Thermodynamics of the Interactions

What quantity describes best the totality of these solute–solvent interactions and how can the various contributions to them be estimated? Let the pure solute B be vaporized in an imaginary process to a gas, and let this gas be very dilute, so that it obeys the ideal gas laws. In this condition each particle of the solute (molecule or ion) is very remote from any neighbor and has no environment with which to interact. If B is polyatomic, it does have its internal degrees of freedom, such as bond vibrations and rotation of the particle.

If this ideal gaseous particle B is introduced into a liquid A at a given temperature and pressure, all of the solute–solvent interactions become “switched on,” the solvation process takes place, and the Gibbs energy of solva-

tion, ∆solvGB, is released. In many cases the process of dissolution of a gaseous

solute in a liquid solvent can, indeed, be carried out experimentally—for in- stance, when propane or carbon dioxide is dissolved in water to give a solution at a given gas pressure.

The Gibbs energy change for the process of dissolution of the gaseous solute B,∆solnGB, is the driving force for the material transfer. When equilibrium

is reached, ∆solnGB becomes zero (since, at equilibrium, no more net transfer

occurs). The following equation then holds:

∆solnGB eq = =∆solnGB+ T ln [cB l cB g eq °

0 R ( ) / ( )] ( . )2 9 where ∆solnG° is the standard molar Gibbs energy of solvation of the solute B,B

defined by Eq. (2.9) and cBis its molar concentration (moles per unit volume) in

assumed for the processes discussed elsewhere in this chapter, and is only em- phasized here for Eq. (2.9) by the subscript eq for ∆solnGBand the ratio of con-

centrations, cB(l)/cB(g). The concentration in the gas phase is, of course, given

by the pressure according to the ideal gas law cB(g)= nB/V= P/RT. The ratio of

the concentrations is known as the Bunsen coefficient, KB(B,A), for the solubility

of the gaseous solute B in liquid A. At low pressures and concentrations KB(B,A)

depends only on the temperature and is independent of the pressure and the concentration.

Whether obtained from an actual experimentally feasible process or from a thought process, ∆solnGB°, which is obtained from Eq. (2.9) by re-arrangement,

pertains to the solvation of the solute and expresses the totality of the solute– solvent interactions. It is a thermodynamic function of state, and so are its deriv- atives with respect to the temperature (the standard molar entropy of solvation) or pressure. This means that it is immaterial how the process is carried out, and only the initial state (the ideal gaseous solute B and the pure liquid solvent) and the final state (the dilute solution of B in the liquid) must be specified.

Because it is a function of state,∆solnGB° may be considered to be made up

additively of the contributions from the various stages in which the transfer of the solute particle from the gaseous state into the liquid solvent has been envis- aged by the foregoing to take place.

The first stage is the creation of the cavity in the liquid to accommodate the solute. Obviously, work must be done against the cohesive forces of the liquid that hold its molecules together. This work should be proportional to the required size of the cavity, and increase as the volume of the solute increases. An expression for this work would be

∆cav cav B A 2

G=A Vδ ( .2 10)

where Acavis a proportionality coefficient, VBis the molar volume of the solute B, and δ2

Ais the cohesive energy density of the solvent A. Thus, in a series of solvents for a given solute, the positive contribution of cavity formation to ∆solnG° increases with the squares of the solubility parameters of the solvents.B For a series of solutes and a given solvent, it increases with the molar volumes of the solutes.

It is more difficult to estimate the contribution from the dispersion forces to the solute–solvent interactions. Their energy increases with the product of the polarizabilities of the partners, but decreases strongly with the distance between them (being proportional to the inverse sixth power of the distance). The polarizability is related to the molar volume, hence, to the third power of the linear dimension of the solute or the solvent. Hence, the product of the polarizabilities depends on the sixth power of the distance between the centers of the interacting molecules. Consequently, these tendencies balance each other. For large molecules (e.g., metal chelates, liquid hydrocarbons), it is bet-

ter to consider the interactions between adjacent segments of neighboring mol- ecules, which are at a constant mean distance from one another in the liquid solution. The contribution from the dispersion forces (negative, because they are attractive) is proportional to the surface areas of the interacting molecules, or to the number N of segments present, and depends on their chemical natures. If A represents again the liquid solvent, then [3]

∆dispG= ∑A N NiBA iB iA ( .2 11)

where AiBAis the (negative) interaction Gibbs energy of a pair of segments of kind i, and the summation extends over all the different kinds of segments. A methylene group, a halogen atom, a−CH=CH− group of an aromatic ring, or some other functional group, generally serves as a segment.

There may be additional, specific interactions between the solute and the solvent. Hydrogen bonds may be formed between them, particularly in protic solvents, i.e., solvents that contain hydrogen atoms bonded to oxygen (more rarely, nitrogen) atoms, such as water, alcohols, carboxylic acids, or acidic phosphoric esters. Hydrogen bonds are formed with solute anions, with hydrated ions in general (having an outer surface of water molecules), and with neutral solutes that have a very basic atom with a lone pair of electrons that can accept a hydrogen bond. Also, donor–acceptor bonds can be formed if the solvent has a very basic atom in its structure donating a pair of unshared elec- trons, if it is suitably exposed, and the solute is a cation or some acidic neutral molecule accepting this pair.

