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 R; where appropriate, the entries also describe subsidiary but related concepts. Refer to the Directory for a full list of all the concepts treated.
Radial Distribution Function
In discussions of atomic structure, the radial distribution function, P(r), gives the probability of finding an electron in the range r to r + dr regardless of orientation (in other words, the total probability of finding the electron between two concentric spherical shells of radius r and r + dr) as P(r)dr, where
$ P n , l (r)= r 2 R n , l ( r ) 2 $
with R_{n,l}(r) the radial wavefunction. In the special case of an s electron (l = 0), the wavefunction itself can be used and
In discussions of the structures of condensed phases, the radial distribution function, g(r), is defined such that g(r)dr is the probability that a particle will be found in the range dr at the distance r from another particle.
Radiationless Decay
Radiationless decay is the loss of excitation energy to the surroundings by the stimulation of thermal motion. In contrast, radiative decay is the loss of excitation by emission of electromagnetic radiation. See fluorescence and phosphorescence.
Radioactivity
Radioactivity is the spontaneous emission of particles or electromagnetic radiation from nuclei. The three most common forms of this radiation and its characteristics are summarized in Table R.1.
Table R.1
Three common types of radioactivity.
Type
.
Composition
.
Typical speed
.
Comment and nuclear transition
.
α
Helium-4 nuclei, $ H 2 4 e 2 + $
0.1c
Not very penetrating, but damaging; A → A – 4, Z → Z – 2
β
Electrons, e^{−}
0.9c
Moderately penetrating; A → A, Z → Z + 1
γ
Short wavelength photon
c
Highly penetrating; Often accompanies other types; Nucleon rearrangement, A → A, Z → Z
Type
.
Composition
.
Typical speed
.
Comment and nuclear transition
.
α
Helium-4 nuclei, $ H 2 4 e 2 + $
0.1c
Not very penetrating, but damaging; A → A – 4, Z → Z – 2
β
Electrons, e^{−}
0.9c
Moderately penetrating; A → A, Z → Z + 1
γ
Short wavelength photon
c
Highly penetrating; Often accompanies other types; Nucleon rearrangement, A → A, Z → Z
The unit of activity, $A$ , of a radioactive source is the becquerel (Bq), which is defined as one disintegration per second. It has replaced the curie (Ci), with 1 Ci ≈ 37 GBq (3.7 × 10^{10} disintegrations per second), but the latter is still commonly encountered. The unit of absorbed dose is the gray (Gy), with 1 Gy = 1 J kg^{−1}. The now unofficial unit rad is also still encountered, with 1 rad = 10^{−2} Gy.
The law of radioactive decay asserts that the activity of a radionuclide decays exponentially, and therefore that
$A(t)=A(0) e \u2212 k r t $
The radioactive decay constant, k_{r} (the analogue of a first-order rate constant) is independent of the conditions; the half-life of radiactive decay, t_{1/2}, is related to it by
$ t 1 / 2 = ln 2 k r $
Radius of Gyration
Care must be taken to distinguish between two definitions of the radius of gyration, R_{g}, of a body, one in mechanics, the other in polymer science. The former is used to establish the rotational characteristics of a body; the latter is used to interpret light-scattering measurements.
In mechanics, the radius of gyration is the distance from an axis that a point with the same mass as the body should be placed to emulate the moment of inertia of the body around that axis (Figure R.2). If the body consists of a series of discrete mass elements m_{i} at distances $ ( x i 2 + y i 2 ) 1 / 2 $ from the axis of rotation, z, and its total mass is m, then
$ R g = ( 1 m \u2211 i m i ( x i 2 + y i 2 ) ) 1 / 2 $
If the body is a continuum of mass density ρ(r), then this expression becomes
$ R g = ( 1 m \u222b ( x 2 + y 2 ) \rho ( r ) d \tau ) 1 / 2 $
For a solid, uniform sphere of radius R
$ R g = ( 2 5 ) 1 / 2 R$
around any axis through its centre. For a long thin uniform rod of length L
$ R g = ( 1 12 ) 1 / 2 L$
around an axis through its centre and perpendicular to its length.
