D

doi:10.1039/BK9781837674244-00067

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Physical chemistry is the part of chemistry that seeks to account for the properties and transformations of matter in terms of concepts, principles, and laws drawn from physics. This glossary is a compilation of definitions, descriptions, formulae, and illustrations of concepts that are encountered throughout the subject. This section describes the concepts that begin with the letter D; where appropriate, the entries also describe subsidiary but related concepts. Refer to the Directory for a full list of all the concepts treated.

D Lines

The D lines are pair of closely spaced lines in the emission spectrum of alkali metal atoms. They arise from the transition ²P_3/2 → ²S_1/2 and ²P_1/2 → ²S_1/2 (Figure D.1). The resulting splitting arises from the spin–orbit coupling in the p¹ configuration. In sodium they differ in wavenumber by 17 cm⁻¹.

Figure D.1

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The transitions leading to D lines.

d–d Transition

A d–d transition is a transition of the electron between d orbitals in a d-metal complex in which the degeneracy of the orbital has been removed by the ligand field. In an octahedral complex the transition is typically e_g → t_2g (Figure D.2). That transition is symmetry-forbidden but acquires intensity from an antisymmetric vibration of the complex, which destroys the centre of symmetry. It is therefore an example of a vibronic transition, a transition that become allowed by coupling to a vibrational excited mode of the complex.

Figure D.2

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A d–d transition in an octahedral complex.

d Orbital

A d orbital is an atomic orbital with l = 2 (Figure D.3).

Figure D.3

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The conventional representation of d orbitals.

Two sets of linear combination of the orbitals shown in Figure D.3 can be formed such that all five of each set have the same shape. One such linear combination is shown in Figure D.4. All five of this set lie along the edges of the double pentagonal pyramid shown in the illustration, which lie at 41.79° to the vertical through the shared apex.

Figure D.4

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One of five equivalent d orbitals.

δ Orbital

A δ orbital is a molecular orbital that is formed by the face-to-face overlap of two d orbitals and which when viewed along the internuclear axis resembles a d orbital. A δ orbital contributes to a quadruple bond, which has the composition σ²π⁴δ² (Figure D.5).

Figure D.5

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A δ orbital.

Daniell Cell

A Daniell cell is a galvanic cell of structure Zn(s)|ZnSO₄(aq)|CuSO₄(aq)|Cu(s). In laboratory studies it would be Zn(s)|ZnSO₄(aq)||CuSO₄(aq)|Cu(s) with a salt bridge to eliminate the junction potential.

Davisson–Germer Experiment

The Davisson–Germer experiment is the observation of the diffraction of electrons from layers of atoms in a nickel crystal. The experiment confirmed that electrons have wavelike properties.

de Broglie Relation

The de Broglie relation states that a particle travelling with linear momentum p has a wavelength λ given by

λ = \frac{h}{p}

where h is Planck’s constant. The relation is justified by the Schrödinger equation, with λ identified with the wavelength of the wavefunction of a free particle. A consequence is that the wavelength of a particle of mass m and charge Q (for an electron, Q = −e) accelerated from rest though a potential difference Δϕ is

λ = \frac{h}{{(2 m | Q | Δ ϕ)}^{1 / 2}}

debye (the unit)

The unit debye, D, is a non-SI unit widely used to report the electric dipole moment of a molecule: 1 D ≈ 3.335 64 × 10⁻³⁰ C m. The original definition was that 1 D was the magnitude of the dipole moment of charges +1 esu and −1 esu separated by 1 Å (1 Å = 10⁻¹⁰ m).

Debye–Hückel Limiting Law

The Debye–Hückel limiting law is an expression for the mean activity coefficient, γ_±, of an ionic solution as the ionic strength, I, approaches zero:

\log γ_{\pm} = - A | z_{+} z_{-} | I^{1 / 2}, A = \frac{F^{2}}{4 π N_{A} \ln 10} {(\frac{ρ}{2 ε^{3} R^{3} T^{3}})}^{1 / 2}

where z_± are the charge numbers of the ions, ε is the permittivity of the solution, and ρ its mass density (note the common logarithm, to base 10). At 298 K and an aqueous solution, A = 0.509. Deviations from the limiting law occur even at low ionic strengths. A better fit to observation is obtained with the empirically inspired Davies equation:

