F

An f orbital is an atomic orbital with l = 3 (Figure F.1). There are seven such orbitals in a given shell of an atom and their occupation gives rise to the f-block of the periodic table (the lanthanoids and the actinoids).

Linear combinations of the orbitals depicted here can be formed to generate seven orthogonal orbitals of the same shape (Figure F.2). They are aligned along the edges of a pair of heptagonal pyramids that share an apex, as shown on the right, and which make an angle of 30.38° to the vertical line through the shared apex.

Faraday’s constant, F, is the magnitude of the charge per mole of electrons, F = N_Ae, and has the value 96.485 332 12 kC mol⁻¹.

A fermion is a particle with a half-integral spin quantum number. Important fermions in chemistry are the electron and proton, both of which are spin-half particles. Fermions obey the Pauli principle, that their wavefunctions change sign under particle exchange, and therefore obey the Pauli exclusion principle, that no more than one can occupy any state (state here including spin, so up to two can occupy a spatial state provided their spins are opposite). A general feature of the universe is that matter can be regarded as fermions (including the constituents of nuclei) bound together by the exchange of bosons.

A ferromagnetic material is one in which domains of electrons are aligned with their spins parallel (Figure F.4). Materials that possess a ferromagnetic phase do so below a critical temperature called the Curie temperature, T_C. In an antiferromagnetic solid adjacent atoms have their electron spins locked together in an antiparallel array. The transition from paramagnetism to antiferromagnetism occurs at the Néel temperature, T_N. A ferrimagnetic solid is like an antiferromagnetic material but the spins on neighbouring atoms are different and the net spin of the sample is nonzero.

The figure axis of a body is the principal n-fold rotational axis with n > 2 (Figure F.5). Some molecules (those classed as tetrahedral, cubic, octahedral, and icosahedral) have more than one principal axis.

The fine structure of an NMR spectrum is the line spitting that arises from coupling between nuclear spins. The fine structure in an atomic spectrum is the line splitting that arises from spin–orbit coupling.

The fine-structure constant can be used to express in a compact form a number of other basic quantities that are related to the strength of the interaction between electrical charges. For example, the Bohr radius is a₀ = ħ/αm_ec ² and the Rydberg constant is $hc R ˜ ∞ = 1 2 α 2 m e c 2$ ⁠.

the First law of thermodynamics represents a combination of the statements of the conservation of energy and the equivalence of heat and work as modes of transferring energy between a system and its surroundings. There are several equivalent statements of the law:

1. The work required to transform a system from a specified initial state to a specified final state along an adiabatic path is independent of the path.

2. A change in the internal energy of a system is an exact differential that can be written in the form dU = dw + dq, where dw is the energy supplied to the system as work and dq is the energy supplied as heat.

3. The internal energy of an isolated system is constant.

4. Perpetual motion of the first kind (the generation of work without the consumption of fuel) is impossible.

Statement 1 implies the existence of a state function and motivates the introduction of the internal energy into thermodynamics. Statement 2 acknowledges the equivalence of heat and work as modes of transferring energy between the system and its surroundings. Statement 3 acknowledges the conservation of energy augmented by recognizing two modes of changing the internal energy (heat and work). Statement 4 is a verbalization of the second and third statements, and is essentially the earliest statement of the law.

Nuclear fission is the decay of a nucleus into two fragments of similar mass. Nuclear fusion is the formation of a nucleus by the coalescence of two smaller nuclei. These processes are often accompanied by the ejection of particles such as neutrons, neutrinos, and γ-ray photons. Fission that can take place without needing to be initiated by the impact of other particles is called spontaneous nuclear fission. That is in contrast to induced nuclear fission, which is initiated by the impact of a neutron on a heavy nucleus. Nuclei that can undergo induced fission are called fissionable. Nuclei that can be nudged into undergoing fission even by slow (thermal) neutrons are classified as fissile. Fissile nuclei include uranium-235, uranium-233, and plutonium-239.

Flash desorption spectroscopy is the observation of the pressure surge of desorbed adsorbate that occurs when a temperature is swept up through a characteristic value. If the adsorbate is present at sites with different binding characteristics, then a series of peaks is observed as the temperature reaches values at which desorption is rapid from each site. The technique is used to determine the activation energy for desorption from a surface.

The Flory θ temperature is the temperature at which the osmotic viral coefficient of a macromolecular solution is zero. A solution that is at its Flory θ temperature is called a θ-solution. Such a solution behaves nearly ideally and so its thermodynamic and structural properties are easier to describe even though its concentration might not be low.

The observational definition of fluorescence is that it is the emission of electromagnetic radiation from an excited state of a molecule that ceases as soon as the exciting radiation is extinguished. The mechanistic definition includes the statement that excitation and emission occur without change of multiplicity. Except in the special case of resonance fluorescence (see below), it occurs at longer wavelengths, lower frequencies, than the exciting radiation and displays the vibrational fine structure of the lower electronic state.

