Non-ideal gases and intermolecular interactions
3.1 Non-ideal behavior
Whereas all gases act ideally at low pressure where they are on average far from each other, all gases act non-ideally as they approach conditions that promote condensation. At low enough T and high enough p, real gases should condense into a liquid. Indeed, at 1 atm and low enough T every element except He condenses into a liquid. He condenses into a solid at 1 atm and requires p = 25 atm to form a liquid. However, if we examine the ideal gas law in these limits we find
Therefore an ideal gas would condense into nonexistence. This is an artifact of assuming that the particles are point-like with no intrinsic volume.
If molecules were neutral, nonpolarizable hard spheres, there would be extremely limited deviations from ideal gas behavior. We can add a term to the ideal gas law to account for molecular volume. However, the revised equation would predict that all molecules would condense at T = 0 K, rather than having the molecule-specific boiling point Tb that we know exists. The more molecules deviate from neutral nonpolarizable spheres, the more we expect them to act non-ideally. And since the intermolecular interactions increase in strength the closer molecules are to one another, the greater the deviations. Therefore as density increases, we expect deviations from ideal behavior to increase.
3.2 Interactions of matter with matter
Since interactions are the root cause of non-ideal behavior, now is a good time to review a few qualitative trends from the Periodic Table that will help us to think about interatomic and intermolecular interactions. As we move to the right and down in the periodic chart, the atomic number Z (= positive charge on the nucleus) increases as does the molar mass M (or the mass of a single atom m). The radius of the atom also tends to increase because the number of electrons in the neutral atom must equal Z (there are notable exceptions as in the case of the lanthanide contraction). As a general rule, the heavier the atom, the stronger its interatomic interactions.
The final column of the periodic chart is a bit different from the rest. These are the noble gases. They only partake in chemical bonding in special cases, if at all, because they have filled electronic shells. We will discuss electronic configurations and orbitals in detail later. You are already familiar with these concepts in a descriptive sense; however, we will eventually describe them with a quantum mechanical treatment. Atoms become bigger as more electrons are added because only two electrons can fit in a given orbital. As subsequent orbitals are added to accommodate more electrons, the orbitals have progressively larger radial extents.
The force that holds the electrons in an atom is the attractive electrostatic interaction between the positive nucleus and the negatively charged electron. The form of the potential energy curve V(r) is shown in Fig. 3.1. The negative value indicates an attractive interaction. The interaction energy falls off as 1/r, in other words, it goes to zero at large electron–nucleus separation r according to
where e is the elementary charge and ϵ0 is the vacuum permittivity. The closer the electron gets to the nucleus, the lower the energy. The conundrum of why the electron does not simply spiral into the nucleus can only be answered quantum mechanically. The farther away the electron is from the nucleus, the more weakly it is bound.
Figure 3.1 A representation of the electrostatic potential that holds an electron in the H atom. The electrostatic interaction that binds the electron to the nucleus behaves classically (obeys classical physics); however, prediction of the energy level structure requires quantum mechanics. In the H atoms, to a first approximation, all electrons with the same principle quantum number are degenerate (at the same energy); whereas this is not true for atoms with more than one electron. a0 is the Bohr radius, the atomic unit of length.
All atoms in their ground state are neutral and spherically symmetric. This symmetry means that they are nonpolar. In order to induce a dipole in them, the charge distribution must be distorted from its spherical symmetry: the atom must be polarized. Higher Z atoms are more polarizable than lighter atoms for two reasons. First, since small orbitals close to the nucleus fill first and larger orbitals farther from the nucleus fill last, the outer electrons tend not to experience the full nuclear charge. This effect is called screening: the reduction of the effective charge that interacts with a particle of interest as a result of intervening charge. Second, since the outer electrons are further away, they have a lower attraction to the nucleus from the 1/r dependence of the electrostatic interaction. Polarizability α is proportional to the volume V occupied by the electronic states of the system (atom or molecule)
where ϵ0 is the vacuum permittivity.
The implication for the noble gases is clear. In going from He to Xe and then Rn, the polarizability will increase substantially. Xe is much more susceptible to spontaneous or induced fluctuations in its electron cloud that push it away from spherical symmetry. These fluctuations lead to transient dipoles that can interact with one another and change the energy of the gas. The effects are quite small for He, but reasonably large for Xe. The trend holds for the rest of the periodic chart: as Z increases down a column or across a row, so does polarizability. This is reflected in the physical state of elements at room temperature and pressure in that gases tend to be located in the upper right, and solids are formed as one moves down in the Periodic Table. The only elements that are liquid or close to being liquid at standard ambient conditions are Ga (Tf = 29.77 °C), Hg (Tf = −38.83 °C) and Br2 (Tf = −7.2 °C). Phosphorus (Tf = 44.1 °C) and the alkali metals, for instance Cs (Tf = 28.5 °C) also have low melting points but are pyrophoric and not encountered in their elemental form (for long) when exposed to air. Recall also that the halogens as well as H2, N2, and O2 exist as diatomic molecules at standard ambient temperature and pressure.
