatic or conjugated amine K BB 5000solute acceptor: -OH; all steroids; solvent donor: - OH ; all steroids Oi refers to the number of K O, K OH interations in polyfunctional molecules

ln  B = A + B + D + F + O + OH (2) Assuming no additional phase transitions between T fus and 298 K A = - (  fus H/R)(1/T-1/T fus ) (3)

B = 0.5  S (V B /V S – 1) ln(  B +  S V B /V S )  S = 1 -  B represents a correction factor for the entropy of mixing accounting for the different sizes of the solute and solvent molecules D = -  S 2 V B (  B -  S ) 2 /[RT(1.0 + max(K OH, K O )  S /V S )] represents the change in the non-hydrogen bonding cohesion forces when fluid solute is mixed with solvent

Solvent  S V S chloroform CCl benzene toluene CH 2 ClCH 2 Cl cyclohexane butyl acetate acetone ethyl acetate hexane octane butanol propanol methanol

F = - r S  S V B /V S +  OHi  S (r S +b i ) represents the structuration of the solvent when the solvent is an alcohol or water b = 0 for alcoholic solvents, O =  Oi ln[(1 + K Oi (  S /V S - Oi  B /V B )] represents the proton acceptor solute-solvent interaction on the solute. It reflects the effect of hydrogen bonding on solubility between a hydroxylic solvent and proton acceptor sites on the solute.

OH represents the proton donor solute solvent interaction describing the the effect on solubility of a proton donor site on the solute which both self associates (K BB ) and interacts with the solvent (K OH ). For non-aqueous solvents OH = 0

Calculation of solubility of methyl hexadecanoate The following is needed: 1.The melting point and the molar enthalpy of fusion in order to calculate A 2.The formula in order to calculate the molar volume 3.The values of K O, K OH 4.The cohesion parameter  B. This is obtained by measuring the solubility of the compound in a solvent that does not form hydrogen bonds such as hexane..

Calculation of  B for methyl hexadecanoate For non-hydrogen bonded solvents K OH, K O are not found, therefore K = 0, r S = 0 ln  B = A + B + D D = ln  B - A - B  S 2 V B (  B -  S ) 2 /RT = ln  B - A - B (  B -  S ) 2 = -RT(ln  B - A – B)/  S 2 V B mol fraction X B = ; V B = 309cm 3 /mol A = V S (hexane) = 131.6cm 3 /mol  S = 14.56(J/cm 3 ).5

Calculation of solubility of methyl hexadecanoate For non-hydrogen bonded solvents K OH, K O are not found, therefore K = 0, r S = 0 For alcohols K O = 110;solute acceptor: ester; solvent donor: -OH r S = 1  fus H (303.8) = kJ mol -1 V B = 309 cm 3 /mol  B = J 0.5 /cm -1.5

Suppose we use experimental solubilities and MOD theory For a known compound with an unknown mp and fusion enthalpy and a known solubility in a known solvent ln  B = A + B + D + F + O + OH We know ln  B ; all the terms in B, all the terms in D except  B ; all the terms in F, O and OH By measuring the solubility in two or more solvents, we have two (or more equations) and two unknowns and we can solve for A and  B

Solvent  SO V SO  Bexpt B D F O ln  Bcalcd  Bexpt chloroform CCl benzene toluene CH 2 ClCH 2 Cl cyclohexane butyl acetate acetone ethyl acetate hexane octane butanol propanol methanol

A = - (  fus H/R)(1/T-1/T fus ) RA = -  fus H/T +  fus S Since T = K  fus S =  tpce S = 2* *7.1* = 173 RA = (8.314)(-0.75) = Then  fus H = T(-RA +  fus S) = ( ) = J mol -1 T fus = 304 K Lit.  fus H(304) = 55.6 kJ mol -1

; b non-associated solvents; c associated solvents; d computed using all experimental solubility data;

Compound T fus expt T fus calc b T fus calcd c T fus calcd d  tpce S estd e  tpce H calcd d  tpce H expt

Figure. A comparison of fusion temperatures calculated from solubility measurements and estimated total phase change entropies with experimental values. Standard deviation K  12 K K  23 K All data (81)  39 K

Figure. A comparison of calculated total phase change enthalpies with experimental values for all 81 compounds in the data base. Standard deviation  6.4 kJ mol -1