The mechanism of fluorescence is that excitation to a higher electronic state of a molecule is followed by radiationless transitions down the vibrational states of that electronic state as it discards energy into its surroundings (Figure F.7). The final transition is by the emission of radiation as the molecule returns from the vibrational ground state of the excited electronic state to various vibrational states of the lower electronic state. The electronic states involved all have the same multiplicity and are typically singlet states. Consequently, there are no slow, spin-forbidden transition involved in the process and it is therefore fast.

In resonance florescence the incident radiation and the resulting fluorescence radiation have the same frequency. The fluorescence is much more intense because the emissive transitions are stimulated by the incident radiation. In delayed fluorescence the emission is delayed after the initial excitation. There are two mechanisms of delayed fluorescence. In one mechanism, the excited molecule forms an excimer or exciplex, which subsequently decays. In the other mechanism, two triplet excited states pool their energy to form an excited singlet state, which then decays radiatively. In X-ray fluorescence radiation is emitted from a sample after an X-ray photon has ejected an electron from an inner shell of an atom or deep in the band structure of a metal. Then emission occurs when an electron of higher energy falls into the vacancy and emits the excess energy as radiation.

The flux, J , of a property is the quantity that passes through a region in an interval divided by the area of the region and the length of the interval (colloquially, the ‘quantity per unit area per unit time’). It includes the flux of matter, of energy as heat, and of electric charge. It is found empirically that the flux of a property is proportional to the gradient of a related property. Thus it is found that the flux of matter is proportional to the concentration gradient and the flux of energy as heat is proportional to the temperature gradient. See diffusion.

A Fourier transform expresses f as a superposition of harmonic (sine and cosine) functions, with g the amplitude of the corresponding harmonic function in the mix. If f is a rapidly changing function (in space or time), then mainly short-wavelength, high-frequency components are needed and g is correspondingly large for those components and small for others. If f is a slowly changing function, then the opposite is true (Figure F.9).

Fourier transform techniques in spectroscopy are effectively physical realizations of this mathematical procedure. In X-ray crystallography, the intensity of the diffraction at a range of angles is related to structure factors, and the spatial Fourier transform gives the electron density distribution in the unit cell (see Fourier synthesis). Spectroscopy makes use of temporal Fourier transforms. In Fourier-transform NMR spectroscopy (FT-NMR) the resonance spectrum is extracted by Fourier transformation of the time-variation of the absorption signal. Thus, in its simplest form, a free-induction decay (FID) signal from a molecule that has been subjected to a radiofrequency pulse emits a range of frequencies as it returns to equilibrium. The Fourier transform of that signal reveals the component frequencies present (that is, the spectrum).

In fractional distillation a liquid mixture is separated into its components by successive vaporizations and condensations in a vertical fractionating column. In terms of a phase diagram, these successive steps result in the vapour highest in the column being richest in the most volatile component. If that fraction, or sample of distillate boiling in a particular temperature range, is withdrawn the next fraction may be removed and so on until the components have been separated (Figure F.10). The efficiency of a particular process is measured by in terms of the number of theoretical plates, or horizontal tie lines, that are encountered between the initial composition and the most highly refined fraction. The separation fails if the mixture forms an azeotrope.

The Franck–Condon principle states that an electronic transition occurs within a stationary nuclear framework. Its classical basis is that the nuclei are much heavier than the electrons, so remain in their initial state while the electronic transition takes place. The implication of the principle is that an electronic transition occurs vertically, in the sense that the nuclei remain in their initial locations in a molecular potential energy diagram (Figure F.11). Moreover, because they are stationary initially, they remain stationary. Consequently, at the end of the electronic transition the nuclei are found at a turning point of the upper electronic molecular potential energy curve. They then begin to vibrate with the corresponding energy. In other words, the Franck–Condon principle accounts for the excitation of vibration as a result of an electronic transition.

In quantum mechanical terms, the principle expresses the fact that the dynamical state of the nuclei (that is, their wavefunction) is preserved as much as possible during an electronic transition. The final nuclear vibrational wavefunction is therefore the one that most resembles the initial nuclear vibrational wavefunction (Figure F.12). The former has a peak near a classical turning point, and the latter has a peak close to the equilibrium nuclear conformation. Thus, the quantum mechanical description captures the essence of the classical description. Moreover, because there are several quantized vibrational states lying close to the intersection of a vertical line with the upper curve there is a probability that any of these nearby states can be excited. The same remarks apply to the downward transition responsible for fluorescence and phosphorescence. The principle is made quantitative by noting that the intensities of the transitions are proportional to the square the overlap integral between the initial and final vibrational wavefunctions, $S v f v i$ (a measure of the resemblance of two wavefunctions). In this context, the squares of these integrals, are referred to as Franck–Condon factors.

The frontier orbitals of a molecule are their highest occupied molecular orbital, the HOMO, and the lowest unoccupied molecular orbital, the LUMO. Frontier orbitals are important because they are largely responsible for the chemical and spectroscopic properties of molecules. The lowest energy electronic transition is typically HOMO → LUMO.