\log γ_{\pm} = - \frac{| z_{+} z_{-} | A I^{1 / 2}}{1 + B I^{1 / 2}} + C I

The limiting law is based on a model of the ionic solution in which an ionic atmosphere of opposite charge surrounds each ion as result of the competition between the attracting effect of the charge of the central ion and the dispersing effect of thermal motion. The charge of the atmosphere lowers the potential energy of the central ion and therefore its chemical potential. The effect of the atmosphere is equivalent to a single point charge at the distance r_D, the Debye length, from the central ion (Figure D.6), where

r_{D} = {(\frac{ε R T}{2 ρ F^{2} I})}^{1 / 2}

Figure D.6

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The Debye length.

The effect of the ionic atmosphere is to change the Coulomb potential due to the central ion to a shielded Coulomb potential with shielding length λ equal to r_D.

Debye–Hückel–Onsager Theory

The Debye–Hückel–Onsager theory (DHO theory) seeks to account for the constant

K

in the Kohlrausch law of ionic conductivity of a strong electrolyte:

K = A + B Λ_{m}^{○}, A = \frac{z^{2} e F^{2}}{3 π η} {(\frac{2}{ε R T})}^{1 / 2}, B = \frac{z^{2} η e F}{24 π ε R T} {(\frac{2}{ε R T})}^{1 / 2}

where z is the charge number of the ion, ε is the electric permittivity, and η is the viscosity of the solution. Note that B = (η ²/8εFRT)A. There are two contributions. One is the relaxation effect, in which the ionic atmosphere becomes asymmetric and the displacement of its centre of charge from the ion exerts a retarding force. The other is the electrophoretic effect, the retarding effect of moving the entire unit of central ion and its atmosphere through the solution.

Debye–Scherrer Method

The Debye–Scherrer method in X-ray diffraction employs a monochromatic beam of X-rays and a powdered solid, initially in a capillary tube and recorded photographically but now spread on a flat plate and recorded electronically.

Degeneracy

A level is degenerate if two or more states have the same energy. Degeneracy can normally be traced to a symmetry of the system that allows the wavefunctions of degenerate states to be converted into one another by a symmetry operation (Figure D.7). It follows from group theoretical arguments that the maximum degeneracy of a level of a system (such as a molecule) is equal to the largest character for the identity operation in the character table of the system’s point group. Thus, a molecule with C₅ symmetry can have levels that are no more than twofold degenerate, and a molecule with octahedral symmetry can have levels that are no more than threefold degenerate.

Figure D.7

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A simple twofold degeneracy.

In certain cases, degeneracy exists even though superficial inspection of the system does not reveal the presence of symmetry operations able to interrelate wavefunctions. The exact (rather than merely numerically similar) degeneracies are then classified as accidental. One example of accidental degeneracy is provided by a hydrogenic atom, in which all orbitals with the same principal quantum number have the same energy. Another is the degeneracy of the states of a particle in a rectangular box with the sides in various rational proportions (Figure D.8). A third is the high degeneracy of the states of a particle in an equilateral triangular region. However, in such cases deeper inspection of the system shows the presence of a hidden symmetry that interrelates the wavefunctions. The illustration shows the internal rotations that account for the double degeneracy of the level of energy 5h ²/8mL ² in a rectangular well corresponding to the states {n_X,n_Y} = {1,4} and {2,2} with one side that is twice as long as the other side. The accidental degeneracy of a hydrogenic atom is due to the SO4 symmetry of the Coulomb potential (its rotational symmetry in four dimensions).

Figure D.8

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Accidental degeneracy and hidden symmetry.

Delocalization Energy

The delocalization energy E_deloc of conjugated hydrocarbon is the difference between the calculated energy of the π electrons, E_π, and the sum of its N individual localized bonds:

E_{deloc} = E_{π} - N E_{π}^{local}

For butadiene, E_deloc = 0.48|β|; for benzene E_deloc = 2|β|.

