1
Pure Liquids
This chapter will be given over to atomic and molecular liquids. A pure molecular liquid is a liquid comprising only one type of non-dissociated molecules. The study of liquids is more difficult than that of gases and solids because they are in an intermediary state, structurally speaking. Indeed, as is the case with solids, we can imagine that in liquids (and this is confirmed by X-ray diffraction), the interactions between molecules are sufficiently powerful to impose a sort of order within a short distance of the molecules. However, the forces involved in these interactions are sufficiently weak for the molecules to have relative mobility and therefore for there to be disorder (no form of order) when they are far apart, as is the case with gases.
1.1. Macroscopic modeling of liquids
In the areas where liquids are typically used, far from the critical conditions, it is often possible to consider liquids to be incompressible - ? meaning that (?V / ?P)T ? 0 - but dilatable. The order of magnitude of a meaning dilation coefficient is 10-3 degrees-1, whereas that of the compressibility coefficient is 10-4 atm-1.
As we approach the critical conditions, this approximation is no longer possible, and the properties of the liquid tend more to be governed by an equation of state. Whilst the "cubic" equations of state for gases do include critical conditions, it is accepted that the properties of liquids often necessitate equations of state that take account of the intervention of forces when more than two bodies are concerned. Additionally, the third- and fourth coefficients of the virial, which can no longer be ignored in the case of liquids, become necessary when these types of forces are at work.
Certain equations of state specific to liquids have been put forward in the literature, including Rocard's, which is written thus:
[1.1] In addition, this equation, expressed as the expansion of the virial, assumes the form:
[1.2] This equation does indeed include the third and fourth coefficients of the virial.
The heat capacities at constant volume and constant pressure are practically identical, around 0.5cal/g, or 2.1kJ/kg.
1.2. Distribution of molecules in a liquid
On a structural level, liquids are classified into two categories: associated liquids and non-associated liquids.
A liquid is said to be non-associated if the intra-molecular degrees of freedom (rotational, vibrational, electronic and nuclear) are not majorly disturbed by the proximity of neighboring molecules. These liquids can be treated, as is the case with gases, with independence between the internal motions and the translation of the molecules.
A liquid is said to be associated if, unlike in the previous case, the molecule's internal degrees of freedom are disturbed by the proximity of other molecules. This disturbance may be so great that, in practical terms, we need to consider associations between molecules, coming together to form dimers, trimers, etc. The new bonds that need to be taken into account are usually hydrogen bonds, whose energy is 4-5 times less than that involved in typical chemical bonds, but which are 4-5 times stronger than intermolecular bonding by van der Waals forces. When the temperature rises, these bonds are broken and, particularly when the thermal agitation energy (kBT) is much greater than the energy in the hydrogen bond, the molecules separate and regain individuality when they are near to the gaseous state.
These associations lend associated liquids very special properties, such as anomalies of the dilation coefficient, high viscosity, low surface tension and a high boiling point. Liquid water belongs to this category. The best way of dealing with these liquids in thermodynamics is to consider them no longer as pure liquids, but rather to treat them as associated solutions, with dimeric, trimeric (etc.) molecules - see section 2.5.
1.2.1. Molecular structure of a non-associated liquid
Hereinafter, we shall focus only on non-associated liquids, and we shall suppose the molecules are spherical. A non-associated liquid is characterized by a local order, or short-distance order. The best illustration of this is of liquid metals. In Figure 1.1, which gives a 2-dimensional image of the arrangement of spherical molecules in a liquid, we can see that the molecules are relatively close together, and that around each molecule, there is an area of order which is illustrated by the circles superimposed on the figure. The short-distance arrangement, within the circles, is almost identical to the molecular arrangement in a solid crystal but, unlike with a crystal, there is no long-distance order. The two circles on Figure 1.1 exhibit no periodicity.
Figure 1.1. Two-dimensional diagram of the distribution of molecules in a liquid
The second difference between a crystal and a liquid is that in the latter, the molecules are in perpetual motion, so Figure 1.1 is representative of the situation only at a given time. Unlike with a solid crystal, the distribution of those molecules would be different an instant later, although we would find similar zones of ordered arrangement.
Hence, in order to accurately describe a liquid, we cannot content ourselves with merely describing the position of a few appropriately-chosen neighboring molecules, as we can with the lattice of the crystal. We would have to define the positions of each of the molecules at every moment in time. In view of the impossibility of the task in a medium with normal dimensions (around a mole, which contains 1023 molecules), we use statistical methods using so-called correlation functions. The paired correlation function which we intend to examine constitutes the first level of this description.
1.2.2. The radial distribution function
Throughout this chapter, we shall suppose that the interactions between N particles of a liquid medium are additive and paired, meaning that the internal energy due to these interactions is merely the sum of the interactions between molecules, two by two. Thus, the internal energy is the sum of the e r. This energy energies between the molecules taken two by two ei,j (ri,j) depends only on the distance between the two molecules. Hence, we have:
[1.3] Consider a molecule chosen at random in the structure (Figure 1.2). Let dN(r) signify the number of molecules whose centers are situated in the crown between the two spheres centered on the chosen molecule, with radii r and r+dr and volume 4p r2 dr. The density of molecules in the crown ?(r), i.e. the number of molecules situated in the crown per unit volume of that spherical crown, at a distance r from the central molecule, is such that:
[1.4] Figure 1.2. Arrangement of molecules of liquid around the center of a cage
The volumetric density ? is defined as the ratio of the total number of molecules in the liquid in question to the volume of that liquid, i.e.:
[1.5] We define the paired correlation factor or the radial distribution function g(r) by the relation:
[1.6] As we can see, this function is the ratio of the mean value of the local density of molecules (mean calculated at the positions, at a given time and over a period of time) to the volumetric density of molecules. The correlation factor g(r) is proportional to the probability of finding a molecule at a distance r + dr from another molecule. Thus, we can write the relation:
[1.7] where di,j is the Kronecker delta, such that:
This ratio [1.7] quantifies the local structure - in other words, the way in which the molecules are arranged in relation to one another.
1.2.3 The curve representative of the radial distribution function
By combining relations [1.4] and [1.6], we see that the mean number of molecules in the coronal volume between the spheres with radii r and r + dr will be:
[1.8] In the solid crystal, only certain distances exist, and the representative curve for the function g(r) exhibits extremely slender peaks for these distances.
In the case of the liquid, the curve representing the function g(r) has the shape shown in Figure 1.3. We obtain a first peak with a breadth ?r/r of several %, which represents the distance between the first neighbors. The next peaks, which represent the second, third (etc.) neighbors, are heavily damped because of the disorder over a long distance. The function g(r) tends toward 1 at a long distance, there is no longer order and therefore, on average, we always find the same number of molecules per unit volume as are present in the overall liquid.
Figure 1.3 can be obtained by neutron diffraction or hard, very penetrating X-ray diffraction, such as those produced by synchrotron sources.
In principle, the distribution g(r) is null for distances less than 0.5 Ä, because there is no chance of finding two molecules that close together, given...