In polymer science, the radius of gyration of a polymer composed of N monomer units each of mass m_{i} at the distance r_{i} from its centre of mass (not the perpendicular axis to an axis of rotation) is defined as
$ R g = ( 1 m \u2211 i m i r i 2 ) 1 / 2 $
If all the monomers are identical, then
$ R g = ( 1 N \u2211 i r i 2 ) 1 / 2 = \u2329 R 2 \u232a 1 / 2 $
If the ‘polymer definition’ were used to calculate the radius of gyration of a uniform sphere of radius R, the outcome would be
$ R g = ( 3 5 ) 1 / 2 R$
Radius-ratio Rule
The radius-ratio rule is the expression of a correlation between the ratio of cation and anion radii, γ, where
$\gamma = r smaller r larger $
and the likely crystal structure of an ionic solid:
Deviations from the rule are sometimes taken to be indicative of the onset of covalence, but the arbitrary character of the values of ionic radio radii is a contribution to its unreliability.
Rainbow Scattering
As the impact parameter, b, in a molecular collision is decreased, the angle of scattering increases, passes through a maximum, and then decreases again (Figure R.3). There is strong constructive interference, referred to as rainbow scattering, between the neighbouring paths at the turning point, the rainbow angle, and hence a pronounced peak in the scattering.
The Raman effect is the inelastic scattering of electromagnetic radiation by molecules. A Raman spectrum is the record of the frequencies present in that scattered radiation. Radiation scattered with a lower frequency than the incident radiation is called Stokes radiation; that scattered with a higher frequency is called anti-Stokes radiation (Figure R.4). The effect arises because an incoming photon may either deposit energy or collect energy from the internal modes of motion of the molecule in the sample and hence emerge with a different frequency. In vibrational Raman spectra in the gas phase, vibrational transitions are accompanied by rotational transitions, which gives the vibrational spectrum a branch structure.
The gross selection rule for rotational Raman transition is that the polarizability of the molecule must be anisotropic. That for vibrational Raman spectra is the the polarization must vary with a normal mode distortion. In group theoretical terms, the normal mode must transform in the same way as a quadratic form (xy, etc). The exclusion rule states that a centrosymmetric molecule cannot be both Raman active and infrared active (it can be inactive in both).
The O-branch of a vibrational transition is due to ΔJ = −2, the Q-branch to ΔJ = 0, and the S-branch to ΔJ = +2.
Further information about the symmetry of the molecule is obtained from the depolarization ratio of the scattered radiation. In resonance Raman spectroscopy, an enhanced intensity is achieved by using incident radiation that has a frequency close to an electronic excitation of the molecule. In stimulated Raman spectroscopy the Stokes and anti-Stokes radiation in a forward direction is strong enough to undergo more scattering. See also CARS.
Random Walk
In a fully general random walk, each step is of random length and in a random direction. However, the term is normally restricted to a much simpler case in which all the steps have the same length but are in random directions. For a one-dimensional random walk along the x-axis, where each step may be in either direction, the probability of being at a distance xl from the origin after N steps each of length l is
$P(x)= ( 2 N \pi ) 1 / 2 e \u2212 x 2 / 2 N $
In three dimensions, the probability that the ends of the walk are a distance between r and r + dr apart after N steps is
$dP(r)=f(r)dr,f(r)=4\pi ( a \pi 1 / 2 ) 1 / 2 e \u2212 a 2 r 2 ,a= ( 3 2 N l 2 ) 1 / 2 $
A random coil of identical monomer units with contour length Nl is the physical realization of a random walk. However, the preceding expressions are relevant only if the coil is self-transparent in the sense that the coil may cross and retrace itself and the monomers may occupy the same region of space. In that case, the mean separation of the ends, 〈R〉, the root-mean-square-separation of the ends, R_{rms} = 〈R^{2}〉^{1/2}, and its (polymeric) radius of gyration, R_{g}, are as follows:
$\u2329R\u232a= ( 8 3 \pi ) 1 / 2 N 1 / 2 l R rms = N 1 / 2 l R g = ( N 6 ) 1 / 2 l$
If the monomers in the chain are constrained to lie at an angle θ between neighbours (Figure R.5), then these average values should be multiplied by a factor F, where
$F= ( 1 \u2212 cos \theta 1 + cos \theta ) 1 / 2 $
The rate constant, k_{r}, of a reaction is the constant of proportionality between the rate of the reaction, $v$ , and an expression involving the concentrations of the reactants and products, as in
$v= k r [A][B],v= k r1 [ A ] [ B ] 1 + k r2 [ C ] $
See reaction rate. The rate constants for forward (k_{r}) and reverse $( k \u2032 r )$ elementary reactions are related by the equilibrium constant, K, for that pair of reactions:
$K= k r k \u2032 r $
When the forward and reverse elementary reactions have different orders, care must be taken to ensure that K is dimensionless. Thus, for A + B ⇌ C,
$K= k r c \u29b5 k r \u2032 $
For a multistep reaction the overall equilibrium constant is the product of these factors for all the steps of the reaction mechanism:
$K= \u220f i k r , i c \u29b5 \Delta n i k r , i \u2032 $
In this expression Δn_{i} is the difference in orders, forward – reverse, for the elementary reaction i.