Density Functional Theory

Density functional theory (DFT) is a procedure for the calculation of the electronic structures of molecules which focusses directly on the electron density, ρ( r ), rather than on the wavefunction. In mathematics a functional is a function of a function. Although the energy depends on quantum mechanical aspects of the electron distribution, such as electron exchange, according to the Hohenberg–Kohn theorem, the energy can be expressed solely in terms of the electron density, a classical concept. Thus, the energy, E, of a molecule is a functional in the sense that it can be expressed as a function of the electron density, which itself is a function of position, and is written E[ρ]. When expressed in terms of the wavefunction ψ( r ) but with the electron density ρ( r ) = 2|ψ( r )|² in mind, the Schrodinger equation becomes the Kohn–Sham equation, which for a two-electron molecule is

{\hat{H}}_{1} ψ (r_{1}) + \frac{e^{2}}{4 π ε_{0}} \int \frac{ρ (r_{2})}{r_{12}} ψ (r_{1}) d τ_{2} + V_{XC} (r_{1}) ψ (r_{1}) = ε ψ (r_{1})

The quantity V_XC(r₁), which lies at the heart of the DFT procedure, is the exchange–correlation potential, and is defined so that if there is a change in electron density δρ( r ) in the molecule, then the resulting change in energy due to electron exchange is

δ E_{XC} [ρ] = \int V_{XC} (r) δ ρ (r) d τ

The varieties of approach to the DFT procedure (which involves a self-consistent field approach to the numerical solution of the Kohn–Sham equation) are distinguished by the choice of an expression for the exchange–correlation potential. One approach uses a model of a uniform electron gas, in which case

E_{XC} [ρ] = A \int ρ {(r)}^{4 / 3} d τ, A = - \frac{9 e^{2} {(3 / π)}^{1 / 2}}{32 π ε_{0}}

Other expressions have been developed that lead to results in accord with a variety of experimental measurements.

Density of States

The density of states, ρ, is the number of states in a small range of energies divided by the width of the energy of the range: ρ = dN/dE (Figure D.9). In other words, the number of states between E and E + dE is ρdE, with ρ evaluated at E.

Figure D.9

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The density of states.

Depolarization Ratio

The depolarization ratio, ρ, of a line in a Raman spectrum is the ratio of the intensities of scattered radiation with polarization perpendicular and parallel to the plane of polarization of the incident radiation (Figure D.10):

ρ = \frac{I_{⊥}}{I_{∥}}

A line is classified as depolarized if ρ is greater than about 0.75. A general rule is that only totally symmetrical vibrations give rise to polarized lines.

Figure D.10

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Raman polarization.

Dialysis

The process of dialysis is the removal of small molecules and ions from solution that also contains macromolecules or colloidal particles by using a membrane that allows the passage of only the smaller molecules. The process is accelerated in electrodialysis, which is dialysis in the presence of an electric field.

Diamagnetism

A diamagnetic substance is one with a negative magnetic susceptibility. Diamagnetic substances tend to move out of a magnetic field. The magnetic flux density inside a diamagnetic substance is less than in a vacuum. All substances have a diamagnetic contribution to their total magnetic susceptibility. It arises from the generation of electronic currents which give rise to a magnetic field that opposes the applied field. In certain cases, especially when a molecule has low-lying excited states, the induced current runs in the opposite direction and augments the applied field, giving rise to temperature-independent paramagnetism (TIP).

Diathermic

A diathermic boundary is a boundary that permits the passage of energy (as heat) when there is a temperature difference across it.

Diatomic Molecules

Schematic molecular orbital energy level diagrams for homonuclear diatomic molecules of Period 2 are shown in Figure D.11, with the uppermost filled orbital (the HOMO) labelled for each species. The change in the diagram between Groups 15 and 16 arises from the changing separation of the atomic s and p orbitals: when the separation is large, they may be treated separately (qualitatively, at least) but not when the separation is small.

Figure D.11

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Homonuclear diatomic energy levels.

Diffraction

Diffraction is the interference that occurs between waves due to an object in their path. The resulting pattern of intensity is called the diffraction pattern. Diffraction occurs when the dimensions of the diffracting object or its detailed structure are comparable to the wavelength of the radiation. The diffraction pattern arising from light incident on a screen with two slits is illustrated in Figure D.12. Diffraction from a stack of semitransparent planes is the basis of Bragg’s law of diffraction, which was used in early X-ray crystallography.

Figure D.12

$Diffraction by two slits.$

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Diffraction by two slits.