The relationship between polarizability and electronic state volume also holds for molecules. Thus, a small molecule such as H2 has α = 0.804 × 10−30 m3 while that of CO is 1.95 × 10−30 m3 and benzene, with its extended π-bonding system, has α = 10.3 × 10−30 m3.
There are a number of ways that matter interacts with other matter. The forces of Nature are electromagnetism, the strong force, the weak force and gravity. Electrical and magnetic forces were unified, that is, shown to be fundamentally related by Maxwell’s theory of electromagnetism. Most intermolecular interactions result from electromagnetic interactions.
Protons and neutrons, unlike electrons, are composite particles that require a force to hold them together. The strong force holds protons and neutrons together as individual nucleons. The field of quantum chromodynamics (QCD) encompasses the study of the strong force. The distance-dependence of the strong force is unlike that of any other force. Over a range of about 0.8 fm, which is about the radius of a nucleon, the strong force falls off with distance, much like other forces. Beyond this distance, the strong force is constant. However, interactions between the subatomic particles within nucleons screen the strong force, which results in a much smaller and roughly exponentially decaying residual strong force outside of nucleons. It is this residual strong force that binds proton and neutrons together to form nuclei. Furthermore, the screening-induced exponentially decaying distance dependence causes the residual strong force to be effective over no more than 3 fm or so. Thus, even though it is the strongest of all forces and roughly 100 times stronger than electromagnetism, the strong force can be ignored outside of the nucleus and in most of chemistry.
All fermions, such as electrons and protons, interact through the weak force. Weak interactions are involved in radioactive decay, including beta decay, and nuclear fusion. As implied by the name, the weak force is about 1000 times weaker than electromagnetism. In addition, the weak interaction is extremely short-ranged. At distances greater than about 0.1 fm it is negligible. Therefore, the weak force can be ignored outside of radiochemistry. Electromagnetism and the weak interaction were unified by the electro-weak theory of Glashow, Salam and Weinberg in 1968. The electro-weak interaction was unified with the strong interaction by the Standard Model in the 1970s and confirmed by the discovery of the Higgs Boson in 2013.
Gravitation is by far the weakest force with a coupling constant roughly 10−37 less than that of electromagnetism. It is also the most ubiquitous force since all matter is attracted by this force according to Newton’s law of universal gravitation
Here, objects with masses m1 and m2 are separated by a distance r. The gravitational constant is G = 6.674 × 10−11 N m2 kg−2. In the (approximately) uniform gravitational field of the Earth, this reduces to
for the gravitational force of the Earth that is responsible for weight. The Earth’s gravitational acceleration is g = 9.81 m s−2. This value varies with altitude, local geography and latitude. The gravitational potential energy of an object raised a height h is
Usually, gravity represents a very small correction that has a negligible effect on gases, but this is not so for very cold gases. In fact, the gravitational force is important for the extremely accurate atomic clocks that incorporate a fountain of cesium atoms. The atoms are first laser cooled, then ‘thrown’ upwards only to fall back down into the fountain under the influence of gravity. The gravitational force can often be neglected for liquids and solids. It is, however, essential for explaining meniscus formation or the operation of a manometer.
3.2.1 Directed practice
The total pressure at a depth z under the surface of water is given by
where p° is atmospheric pressure at the surface of the water (taken here to be the standard pressure of 1 bar), ρ is the density of water and g the gravitational acceleration. Calculate the total pressure at a depth of 10 m.
[Answer: 200 kPa, which leads to the rule that for every 10 m dived, the total pressure increases by 1 bar.]
3.3 Intermolecular interactions
We now concentrate on the interactions most important for molecules. As we shall see, these are all the result of electromagnetism and the spatial distribution of electrons relative to the positive nucleus, including how electrons fill atomic and molecular orbitals. We will divide the discussion along the lines of the range of the interactions beginning with long-range interactions.
3.3.1 Long-range interactions
There are three types of long-range interactions: electrostatic, induction, and dispersion. The electrostatic energy depends on the magnitude and position of charge. Induction and dispersion are polarization energies that depend on how electrons in a molecule respond to an external electric field.
