- Related Topics:
- liquid crystal
- solution
- pure liquid
- nonionic liquid
- liquid mixture
- On the Web:
- PNAS - Significant structures in the liquid state (PDF) (Mar. 29, 2025)
While viscosity is concerned with the transfer of momentum and thermal conductivity with the transfer of heat, diffusivity is concerned with the transport of molecules in a mixture. If a lump of sugar is put into a cup of coffee, the sugar molecules travel from the surface of the lump into the coffee at a speed determined by the temperature and by the pertinent intermolecular forces. The characteristic property that determines this speed is called diffusivity—i.e., the ability of a molecule to diffuse through a sea of other molecules. Diffusivities in solids are extremely small, and those in liquids are much smaller than those in gases. For this reason, a spoon is used to stir the coffee to speed up the motion of the sugar molecules, but, if the odour of cigarette smoke fills a room, little effort is needed to clear the air—opening the windows for a few minutes is sufficient.
In order to define diffusivity, it is necessary to consider a binary fluid mixture in which the concentration of solute molecules is c1 at position 1 and c2 at position 2, which is l centimetres from position 1; if c1 is larger than c2, then the concentration gradient (change with respect to distance), given by (c2 - c1)/l, is a negative number, indicating that molecules of solute spontaneously diffuse from position 1 to position 2. The number of solute molecules that pass through an area of one square centimetre perpendicular to l, per second, is called the flux J (expressed in molecules per second per square centimetre). The diffusivity D is given by the formula
The leading minus sign is introduced because, when the gradient is positive, J is negative, and, by convention, D is a positive number. In binary gaseous mixtures, diffusivity depends only weakly on the composition, and, therefore, to a good approximation, the diffusivity of gas A in gas B is the same as that of gas B in gas A. In liquid systems, however, the diffusivity of solute A in solvent B may be significantly different from that of solute B in solvent A. In a very viscous fluid, molecules cannot rapidly move from one place to another. Therefore, in liquid systems, the diffusivity of solute A depends strongly on the viscosity of solvent B and vice versa. While the letter D is always used for diffusivity, viscosity is commonly given the symbol η: in many liquid solutions it is observed that, as the composition changes (as long as the temperature remains constant), the product Dη remains nearly the same.
Thermodynamics and intermolecular forces in solutions
The properties of solutions depend, essentially, on two characteristics: first, the manner in which the molecules arrange themselves (that is, the geometric array in which the molecules occupy space) and, second, the nature and strength of the forces operating between the molecules.
Energy considerations
The first characteristic is reflected primarily in the thermodynamic quantity S, called entropy, which is a measure of disorder, and the second characteristic is reflected in the thermodynamic quantity H, called enthalpy, which is a measure of potential energy—i.e., the energy that must be supplied to separate all the molecules from one another. Enthalpy minus the product of the absolute temperature T and entropy equals a thermodynamic quantity G, called Gibbs energy (also called free energy):
From the second law of thermodynamics, it can be shown that, at constant temperature and pressure, any spontaneous process is accompanied by a decrease in Gibbs energy. The change in G that results from mixing is designated by ΔG, which, in turn, is related to changes in H and S at constant temperature by the equation
At a fixed temperature and pressure, two substances mix spontaneously whenever ΔG is negative; that is, mixing (either partial or complete) occurs whenever the Gibbs energy of the substances after mixing is less than that before mixing.
The two characteristics that determine solution behaviour, structure and intermolecular forces, are, unfortunately, not independent, because the structure is influenced by the intermolecular forces and because the potential energy of the mixture depends on the structure. Only in limiting cases is it possible, on the one hand, to calculate ΔS (the entropy change upon mixing) from structural considerations alone and, on the other, to calculate ΔH (the enthalpy change of mixing) exclusively from relations describing intermolecular forces. Nevertheless, such calculations have proved to be useful for establishing models that approximate solution behaviour and that serve as guides in interpreting experimental measurements. Solutions for which structural considerations are dominant are called athermal solutions, and those for which the effects of intermolecular forces are more important than those of structure are called regular solutions (see below Regular and athermal solutions).
Effects of molecular structure
A variety of forces operate between molecules, and there is a qualitative relation between the properties of a solution and the types of intermolecular forces that operate within it. The volume occupied by a solution is determined primarily by repulsive forces. When two molecules are extremely close to one another, they must necessarily exert a repulsive force on each other since two molecules of finite dimensions cannot occupy the same space; two molecules in very close proximity resist attempts to shorten the distance between them.