Diffusion

Diffusion is the migration of particles down a concentration gradient. Fick’s first law of diffusion states that the flux, J, the number of molecules passing through a window perpendicular to the flow in an interval divided by the area of the window and the length of the interval (colloquially, ‘the number passing per unit area per unit time’), is proportional to the concentration gradient:

J_{x} = - D \frac{\partial N}{\partial x}

In this expression,

N

is the number density (the number of molecules in a region divided by the volume of the region) and the constant of proportionality, D, is the diffusion coefficient. Note that J_x > 0 (flow towards positive x) if the concentration gradient is negative (down, towards positive x). Fick’s second law of diffusion, which is now almost universally referred to as the diffusion equation, expresses the rate of change of concentration in a region as

\frac{\partial N}{\partial t} = D \frac{\partial^{2} N}{\partial x^{2}}

The three-dimensional versions of these equations are similar:

J = - D \nabla N, \frac{\partial N}{\partial t} = D \nabla^{2} N = - \nabla \cdot J

If convection, the motion of a streaming fluid, also occurs, then the diffusion equation becomes the generalized diffusion equation:

1 D : \frac{\partial N}{\partial t} = D \frac{\partial^{2} N}{\partial x^{2}} - v \frac{\partial N}{\partial x}, 3 D : \frac{\partial N}{\partial t} = - \nabla \cdot J - v \cdot \nabla N

where

v

is the velocity of convective flow. Finally, if chemical reaction takes place, then a term representing the appearance or disappearance of a substance must be included. The resulting equation is called the material balance equation. Thus, if a substance with number density

N

disappears by a first-order reaction with rate constant k_r, the material balance equation in one dimension is

\frac{\partial N}{\partial t} = D \frac{\partial^{2} N}{\partial x^{2}} - v \frac{\partial N}{\partial x} - k_{r} N

Analytical solutions of these equations subject to a variety of boundary and initial conditions are available. Thus, for a solution in which the solute J is initially present as an amount n (and therefore number nN_A) in a thin layer of area A, then after a time t its concentration profile is

[J] (x, t) = \frac{n e^{- x^{2} / 4 D t}}{A {(π D t)}^{1 / 2}}

The corresponding concentration distribution when J is removed by a first-order reaction is

{[J]}^{⁎} (x, t) = k_{r} \int_{0}^{t} [J] (x, t) e^{- k_{r} t} d t + [J] (x, t) e^{- k_{r} t}

The diffusion coefficient can be interpreted in terms of various properties of the solute and the solution. For the self-diffusion of gas molecules (the migration of a gas molecule into regions of lower concentration in a nonuniform sample of a single gas), kinetic theory gives

D = \frac{1}{3} λ v_{mean}

where λ is the mean free path and

v_{mean}

is the mean speed of the molecules. Thus, D increases with temperature and decreases as the sample is compressed isothermally (and the mean free path is thereby reduced). In solution, the diffusion coefficient is related to the temperature and characteristics of the solution by the Stokes–Einstein equation

D = \frac{k T}{f}

where f is the frictional coefficient, the constant of proportionality between the retarding force it experiences and its speed through the solution. If the molecule obeys Stokes’ law, that the retarding force is equal to

6 π η a v

⁠, where η the viscosity and a is its hydrodynamic radius, then

D = \frac{k T}{6 π η a}

If diffusion is regarded as a random walk in which the particle jumps through a distance λ in a time τ, then the diffusion coefficient is given by the Einstein–Smoluchowski equation:

D = \frac{λ^{2}}{2 τ}

A model-free result is that, provided diffusion is an activated process, the temperature dependence of the diffusion coefficient over a moderate range is

D (T) = D_{0} e^{- E_{a} / R T}

where E_a is the activation energy and D₀ is a constant. See viscosity.

Diffusion-controlled Reaction

A reaction in solution can be regarded as the outcome of two stages. One is the encounter of the two reactant species. That encounter is followed by the activated reaction itself. If the rate-determining step is the former, the reaction is termed diffusion-controlled; if the latter step is rate-determining, the reaction is activation controlled. In a diffusion-controlled second-order reaction

v = k_{r} [A] [B], k_{r} = 4 π R^{⁎} D N_{A}, D = D_{A} + D_{B}

where R* is the separation of A and B at which reaction occurs. If A and B obey the Stokes–Einstein equation for the diffusion coefficient, then

k_{r} = \frac{8 R T}{3 η}

where η is the viscosity of the medium. This result suggests that the rate constant depends on the properties of the medium and is independent of the size of the reactants.