To calculate the electrostatic potential a priori requires a solution of the Schrödinger equation. However, a multipole expansion can be used to approximate its value. This is only valid where there is no orbital overlap. The zeroth moment of the charge distribution can be taken as the charge q itself. The first moment is the dipole moment given by the product of the charge and the distance r by which two charges −q and +q are separated
Note that 
and R are vectors with the radius vector drawn from the negative to the positive charge. The common unit of dipole moments is the debye, abbreviated D (1 D = 3.336 × 10−30 C m). To get some feeling of the magnitude of 1 D, separating charges of ±e by 1 Å leads to a dipole of 4.80 D. The dipole moment of H2O is 1.85 D, while that of HCl is 1.11 D. Selected values of dipole moments are listed in Table 3.1. Zwitterionic molecules undergo intramolecular proton transfer in aqueous solution, and this produces a positive and a negative charge within one molecule and correspondingly large dipole moments. The values shown in Table 3.1 loosely conform to the expectation that greater electronegativity differences lead to larger dipoles. However, dipoles are the result of charge distributions in molecular orbitals and the product of charge times distance. Thus, CO has a small dipole (that also points in the ‘wrong’ direction – that is, with more charge density on the C end of the molecule) and NaF has a smaller dipole moment than NaI.
Table 3.1 Dipole moments of selected molecules. Data taken from the 96th edition of the CRC Handbook of Chemistry and Physics.
| Name | Formula | μ/D |
|---|---|---|
| Ammonia | NH3 | 1.4718 |
| Hydrogen fluoride | HF | 1.826178 |
| Hydrogen chloride | HCl | 1.1086 |
| Hydrogen bromide | HBr | 0.8272 |
| Hydrogen iodide | HI | 0.448 |
| Imidogen | HN | 1.39 |
| Hydroxyl | OH | 1.655 |
| Mercapto | HS | 0.7580 |
| Carbon monoxide | CO | 0.10980 |
| Nitric oxide | NO | 0.15872 |
| Sodium fluoride | NaF | 8.156 |
| Sodium chloride | NaCl | 9.00117 |
| Sodium bromide | NaBr | 9.1183 |
| Sodium iodide | NaI | 9.236 |
| Water | H2O | 1.8546 |
| Methanol | CH3OH | 1.70 |
| Ethanol | CH3CH2OH | 1.69 |
| Phenol | C6H5OH | 1.224 |
| Acetic acid | CH3COOH | 1.70 |
| Acetonitrile | CH3CN | 3.92519 |
Next come the quadrupole Θ, octopole and hexadecapole moments. A cigar- or disk-shaped charge distribution gives rise to a quadrupole moment; a nearly spherical molecule will have either an octopole (e.g., CH4) or hexadecapole (e.g., SF6) as its first nonzero moment. In a completely spherical charge distribution such as that of a ground-state closed shell atom all moments are zero.
The multipole expansion of the electrostatic energy of two molecules is an expansion in terms of powers of R, where R is the separation of the centers of mass of the two molecules. The interaction energy falls off with distance according to R−(m+n+1), where m and n are the two moments that are interacting. Thus, if the molecules A and B have charges qA and qB, the first term in the expansion is the Coulomb energy
The second term is that between the charge on one molecule and the dipole of the other
Next comes the dipole–dipole interaction
The coordinate system used to describe multipole interactions is defined in Fig. 3.2. The energy of interaction between dipoles depends on their mutual orientation, as is shown in Fig. 3.3. The most favorable interaction is when the dipoles are aligned collinearly head to tail followed by opposite to one another side-by-side. The least favorable is when they are aligned head-to-head and tail-to-tail. In general, the lowest order nonvanishing term will dominate the interaction energy. However, several terms are required to obtain a truly accurate energy. In other words, the multipole expansion approximates the interaction energy of two charge distributions as a series expansion. Whereas, the first nonzero term in the expansion is generally by far the greatest, the series may require several terms to converge on its asymptotic value for highly accurate calculations.
Figure 3.2 The coordinate system for multipole interactions. The centers of mass of the multipoles are separated by a distance R. The multipoles A and B are inclined at angles θA and θB with respect to this axis. The angle φ describes the angle between the planes formed by the multipole axes and the line of centers. A is specified as a dipole to indicate the IUPAC convention.
Figure 3.3 (a) Dipole–charge interaction as a function of orientation at fixed separation. (b) Dipole–dipole interaction as a function of orientation in a collinear configuration. (c) Dipole–dipole interaction in a side-by-side orientation. Note that an in-line head-to-tail arrangement in panel (b) is a more favorable configuration than the side-by-side head-to-tail arrangement in panel (c).
Figure 3.4 The distance-dependence of interactions with different ranges is displayed. Note that all are shown as repulsions (increasing energy with decreasing distance). Dispersion and induction, both of which are always attractive, follow R−6 behavior. The absolute value is shown to facilitate comparison on the same scale. Whether charge–charge (R−1), dipole–charge (R−2) and dipole–dipole (R−3) interactions are attractive (negative) or repulsive (positive) depends on the orientation and the sign of the charges. The exchange interaction (Pauli repulsion) is always repulsive.
Figure 3.5 Hydrogen bonding in (a) the water dimer and (b) liquid water.
The electric field of strength E generated by a charged particle or dipole induces a dipole moment in a neighboring molecule in direct proportion to the polarizability α of the molecule
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