A given (old) aqueous solution is prepared using \(\mathrm{n}_{1}\)(old) moles of water(\(\lambda\)) and \(\mathrm{n}_{j}\) moles of a simple neutral solute at fixed \(\mathrm{T}\) and \(\mathrm{p}\). The enthalpy \(\mathrm{H}(\mathrm{aq} ; \mathrm{old})\) of this solution is expressed in terms of the molar enthalpy of water(\(\lambda\)), \(\mathrm{H}_{1}^{*}(\lambda)\) and the apparent molar enthalpy of the solute \(\phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right)\). \[\mathrm{H}(\mathrm{aq} ; \text { old })=\mathrm{n}_{1}(\text { old }) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right) \nonumber \] We use the description ‘old’ because we envisage preparing a ‘new’ solution by adding \(\mathrm{n}_{1}\)(added) moles of water, enthalpy \(\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda)\). \[\mathrm{H}(\text { added })=\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda) \nonumber \] The enthalpy of the resultant solution is \(\mathrm{H}(\mathrm{aq} ; \text { new })\); \[\mathrm{H}(\text { aq; new })=\left[\mathrm{n}_{1}(\mathrm{old})+\mathrm{n}_{1}(\text { added })\right] \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right) \nonumber \] In effect the ‘old’ solution has been diluted. \[\left.\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{H}(\mathrm{aq} ; \text { new })-\mathrm{H}(\text { aq } ; \text { old })-\left[\mathrm{n}_{1} \text { (added }\right) \, \mathrm{H}_{1}^{*}(\lambda)\right] \nonumber \] Hence, \[\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=\mathrm{n}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}} \text { (new }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} \text { (old }\right)\right] \nonumber \] An isobaric calorimeter measures heat \(\mathrm{q}\) characterising the dilution. \[\begin{aligned} \mathrm{q} / \mathrm{n}_{\mathrm{j}} &=\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{old}) \rightarrow \mathrm{m}_{\mathrm{j}}(\text { new })\right] \\ &=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{\mathrm{j}}=\phi\left(\mathrm{H}_{\mathrm{j}}(\mathrm{j} ; \text { new })\right)-\phi\left(\mathrm{H}_{\mathrm{j}}(\mathrm{j} ; \mathrm{old})\right) \end{aligned} \nonumber \] We imagine a series of experiments in which the molality of solute at the start of the experiment is \(\mathrm{m}_{j}\)(I). Following dilution the molality is \(\mathrm{m}_{j}\)(II). \[\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{II}\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{I}\right) \nonumber \] In a calorimetric experiment we record heat \(\mathrm{q}\) accompanying a second dilution. Hence, \[\Delta_{\mathrm{dll}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{II}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{III})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { III }\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { II }\right) \nonumber \] In a third dilution we have that \[\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{III}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{IV})\right]=\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{IV}\right)-\phi\left(\mathrm{H}_{\mathrm{j}} ; \mathrm{m}_{\mathrm{j}} ; \text { III }\right) \nonumber \] In this experiment the molality of the solution in the sample cell is gradually falling. Combination of the results described by equations (g), (h) and (i) yields the set, \(\Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{II})\right], \Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}} \text { (II) } \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{III})\right], \Delta_{\mathrm{dil}} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{III}) \rightarrow \mathrm{m}_{\mathrm{j}}(\mathrm{IV})\right] \ldots\) This set is expanded with further dilutions until by extrapolation we obtain for solution \(\Delta_{\text {dil }} \mathrm{H}\left[\mathrm{m}_{\mathrm{j}}(\mathrm{I}) \rightarrow \text { infinite dilution }\right]\). We obtain the enthalpies of dilution for all dilutions in a given set of experiments; i.e. for dilution for solutions II, III, IV… Alternatively a given