Dihedral Plane

A dihedral plane, σ_d, is a mirror plane that bisects the angle between two twofold axes of symmetry (Figure D.13).

Figure D.13

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A dihedral plane.

Direct and Complex Reaction

In a direct reaction of the form A + BC → AB + C, the replacement of C by A takes place very quickly by an immediate exchange as soon as the reactants come within reactive range. In a complex reaction, the ABC cluster survives for a period before ejecting C.

Direct Method

The direct method of analyzing X-ray diffraction data is a statistical procedure for tackling the phase problem by computing the probabilities that phases have specific values.

Dislocation

A dislocation in a crystal is a discontinuity in the regularity of the lattice. In a screw dislocation (Figure D.14) the unit cells in one region are pushed up relative to those in a neighbouring region. The result is a path encircling an axis and forming a continuous spiral.

Figure D.14

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A screw dislocation.

Dispersion

The dispersion of a property, such as refractive index, is its variation with frequency (see polarizability).

Disproportionation Reaction

In a disproportionation reaction an atom of an element undergoes both reduction and oxidation. An example is the reaction 2 Cu⁺(aq) → Cu(s) + Cu²⁺(aq). The reverse of disproportionation is comproportionation. An example is Ag²⁺(aq) + Ag(s) → 2 Ag⁺(aq).

Donnan Equilibrium

The Donnan equilibrium is the distribution of ions in two compartments connected by a semipermeable membrane on one side of which there is a polyelectrolyte. If a polyelectrolyte M_νP is present in the compartment L at a concentration [M_νP] and a salt MX is added to both compartments L and R, then at equilibrium the concentrations of the ions are

{[M^{+}]}_{L} - {[M^{+}]}_{R} = \frac{ν [P] {[M^{+}]}_{L}}{2 [X^{-}] + ν [P]}, {[X^{-}]}_{L} - {[X^{-}]}_{R} = - \frac{ν [P] {[X^{-}]}_{L}}{2 [X^{-}]}

Doppler Effect

In the Doppler effect, radiation (electromagnetic or sonic) is shifted in frequency when the source is moving away from or towards the observer. If the source recedes from the observer with a speed

v

⁠, then the observer detects radiation of frequency

ν_{Receding} = \frac{ν}{1 + v / c}, ν_{Approaching} = \frac{ν}{1 - v / c}

where c is the speed of light (or sound) and

v

is the speed of the source. The Doppler effect is responsible for the contribution to spectral line broadening called Doppler broadening. This broadening arises because in the gas molecules have a wide range of velocities and hence the absorption and emission frequencies are spread over corresponding range of Doppler shifted frequencies. When the temperature is T the width of the Doppler broadened line at half height is

δ ν = \frac{2 ν}{c} {(\frac{2 k T \ln 2}{m})}^{1 / 2}, δ λ = \frac{2 λ}{c} {(\frac{2 k T \ln 2}{m})}^{1 / 2}

Duality

Duality is the possession by particles of wavelike properties and vice versa. Duality is the essence of the Schrödinger equation, which provides a means to calculate the wavefunctions that describe the dynamical properties of particles. Duality is expressed by the de Broglie equation, λ = h/p, by which a wavelength is ascribed to a particle with nonzero linear momentum. Experimental evidence for duality includes the diffraction of particles and the photoelectric effect.

Dulong and Petit’s Law

According to Dulong and Petit’s law, all monatomic nonmetallic solids should have a molar heat capacity equal to 3R. This value would be obtained by the atoms acting as classical oscillators, but due to their quantized character, not all vibrations can be stimulated at low temperatures, so the molar heat capacity falls below 3R. See heat capacity.

Log in

D

D Lines

d–d Transition

d Orbital

δ Orbital

Daniell Cell

Davisson–Germer Experiment

de Broglie Relation

debye (the unit)

Debye–Hückel Limiting Law

Debye–Hückel–Onsager Theory

Debye–Scherrer Method

Degeneracy

Delocalization Energy

Density Functional Theory

Density of States

Depolarization Ratio

Dialysis

Diamagnetism

Diathermic

Diatomic Molecules

Diffraction

Diffusion

Diffusion-controlled Reaction

Dihedral Plane

Direct and Complex Reaction

Direct Method

Dislocation

Dispersion

Disproportionation Reaction

Donnan Equilibrium

Doppler Effect

Duality

Dulong and Petit’s Law

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