At larger distances of separation, molecules may attract or repel each other depending on the sign (plus or minus) and distribution of their electrical charge. Two ions attract one another if the charge on one is positive and that on the other is negative; they repel when both carry charges of the same sign. Forces between ions are called Coulomb forces and are characterized by their long range; the force (F) between two ions is inversely proportional to the square of the distance between them; i.e., F varies as 1/r2. Noncoulombic physical forces between molecules decay more rapidly with distance; i.e., in general F varies as 1/rn, n being larger than 2 for intermolecular forces other than those between ions.
The Coulomb force (F) equals the product of the magnitude of the charge on one ion (e1) and that on the other (e2) divided by the product of the distance squared (r2) and the dielectric constant (ε):
If both e1 and e2 are positive, F is positive and the force is repulsive. If either e1 or e2 is positive while the other is negative, F is negative and the force is attractive. Coulomb forces are dominant in electrolyte solutions.
Molecular structure and charge distribution
If a molecule has no net electrical charge, its negative charge is equal to its positive charge. The forces experienced by such molecules depend on how the positive and negative charges are arranged in space. If the arrangement is spherically symmetric, the molecule is said to be nonpolar; if there is an excess of positive charge on one end of the molecule and an excess of negative charge on the other, the molecule has a dipole moment (i.e., a measurable tendency to rotate in an electric or magnetic field) and is therefore called polar. The dipole moment (μ) is defined as the product of the magnitude of the charge, e, and the distance separating the positive and negative charges, l: μ = el. Electrical charge is measured in electrostatic units (esu), and the typical charge at one end of a molecule is of the order of 10-10 esu; the distance between charges is of the order of 10-8 centimetres (cm). Dipole moments, therefore, usually are measured in debyes (one debye is 10-18 esu-cm). For nonpolar molecules, μ = 0.
Polar molecules
The force F between two polar molecules is directly proportional to the product of the two dipole moments (μ1 and μ2) and inversely proportional to the fourth power of the distance between them (r4): that is, F varies as μ1μ2/r4. The equation for this relationship contains a constant of proportionality (F = kμ1μ2/r4), the sign and magnitude of which depend on the mutual orientation of the two dipoles; if the positive end of one faces the negative end of the other, the constant of proportionality is negative (meaning that an attractive force exists), while it is positive (meaning that a repulsive force exists) when the positive end of one faces the positive end of the other. When polar molecules are free to rotate, they tend to favour those orientations that lead to attractive forces. To a first approximation, the force (averaged over all orientations) is inversely proportional to the temperature and to the seventh power of the distance of separation. Mixtures of polar molecules often exhibit only mild deviations from ideality, but mixtures containing polar and nonpolar molecules are frequently strongly nonideal. Because of the qualitative and quantitative differences in intermolecular forces, the molecules segregate: the polar molecules prefer to be with each other, and so do the nonpolar ones. Only at higher temperatures, such that the thermal energy of the molecules offsets the cohesion between identical molecules, do the two liquids mix in all proportions. In mixtures containing both polar and nonpolar components, deviations from Raoult’s law diminish as temperature rises.
Nonpolar molecules
A nonpolar molecule is one whose charge distribution is spherically symmetric when averaged over time; since the charges oscillate, a temporary dipole moment exists at any given instant in a so-called nonpolar molecule. These temporary dipole moments fluctuate rapidly in magnitude and direction, giving rise to intermolecular forces of attraction called London (or dispersion) forces. All molecules, charged or not, polar or not, interact by London forces. To a first approximation, the London force between two molecules is inversely proportional to the seventh power of the distance of separation; it is therefore short-range, decreasing rapidly as one molecule moves away from the other. The London theory indicates that for simple molecules positive deviations from Raoult’s law may be expected (i.e., the activity coefficient γi is greater than 1, as explained previously). Since the London theory suggests that the attractive forces between unlike simple molecules are smaller than those corresponding to an ideal solution, the escaping tendency of the molecules in solution is larger than that calculated by Raoult’s law. As a result, mixing of small nonpolar molecules is endothermic (absorbing heat from the surroundings) and the volume occupied by the liquid solution often exceeds that of the unmixed components—that is, the components expand on mixing.
In addition to the forces listed above, there are so-called induction forces set up when a charged or polar molecule induces a dipole in another molecule: the electric field of the inducing molecule distorts the charge distribution in the other. When a charged molecule induces a dipole in another, the force is always attractive and is inversely proportional to the fifth power of the distance of separation. When a polar molecule induces a dipole in another molecule, the force is also attractive and is inversely proportional to the seventh power of separation. Induction forces are usually small but may make a significant contribution to the energy of a mixture of molecules that are strongly dissimilar.
