P

A p orbital is an atomic orbital with l = 1. There are three p orbitals in each subshell (m_l = −1, 0, +1). Real linear combinations are denoted p_x, p_y, p_z according to their orientation with respect to the three Cartesian axes, each one having one angular nodal plane perpendicular to its axis and passing through the nucleus (Figure P.1). A p orbital has zero amplitude at r = 0, and in a shell with quantum number n has n – 2 radial nodes at r > 0.

In molecular orbital theory, a π orbital is a molecular orbital with a nodal plane lying along the internuclear distance (Figure P.2). When occupied by one or two electrons it represents a π bond. In valence bond theory a π bond consists of two paired electrons in overlapping p orbitals, also with a nodal plane that contains the internuclear distance. In a linear molecule, degenerate π orbitals occur in pairs, with angular momentum of magnitude ħ in opposite directions around the internuclear axis. When occupied, the electron density of the pair of orbitals has cylindrical symmetry. In nonlinear molecules, a π bond supplies torsional rigidity to a double bond because rotation around the bond decreases the overlap of the p orbitals and thereby weakens the bond they form.

Two electron spins are said to be paired if the total spin angular momentum is zero. Paired spins are denoted ↑↓. Specifically, the spin wavefunction for two paired spins is (1/2)^1/2{α(1)β(2) − β(1)α(2)} and is antisymmetric under particle exchange. This combination corresponds to S = 0, M_S = 0 (see vector model). According to the Pauli exclusion principle, two electrons must have paired spins if they occupy the same orbital.

A parabolic potential energy is one that varies with distance as x ², where x is the displacement from equilibrium. See harmonic oscillator.

A parallel band in infrared spectroscopy is a series of lines that arise from a vibrational transition in which the dipole moment of the molecule changes parallel to the axis of symmetry, as in the antisymmetric stretch of CO₂.

A paramagnetic material is one with a positive magnetic susceptibility (see magnetic susceptibility); it tends to move into a magnetic field. The magnetic flux density is greater inside a paramagnetic sample than in a vacuum. Paramagnetism is normally due to the presence of unpaired electron spins. In some instances it can arise from the presence of low-lying excited orbital states and is due to orbital magnetism. In the latter case it is known as temperature-independent paramagnetism (TIP).

Parity is the behaviour of an entity under the operation of inversion. See centre of inversion and g and u.

Two liquids are classified as partially miscible if the temperature–composition phase diagram possesses a region in which the system forms two liquid phases (Figure P.3). The relative composition of the phases in equilibrium is given by the lever rule.

The numerical value of a molecular partition function is an indication of the number of states that are thermally accessible at the temperature of interest; typically $q=1$ at T = 0, when only the nondegenerate ground state is accessible, and $q$ rises towards the total number of states in the molecule (which is typically infinite) as T rises to infinity (Figure P.8).

The pascal (Pa) is the SI unit of pressure, with 1 Pa = 1 N m⁻². A convenient related unit is 1 bar = 10⁵ Pa.

Pascal’s triangle is an array of numbers formed from the coefficients in the binomial expansion of (1 + x)ⁿ . The first 10 rows are as follows:

The Paschen series is a set of lines in the spectrum of atomic hydrogen arising from the transitions between n = 4, 5,… and n = 3. All the lines lie in the infrared.

Passivation is the protection of a surface of a metal by an adhering, impervious, stable film, usually an oxide.

The Pauli exclusion principle states that no more than two electrons can occupy a single orbital, and if two electrons do occupy it, then their spins must be paired. The principle is a consequence of the more general Pauli principle and is at the heart of the building-up principle for the explanation of the electron configurations of atoms (and molecules) and therefore of the periodic table.

Penetration is the ability of an outer electron to be found inside the inner shells of atoms and hence to experience the full Coulombic attraction of the nucleus. The most penetrating electrons are those in s orbitals, which have nonzero probability density at the nucleus. They penetrate more than other electrons of the same shell, which due to their orbital angular momentum have amplitudes that vary as r ^l close to the nucleus (Figure P.10); consequently they experience a less shielded nuclear charge and therefore lie lower in energy than the p and d electrons of the same shell. This lowering of the energy is used in the building-up principle to account for the structure of the periodic table.

A perpendicular band in infrared spectroscopy is an absorption arising from a transition in which the electric dipole moment of the molecule changes in a direction perpendicular to the principal axis of the molecule. An example is the bending mode of CO₂. The perpendicular band of a linear molecule may also show a Q-branch of rotational structure.

A phase diagram is a map showing the regions of the intensive variables (commonly pressure and temperature or relative composition and temperature) where each phase of a system is thermodynamically the most stable. The coexistence curves (or phase boundaries), the lines separating the regions, show the conditions under which the two phases separated by the line are in equilibrium. The locations of the coexistence curves can be discussed in terms of the Clapeyron equation and the related Clausius–Clapeyron equation. A triple point is a point in a phase diagram showing the conditions under which three phases are mutually in equilibrium. Three examples of phase diagrams and their interpretation are shown in Figures P.13–P.15.