solution is diluted by increasing amounts of solvent; e.g. adding ethanol(\(\lambda\)) to a solution of urea in ethanol [1]. In the analysis of enthalpies of solutions simplification of the algebra is achieved by defining a number of L-variables, signalling differences in enthalpies. The relative enthalpy L describes the difference between the enthalpies of real and ideal solutions. For a solution prepared using \(\mathrm{w}_{1} \mathrm{~kg}\) of solvent (e.g. water), \[\mathrm{L}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1} \mathrm{~kg} ; \mathrm{id}\right) \nonumber \] By definition for the solvent, \[\mathrm{L}_{1}(\mathrm{aq})=\mathrm{H}_{1}(\mathrm{aq})-\mathrm{H}_{1}^{*}(\lambda) \nonumber \] For the solute \(j\), \[\mathrm{L}_{\mathrm{j}}(\mathrm{aq})=\mathrm{H}_{\mathrm{j}}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \] \(\mathrm{L}_{1}(\mathrm{aq})\) and \(\mathrm{L}_{j}(\mathrm{aq})\) are the relative partial molar enthalpies of solvent and solute respectively. Similarly in terms of apparent properties, \[\phi\left(\mathrm{L}_{\mathrm{j}}\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty} \nonumber \] \[\mathrm{L}=\mathrm{n}_{1} \, \mathrm{L}_{1}+\mathrm{n}_{\mathrm{j}} \, \mathrm{L}_{\mathrm{j}}=\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right) \nonumber \] \(\phi(\mathrm{L}_{j})\) is the apparent relative molar enthalpy of solute \(j\) in solution at molality \(\mathrm{m}_{j}\), describing the difference between the apparent molar enthalpies of solute \(j\) in real and ideal solutions. In other words we have a direct probe of the role of solute-solute interactions in solution. Both \(\mathrm{L}\) and \(\phi(\mathrm{L}_{j})\) are (by definition) zero for solutions where the thermodynamic properties are ideal. This galaxy of variables is clarified if we return to a calorimetric experiment where a solution is diluted. An (old) aqueous solution is prepared using \(\mathrm{n}_{1}\) moles of water(\(\lambda\)) and \(\mathrm{n}_{j}\) moles of solute producing a solution having enthalpy \(\mathrm{H}(\mathrm{aq} ; \mathrm{old})\). \[\mathrm{H}(\mathrm{aq} ; \text { old })=\mathrm{n}_{1}(\text { old }) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { old }\right) \nonumber \] To this solution we add (at the same \(\mathrm{T}\) and \(\mathrm{p}\)) \(\mathrm{n}_{1}\)(added) moles of water(\(\lambda\)). \[\mathrm{H}(\text { added })=\mathrm{n}_{1}(\text { added }) \, \mathrm{H}_{1}^{*}(\lambda) \nonumber \] in the limit that \(\mathrm{n}_{1}\)(added) is sufficiently large that the molality \(\mathrm{m}_{j}\) of the ‘new’ solution is negligibly small, then \[\operatorname{limit}\left(\text { new } ; \mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{H}_{\mathrm{j}} ; \text { new }\right)=\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty} \nonumber \] \[\Delta \mathrm{H}(\text { old } \rightarrow \text { new })=-\mathrm{n}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right) \nonumber \] By definition, \[\Delta_{\text {dil }} \mathrm{H}=\Delta \mathrm{H}(\text { old } \rightarrow \text { new }) / \mathrm{n}_{j} \nonumber \] \[\Delta_{\mathrm{dil}} \mathrm{H}=-\phi\left(\mathrm{L}_{\mathrm{j}}\right) \nonumber \] Consistent with our definitions of heat \(\mathrm{q}\) and enthalpy change, a positive \(\Delta_{\mathrm{dil}}\mathrm{H}\) indicates that dilution is endothermic. We have not commented on the dependence of either \(\phi\left(\mathrm{L}_{\mathrm{j}}\right)\) or \(\phi\left(\mathrm{H}_{\mathrm{j}}\right)\) on molality of solute. In order to say something about these variables we need explicit equations for these dependences on composition of solution. An important approach to the description of the properties of solutions uses excess thermodynamic functions. The quantity \(\mathrm{L}(\mathrm{aq})\) defined in equation (n) refers to a solutions prepared using \(\mathrm{n}_{1}\) moles of solvent and \(\mathrm{n}_{j}\) moles of solute, contrasting the properties of real and ideal solutions. The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\) refers to the corresponding solutions prepared using \(1 \mathrm{~kg}\) of water and \(\mathrm{m}_{j}\) moles of solute \(j\). \[\mathrm{H}^{\mathrm{E}}=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}}\right)-\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{m}_{\mathrm{j}} ; \mathrm{id}\right) \nonumber \] thus, \[\mathrm{H}^{\mathrm{E}}= \left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)\right]-\left[\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right] \nonumber \] Or, \[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \,\left[\phi\left(\mathrm{H}_{\mathrm{j}}\right)-\phi\left(\mathrm{H}_{\mathrm{j}}\right)^{\infty}\right] \nonumber \] Therefore \[\mathrm{H}^{\mathrm{E}}=\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{L}_{\mathrm{j}}\right) \nonumber \] Again the development of equation (x) reflects our continuing interest in differences with respect to enthalpies. Nevertheless the key isobaric calorimetric equation requires that the measured ratio (\(\mathrm{q} / \mathrm{~n}_{j}\)) for the process solvent + solute forming an ideal solution (at fixed \(\mathrm{T}\) and \(\mathrm{p}\)) equals the standard enthalpy of solution for pure substance \(j\), \(\Delta_{s \ln } \mathrm{H}^{0}\). For neutral solutes, the dependence of partial molar enthalpy of solute \(\mathrm{H}_{j}(\mathrm{aq})\) on solute molality mj is small such that the recorded (\(\mathrm{q} / \mathrm{~n}_{j}\)) for real solutions can often be equated to the corresponding limiting enthalpy of solution, \(\Delta_{s \ln } \mathrm{H}^{0}\) because in an ideal solution the standard partial molar enthalpy of a solute equals the partial molar enthalpy of the solute at infinite dilution. For solute \(j\), \[\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{0}=\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}=\mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq})-\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~s}) \nonumber \] Significantly modern calorimeters are sufficiently sensitive to measure heat \(\mathrm{q}\) when a known but small amount of substance \(j\) is dissolved in a known amount of solvent. In many cases the dependence of \(\Delta_{\sin } \mathrm{H}\) ion solute molality is, for small neutral solutes, negligibly small such that \(\Delta_{\sin } \mathrm{H}\) is assumed to equal \(\Delta_{\sin } \mathrm{H}^{0}\) [2]. Heats of solution can be analysed in terms of group contributions to the enthalpy of solution for a given series of solutes [3]. Moreover the dependence of \(\Delta_{\sin } \mathrm{H}^{\infty}\) for a given solute on temperature yields the corresponding limiting isobaric heat capacity of solution, \(\Delta_{s \ln } C_{p}^{\infty}\) [4]. In fact by measuring \(\Delta_{\sin } \mathrm{H}^{\infty}\) for solutes in two solvents, the derived property is the standard enthalpy of transfer [5]. \[\begin{aligned} \Delta_{s \ln } \mathrm{H}_{\mathrm{j}}^{\infty} &\text { solvent } \mathrm{B} \rightarrow \text { solvent } \mathrm{A}) \\ &=\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}(\text { solvent } \mathrm{A})-\Delta_{\mathrm{s} \ln } \mathrm{H}_{\mathrm{j}}^{\infty}(\text { solvent } \mathrm{B}) \end{aligned} \nonumber \] Such a study identified a quite striking extremum for limiting partial molar enthalpies of solution for \(\mathrm{NaBH}_{4}\) in water + 2-methylpropan-2-ol mixtures at low alcohol mole fractions and \(298.2 \mathrm{~K}\) and hence a quite striking reversal of sign in limiting partial molar isobaric heat capacities for \(\mathrm{NaBPh}_{4}\) in this binary aqueous mixture [5]; see also data for dialkyl sulfonates in alcohol + water mixtures [6] and tri-n-alkyl phosphates in water + DMF binary mixtures [7]. Indeed an extensive literature describes the enthalpies of solution for neutral solutes and, where the results concern one solute in two or more solvents, the corresponding enthalpy of transfer; cf. equation (z) [8]. Where the results describe a series of closely related neutral solutes, it is often possible to estimate contributions from individual groups (e.g. \(\mathrm{CH}_{2}\) and \(\mathrm{OH}\)) to a given limiting enthalpy of transfer [9]. In many reports, the results of calorimetric experiments show clear evidence of a dependence of partial molar enthalpy of a given solute on molality of the solution. One of the first reports of such a dependence for neutral solutes was published in 1940 [10]. Hence a direct signal is obtained of enthalpic solute-solute interactions in solution. An aqueous solution is prepared using water(\(\mathrm{w}_{1} = 1 \mathrm{~kg}\)) and \(\mathrm{m}_{j}\) moles of solute \(j\) at defined \(\mathrm{T}\) and \(\mathrm{p}\). \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}(\mathrm{aq}) \nonumber \] In the event that the thermodynamic properties of this solutions are ideal the enthalpy of the solution is given by equation (zb). \[\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg} ; \mathrm{id}\right)=\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)+\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \] The excess enthalpy \(\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)\) is given by equation (zc). \[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{H}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)-\left(1 / \mathrm{M}_{1}\right) \, \mathrm{H}_{1}^{*}(\lambda)-\mathrm{m}_{\mathrm{j}} \, \mathrm{H}_{\mathrm{j}}^{\infty}(\mathrm{aq}) \nonumber \] \(\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)\) can be expressed as a power series in molality \(\mathrm{m}_{j}\). \[\mathrm{H}^{\mathrm{E}}\left(\mathrm{aq} ; \mathrm{m}_{\mathrm{j}}\right)=\mathrm{h}_{\mathrm{jj}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{2}+\mathrm{h}_{\mathrm{ij}} \,\left(\mathrm{m}_{\mathrm{j}} / \mathrm{m}^{0}\right)^{3} \nonumber \] A given solution contains solute \(j\) such that an isobaric calorimeter is used to measure the heat of dilution. We obtain the enthalpy per mole of solute on going from molality \(\mathrm{m}_{j}\)(initial) to \(\mathrm{m}_{j}\)(final), \(\Delta_{\mathrm{dil}} \mathrm{H}\). \[\Delta_{\mathrm{dil}} \mathrm{H}=\mathrm{H}^{\mathrm{E}}\left(\mathrm{m}_{\mathrm{j}} \text { - final }\right) / \mathrm{m}_{\mathrm{j}}(\text { final })-\mathrm{H}^{\mathrm{E}}\left(\mathrm{m}_{\mathrm{j}}-\text { initial }\right) / \mathrm{m}_{\mathrm{j}} \text { (initial) } \nonumber \] Equations (zd) and (ze) yield an equation for measured \(\Delta_{\mathrm{dil}} \mathrm{H}\) in terms of enthalpic solute-solute pairwise and triplet interaction parameters. \[\begin{aligned} \Delta_{\mathrm{dil}} \mathrm{H}=\mathrm{h}_{\mathrm{ij}} \,\left[\mathrm{m}_{\mathrm{j}}(\text { final })\right.&\left.-\mathrm{m}_{\mathrm{j}}(\text { initial })\right] / \mathrm{m}^{0} \\ &+\mathrm{h}_{\mathrm{jij}} \,\left[\left\{\mathrm{m}_{\mathrm{j}}(\text { final })\right\}^{2}-\left\{\mathrm{m}_{\mathrm{j}}(\text { initial })\right\}^{2}\right] /\left(\mathrm{m}^{0}\right)^{2} \end{aligned} \nonumber \] In most cases, authors concentrate attention on pairwise interaction parameters [11] between identical (homotactic) and different (heterotactic) solute molecules in a given solution [12]. 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Faraday Soc., 1971, 67 , 1943.; cyclic ethers(aq). S. Cabani, G. Conti and L. Lepori, Trans. Faraday Soc., 1971, 67 , 1933; cyclic amines. K. P. Murphy and S. J. Gill, Thermochim. Acta, 1989, 139 ,279; diketopiperazine(aq). J. M. Corkhill, J. F. Goodman and J. R. Tate, Trans. Faraday Soc., 1969, 65 , 1742 [4] S.-O. Nilsson and I. Wadso, J. Chem. Thermodyn.,1986, 18 ,673; esters(aq). S. J. Gill, N. F. Nichols and I. Wadso, J. Chem. Thermodyn., 1975, 7 , 175. [5] E. M. Arnett and D. R. McKelvey, J. Am. Chem. Soc., 1966, 88 , 5031 [6] C. V Krishnan and H. L. Friedman, J Solution Chem.,1973, 2 ,37. [7] C. de Visser, H. J. M. Grunbauer and G. Somsen, Z. Phys. Chem. Neue Folge, 1975, 97 ,69. S. Lindenbaum, D. Stevenson and J. H. Rytting, J. Solution Chem., 1975, 4 ,893. [8] S. Cabani, G. Conti, V. Moliica and L. Bernazzani, J. Chem. Soc. Faraday Trans.,1991, 87 ,2433; neutral solutes in aq. soln and in solution in octanol. W. Riebesehl, E. Tomlinson and H. M. Grunbauer, J. Phys. 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