The temperature dependence of a rate constant is expressed by the Arrhenius equation in terms of the activation energy, E_{a}:
$ k r =A e \u2212 E a / R T $
For a first-order reaction, the rate constant is related to the half-life of the reactant and the time constant, τ, for the decay of its concentration by
$ k r = ln 2 t 1 / 2 , k r = 1 \tau $
Rate constants may be estimated on the basis of transition-state theory, collision theory, and molecular reaction dynamics. See also diffusion-controlled reactions, kinetic isotope effect, and kinetic salt effect.
Rate Determining Step
The rate determining step in a proposed reaction mechanism is the step with the smallest rate constant in a pathway that controls the rate of the overall reaction. Although it is commonly identified as the slowest step in the reaction, that criterion must be used with care, for all the forward steps take place at the same rate. It is the slowest step in the sense that, as a result of having a small rate constant, when taking place in isolation the reaction is typically slow.
Rate Law
A rate law is the empirically determined relation between the rate of a reaction, $v$ , and the concentrations of the reactants and products in the overall chemical equation. It is a differential equation for the rate of change of concentrations. In certain cases it may be integrated analytically to give the integrated rate law, an expression for the time dependence of the concentration of a reactant or product. Two examples are
$ First - order reaction : d [ A ] d t = \u2212 k r [ A ] , [ A ] ( t ) = [ A ] 0 e \u2212 k r t Second - order reaction : d [ A ] d t = \u2212 k r [ A ] 2 , [ A ] ( t ) = [ A ] 0 1 + k r t [ A ] 0 $
Note that these rate laws have been written in terms of d[J]/dt, and that they might be different if written in terms of $v= ( 1 / v J ) d [ J ] / d t $ . For an overall second-order reaction of stoichiometry A + B → P,
$ d [ P ] d t = k r [A][B],[P](t)= [ A ] 0 [ B ] 0 ( 1 \u2212 e ( [ B ] 0 \u2212 [ A ] 0 ) k r t ) [ A ] 0 \u2212 [ B ] 0 e ( [ B ] 0 \u2212 [ A ] 0 ) k r t $
This integrated rate law applies even if [A]_{0} = [B]_{0} provided the limit is taken as [B]_{0} → [A]_{0}.
The rate law of an overall reaction can be constructed from the rate laws of the sequence of unimolecular and bimolecular elementary reactions proposed for its mechanism. To do so, it is common to suppose either that there is a pre-equilibrium step or to invoke the steady-state approximation. It is now increasingly common for rate laws to be investigated by numerical integration on computers.
Reaction Coordinate
A reaction coordinate is a notional axis used to represents the course of a reaction (Figure R.6). If the reaction is elementary, then the reaction coordinate expresses the reactants on the left, an activated complex close to the middle, and the products on the right. If the reaction takes place in a series of steps, then intermediate points represent a series of reaction intermediates. In certain simple cases it is possible to give the reaction coordinate a precise analytical significance. For instance, if the step is a collinear reactive collision of three atoms, then the reaction coordinate can be identified with a path along the floor of the valley and across the saddle point of a potential energy surface.
A reaction intermediate is a species that occurs as in an elementary step in a reaction mechanism but not in the overall chemical equation for the reaction.