Empirically, phosphorescence is the emission of visible light when the substance is illuminated with higher energy electromagnetic radiation and which persists for at least short times after the source of illumination is removed. Mechanistically, it is the emission of visible light after excitation and steps that include intersystem crossing, commonly from a singlet to a triplet state.

In more detail, the mechanism of phosphorescence is as follows (Figure P.16). First, a molecule is excited electronically into a vibrationally excited state of an upper electronic state. Then, as radiationless decay of the vibrational excitation occurs, the excited electronic state undergoes intersystem crossing under the influence of spin–orbit coupling and turns into an excited state with a different multiplicity, commonly a triplet (S = 1). Radiationless decay of the vibrational excitation now continues in the new electronic state until the molecule reaches the vibrational ground state of that state. There is then a delay as the state undergoes a spin-forbidden emission transition to the ground singlet state.

Intersystem crossing depends on the presence of spin–orbit coupling, so the phosphorescence may be enhanced by the presence of atoms of a heavy element (such as sulfur). Phosphorescent radiation has a lower frequency than the incident radiation because vibrational energy has been discarded and because the intersystem crossing generally takes place to a lower lying electronic state than the initially excited state.

The photoelectric effect is the ejection of electrons from a solid, usually a metal, when it is illuminated with high energy electromagnetic radiation, typically in the ultraviolet region. The following characteristics are observed:

1. No electrons are ejected, regardless of the intensity of the radiation, unless the frequency exceeds a threshold value characteristic of the solid.

2. The kinetic energy of the ejected electrons varies linearly with the frequency of the incident radiation but is independent of its intensity.

3. Even at low light intensities electrons are rejected immediately if the frequency is above threshold.

The ultraviolet technique yields information about the valence shells of molecules. The X-ray technique allows inner electrons to be studied. These electrons are largely but not entirely independent of the state of bonding of the atom, so the kinetic energy of the photoejected electron can be used to identify the element present in the sample (hence the name ESCA). The ultraviolet technique is enriched by the observation of vibrational structure in the spectrum because some of the energy of the incident photon is left in an excited state of vibration of the cation formed by ionization.

A photon is a quantum of electromagnetic radiation. More generally, a photon is a boson responsible for conveying the electromagnetic force between electrically charged particles. A photon of radiation of frequency $ν$ has an energy $hν$ ⁠; it is massless and is a spin-1 boson. The component of spin angular momentum around its direction of travel is designated by its helicity, σ, which may be ±1, corresponding to left and right circularly polarized radiation, respectively. In a vacuum, photons travel at the speed of light, c, which has the exact, defined value c = 299 792 458 m s⁻¹.

A plait point is a critical point on a triangular phase diagram at which two phases in equilibrium have the same composition (Figure P.17).

Planck’s constant, h, is the fundamental constant introduced at the inception of quantum theory and occurs whenever quantum phenomena are important. Its exact, defined value is h = 6.626 070 15 × 10⁻³⁴ J s. A closely related quantity, which is particularly useful for the discussion of the of angular momenta and circular motion in general, is the reduced Planck’s constant, ħ = h/2π. Its value is ħ ≈ 1.054 571 817 × 10⁻³⁴ J s.

A polar molecule is a molecule with a permanent electric dipole moment. For a molecule to be polar it must belong to one of the groups C_n; in which case the dipole moment lies parallel to the symmetry axis.

The polarizability of a molecule varies with frequency of the applied field (Figure P.18). At low frequencies and if the molecule is polar, the polarizability includes a contribution from orientation distortion, the alignment of the entire molecules in the applied field. This contribution is lost when the frequency is greater than the rotational frequency of the molecule. The next contribution to disappear is the distortion polarization, the polarization that stems from the distortion of the shape of the molecule. This contribution disappears when the frequency exceeds the vibrational frequency of the molecule. At optical frequencies, only the electronic polarization, the polarization stemming from the shift in positions of electrons, survives.

The polarization mechanism is a means of coupling the magnetic moments of nuclei and electrons through the bonds that link them. It is responsible for aspects of the hyperfine structure of EPR spectra and the fine structure of NMR spectra.

The mechanism of the interaction between the two protons in an H–C–H group is as follows (Figure P.19). Suppose the spin of one proton is α. There is a slight advantage, due to the Fermi contact interaction, for the electron in the bond to it to be α too. Therefore, the other electron in the bond must be β, and will be close to the C atom. Hund’s rule favours parallel spins in orthogonal orbitals on atoms, so there will be a slight advantage in the electron close to the C atom in the second bond to be β too. Therefore, the second electron of the second bond will be α and close to the second proton. That proton interacts with it by another Fermi contact interaction, and has a lower energy if it too is α and a higher energy if it is β. Thus, there is a coupling between the two protons. A similar line of reasoning applies to the hyperfine coupling of a π electron in a C–H fragment as expressed by the McConnell equation.