A generalized equation for∆solvGB° is [4]

∆solvGB A A A A A °_{= + + + +} 0 2 2 12 ππ* αα ββ δδ ( . )

which describes the value of∆solvG° for a given solute (characterized by AB 0, Aπ,

Aα, Aβ, and Aδ) with a series of solvents. The solvents are characterized by their Taft–Kamlet solvatochromic parameters:π*

for polarity–polarizability,α for hydrogen bond donation acidity, andβ for hydrogen bond acceptance basicity. Values of these solvatochromic parameters have been tabulated for many sol- vents. (Table 2.3 gives the values for selected solvents.) As before,δ is the solubility parameter. The first two terms on the right-hand side of Eq. (2.12) express∆dispG, the next two the hydrogen-bonding interactions, and the last one ∆cavG.

As an example, consider phenol as the solute and water and toluene as two solvents. The parameters for phenol are Aπ= 5.7, Aα= −12.9, Aβ= −18.3, and Aδ= 0.0091, whereas A0is unspecified, but a negative quantity. With the solvent parameters fromTables 2.1and 2.3, the standard Gibbs energy of sol- vation of phenol in water becomes A0+ 3.39, and in toluene A0+ 4.11 kJ mol−1.

It is seen that∆solvGB° is lower in water than in toluene, so that the transfer of phenol from water to toluene entails an increase in∆solvGB°. The consequence of this is that phenol prefers water over toluene, since work would be required to make this transfer. It should be remembered that the standard Gibbs energies of solvation refer to the state of infinite dilution of the solute (solute–solute

Table 2.3 Solvatochromic Parameters for Some Solvents and (Monomeric) Solutes (in Parentheses)

Substance π* α β Substance π* α β

c-Hexane 0 0 0 2-Butanone 0.60 0.06 0.48

n-Hexane −0.11 0 0 Cyclohexanone 0.68 0 0.53

Benzene 0.55 0 0.10 Acetophenone 0.81 0.04 0.49

Toluene 0.49 0 0.11 Acetic acid 0.64 1.12 0.45

p-Xylene 0.45 0 0.12 Hexanoic acid 0.52 1.22 0.45

Dichloromethane 0.82 0.13 0.10 Benzoic acid 0.80 (0.87) (0.40)

Chloroform 0.58 0.20 0.10 Ethyl acetate 0.45 0 0.45

Carbon tetrachloride 0.21 0 0.10 Butyl acetate 0.46 0 0.45

1,2-Dichloroethane 0.73 0 0.10 Propylene carbonate 0.83 0 0.40

Chlorobenzene 0.68 0 0.07 Formamide 0.97 0.71 0.48

Water 1.09 1.17 0.47 Dimethylformamide 0.88 0 0.69

Water monomer (0.39) (0.32) (0.15) Tetramethylurea 0.79 0 0.80

Methanol 0.60 0.98 0.66 Hexamethyl phosphoramide 0.87 0 1.00

Methanol monomer (0.39) (0.38) (0.41) Acetonitrile 0.66 0.19 0.40

Ethanol 0.54 0.86 0.75 Benzonitrile 0.88 0 0.37

Ethanol monomer (0.39) (0.36) (0.47) Nitromethane 0.75 0.22 0.06

1-Propanol 0.52 0.84 0.90 Nitrobenzene 0.86 0 0.30

2-Propanol 0.48 0.76 0.84 Triethylamine 0.09 00.71

1-Hexanol 0.40 0.80 0.84 Tri-n-butylamine 0.06 0 0.62

Trifluoroethanol 0.73 1.51 0 Dimethyl sulfoxide 1.00 0 0.76

Trifluoroethanol monomer (0.59) (0.10) Diphenyl sulfoxide 0 0.70

Phenol 0.72 1.65 0.30 Sulfolane 0.90 0 0.39

Diethyl ether 0.24 0 0.47 Triethyl phosphate 0.69 0 0.77

Diisopropyl ether 0.19 0 0.49 Tributyl phosphate 0.63 0 0.80

Tetrahydrofuran 0.55 0 0.55 Triethyl phosphine oxide 0 1.05

Dioxane 0.49 0 0.37 Triphenyl phosphine oxide 0 0.94

Anisole 0.70 0 0.32 Pyridine 0.87 0 0.64

1,2-Dimethoxyethane 0.53 0 0.41 Quinoline 0.93 0 0.64

Acetone 0.62 0.08 0.48

interactions being, therefore, absent), and that a reaction such as ionic dissocia- tion of the phenol in water is ignored in this example.

Extensive tables of solute parameters are beyond the scope of this book. Equations (2.10) to (2.12) are meant to show the nature of the dependencies of the additive terms on various quantities. They enable the prediction of tenden- cies of solute–solvent interactions for a given solute with a series of solvents or for a series of solutes with a given solvent.