Reaction Quotient
A reaction quotient, Q, for a reaction written as
$0= \u2211 J \nu J s J ,0=v\u22c5s$
(with s_{J} the chemical formula of the reactant or product J) is defined as
$Q= \u220f J a J \nu J $
That is, it has the form
$Q= activities of products raised to their powers activities of reactants raised to their powers $
In elementary applications, the activities a_{J} are replaced by [J]/c^{⦵} or, for gases, p_{J}/p^{⦵}. Note that Q is dimensionless. The reaction quotient occurs in the relation between the Gibbs energy of reaction and its standard value:
$ \Delta r G= \Delta r G \u29b5 +RTlnQ$
The value of Q when the reaction mixture has its equilibrium composition is called the equilibrium constant, K, of the reaction.
Reaction Rate
The universal reaction rate, $v$ , is defined in terms of the stoichiometric numbers (the signed stoichiometric coefficients, positive for products, negative for reactants) that occur in the chemical equation for the reaction written as
$0= \u2211 J \nu J s J , 0 =v\u22c5s$
(with s_{J} the chemical formula of the reactant or product J) and the rate of change of molar concentration of a participant:
$v= 1 \nu J d [ J ] d t $
With this definition, there is a single reaction rate for the reaction regardless of the species J used in the expression. It is also common for the rate to be expressed in terms of the consumption or formation of specified participant, in which case the specific rate is
$ v J =\xb1 d [ J ] d t $
with the sign chosen so that the rate is positive. The specific rates for different choices of J may be different for the same reaction, depending on its stoichiometry.
Redox Reaction
A redox reaction is a chemical reaction in which the reactants undergo reduction and oxidation. The reaction involves the transfer of an electron from the reductant (the reducing agent) and its acceptance by the oxidant (the oxidizing agent). Thus, the reductant is oxidized and the oxidant is reduced. As electrons are so intimately involved in bonding, electron transfer is commonly accompanied by atom transfer. A redox reaction can be identified by noting a change in the oxidation numbers of the reactants. In aqueous solution, a reactant is capable of reducing an oxidant if its standard potential is more negative than that of the latter.
Reduced Mass
If two bodies are linked by a centrosymmetric force, then their relative motion can be represented by a point of reduced mass, μ, around their centre of mass (Figure R.7), with
$ 1 \mu = 1 m A + 1 m B ,\mu = m A m B m A + m B $
The term ‘reduced mass’ is sometimes used in other contexts (particularly the vibration of a diatomic molecule, where the same expressions apply), but in those contexts a more appropriate and more general term is effective mass.
A reduced variable, X_{r}, is a physical quantity expressed as a multiple of a related characteristic property of the substance. It is used in the discussion of the properties of gases, where the characteristic properties are taken to be the critical constants, p_{c}, V_{c}, and T_{c}, of the gas. Thus, the reduced pressure is p_{r} = p/p_{c}, the reduced molar volume is V_{r} = V_{m}/V_{c}, and the reduced temperature is T_{r} = T/T_{c}. When certain equations of state (those with two variable parameters, such as the a and b in the van der Waals equation) are expressed in terms of reduced variables, the parameters disappear and the equation becomes ‘universal’, as in
$ p r = 8 T r 3 V r \u2212 1 \u2212 3 V r 2 $
Reduction
Reduction is the addition of an electron to a species, as in the half-reaction
$Ox(state)+ e \u2212 \u2192Red(state)$
where Ox is the oxidized form of the species and Red the corresponding reduced form, the two jointly forming the Ox/Red couple. The electrons are regarded as stateless in this representation. Many reductions are also accompanied by the transfer of atoms. To identify reduction, note whether the oxidation number of an element in a reactant has decreased in the reaction. The potency of a reactant to bring about reduction in aqueous solution is measured by the standard potential of the Ox/Red couple: the more negative this potential, the more potent is the reactant as a reductant.
Reference State
The reference state of an element is its thermodynamically most stable state at the temperature of interest. The exception is phosphorus, where the white allotrope is used because it is more reproducible than its other allotropes. A reference state is used to define the enthalpy and Gibbs energy of formation of a compound, which refer to the formation of a compound from its elements in their reference states. At 298 K and 1 bar, the reference states of hydrogen and carbon are hydrogen gas and graphite, respectively.
Refinement
In X-ray crystallography, refinement means mathematical techniques used to improve the electron density obtained from an imperfectly phased Fourier synthesis.