Light is plane polarized if its electric field oscillates in a single plane (and its magnetic component oscillates in a perpendicular plane). It is circularly polarized when the electric vector rotates around the direction of propagation (Figure P.20). Left-circularly polarized light rotates in a counterclockwise direction as perceived by a viewer facing the oncoming ray (and consists of photons of helicity σ = +1). Right-circularly polarized light rotates in a clockwise direction from the same viewpoint (and consists of photons of helicity σ = −1). Plane polarized light can be regarded as a superposition of two counter-rotating circularly polarized rays with the same amplitude. Elliptically polarized light is obtained when the superimposed circular components have different amplitudes.

Positronium, Ps, is a hydrogenic species consisting of an electron, e⁻, and its antiparticle, a positron, e⁺. The ground state is para-positronium (p-Ps), with paired spins (S = 0, M_S = 0); its lifetime is 0.12 ns and decays with the emission of two γ-ray photons. ortho-Positronium (o-Ps), with S = 1, M_S = 0, ±1, lies 1 meV above the singlet, lives for 142 ns, and decays into three γ-ray photons.

A potential energy surface is a plot of the potential energy of a collection of atoms as their relative coordinates are allowed to range over all positions. It is common for visual portrayals to exhibit slices through the full surface; the slices correspond to a variety of constraints applied to the atoms. Thus, in the discussion of a triatomic system the three atoms may be constrained to be collinear (Figure P.21). The trajectory of least potential energy through a surface is identified with the reaction coordinate and the energy requirements and optimum mode of motion, including the value of the activation energy, are assessed on the basis of trajectories over the surface. For a correct mechanistic portrayal of the behaviour of a point on the surface, the axes are slanted (at 60° for a homonuclear triatomic system), for otherwise the effective mass of the system would depend on the direction in which it was travelling. For the classification of surfaces as attractive and repulsive, see that entry.

The powder method of X-ray diffraction makes use of a monochromatic beam of X-rays and a powder sample. The technique is used to recognize the composition of samples from their characteristic diffraction patterns and to determine the symmetry and dimensions of the unit cell.

Power, P, is the rate at which energy is supplied. It is measured in watts, W, with 1 W = 1 J s⁻¹.

Precession is the migration of the axis of rotation of a body on a cone around a fixed axis (Figure P.22). Classically, a magnetic dipole precesses around the direction of an applied magnetic field. In quantum mechanics the concept of precession is preserved in the vector model of angular momentum. In this model, an angular momentum is represented by a vector with a length proportional to the magnitude of the momentum, {j(j + 1)}^1/2, and a component on the z-axis proportional to the quantum number m_j. In the absence of a magnetic field the vector lies stationary at an unspecified angle on a cone. In the presence of a magnetic field, it is supposed to process on this cone at its Larmor frequency. The stronger the field, the faster is the procession.

In the phenomenon of predissociation it is found that the sharp vibrational structure of an electronic transition gives way to a broad featureless structure before the sharp structure resumes and the continuum characteristic of dissociation is reached. The explanation is that the upper electronic molecular potential energy curve is crossed by a dissociative state (Figure P.23). An internal conversion occurs at the intersection, and the upper state takes on a dissociative character at energies that correspond to the crossing. That dissociative character shortens the lifetime of vibrational states, and as a result of lifetime broadening transitions to these states become blurred.

The principal axis is the axis of symmetry of highest order; the C_n axis with the highest value of n.

Promotion is a concept employed in valence bond theory to account for the number of bonds that an atom can form. It is supposed that the overall lowering of energy on bond formation involves an investment in energy in which one or more electrons are transferred from filled valence-shell orbitals into empty valence-shell orbitals, so making more unpaired electrons available for bond formation. Thus, the ground state configuration of carbon, which being $[He]2 s 2 2 p x 1 2 p y 1$ can form only two bonds, is regarded as promoted to $[He]2 s 1 2 p x 1 2 p y 1 2 p z 1$ ⁠, which can form four bonds. The notional investment of energy is recovered from the energy released by the formation of more bonds.

Proton decoupling is a technique used in NMR to simplify the appearance of carbon-13 spectra. The fine structure of the carbon-13 resonance due to their spin-coupling to protons is effectively eliminated by irradiating the latter at their resonance frequencies so that they undergo a rapid spin reorientation. As a result, the coupling to them is averaged to zero.

A pVT surface is a graphical depiction of the equation of state in which one variable is plotted against the other two. The only states in which the substance can exist correspond to points on the surface. The surfaces shown in Figures P.24 and P.25 are for a perfect gas and a van der Waals gas (the labels on the latter are reduced temperatures).