Refractive Index
The refractive index, n_{r}, of a medium is defined as
$ n r = c c \u2032 $
where c′ is the speed of light in the medium and c its speed in a vacuum. The refractive index and the relative permittivity at the frequency of the radiation are related by
$ n r (\omega )= \epsilon r ( \omega ) 1 / 2 $
At optical frequencies, only the electronic contribution to the permittivity survives. Provided the frequency is below an absorption frequency, the refractive index typically increases with frequency; thus, blue light is refracted more than red light. See dispersion.
Relative Permittivity
The relative permittivity, ε_{r}, of a medium is the ratio of its permittivity, ε, to the electric constant (the vacuum permittivity), ε_{0}:
$ \epsilon r = \epsilon \epsilon 0 $
The relative permittivity is still often called the dielectric constant of the medium. For the relation between the relative permittivity and the polarizability and dipole moment of the molecules, see the Clausius–Mosotti equation. The practical importance of the relative permittivity is that the electric potential due to a point charge (and any derived quantities, such as a dipole interaction) is reduced by a factor of ε_{r} from its value in a vacuum:
$ \varphi vacuum (r)= Q 4 \pi \epsilon 0 r , \varphi medium (r)= Q 4 \pi \epsilon r = 1 \epsilon r \varphi vacuum (r)$
Relaxation
Relaxation denotes the return to equilibrium. The return is typically but not universally exponential (Figure R.8), with
$X(t)\u2212 X equilibrium \u221d e \u2212 t / \tau $
where τ is the relaxation time. In chemical kinetics, a reaction mixture in which the forward and reverse reactions are both first-order with rate constants k_{r} and $ k \u2032 r $ , the relaxation time is given by
$ 1 \tau = k r + k \u2032 r $
The rate constants are those for the currently prevailing conditions (such as at the new temperature in a temperature-jump experiment). If one of the reactions is second-order, the relaxation is exponential only for small departure from the new equilibrium.
In magnetic resonance, spin relaxation needs to take into account two aspects of equilibrium: the spin states have both a Boltzmann distribution of populations and random relative phases (Figure R.9). The longitudinal relaxation time (or spin–lattice relaxation time), T_{1}, is the time constant for the return of the population of spin states to equilibrium expressed in terms of the z-component of the magnetization, M_{z}:
$ M z (t)\u2212 M z , equilibrium \u221d e \u2212 t / T 1 $
The relaxation to random phases of the spins is equivalent to the return of the x-component of magnetization to zero. It occurs with the time constant T_{2}, the transverse relaxation time (or the spin–spin relaxation time)
$ M x (t)\u221d e \u2212 t / T 2 $
It is as a result of the transverse relaxation time that a resonance line acquires a linewidth, given by
A representation in group theory is a set of matrices that multiply in the same way as the elements of the group; that is, the relations exhibited by the two sets under multiplication are homomorphic. A single member of a representation is a representative of the element in question. The representation is reducible if a symmetry transformation of the basis can be found that simultaneously converts all the representatives to block diagonal form. See character. It is irreducible (an ‘irrep’) if it cannot be so reduced. Irreducible representations are labelled A or B if the matrices are one-dimensional, E if two-dimensional, and T if three-dimensional. Various subscripts are applied to these letters according to other features of the matrices.
Repulsive Interaction
The repulsive interaction between two closed-shell species, the rise in their potential energy as they come into contact, arises from the overlap of their wavefunctions and the demands of the Pauli exclusion principle. The latter ensures that the four electrons available from the two occupied orbitals enter both the bonding and antibonding combinations of the two orbitals and therefore result in a net increase in energy. The Lennard Jones (6,12)-potential energy represents the energy of interaction by a term proportional to 1/r^{12} but an exponential dependence on the internuclear distance r is probably more realistic.
Residual Entropy
The residual entropy of a substance is its nonzero molar entropy at T = 0. It is commonly identified by noting that the measured Third-law molar entropy (which is based on the entropy being zero at T = 0) is smaller than a calculated value. If there is residual orientational disorder, in which molecules can adopt s orientations with negligibly different energy (Figure R.10), then
$ S m (0)=Rlns$
The residual entropy of ice (3.4 J K^{−1} mol^{−1}) arises from the randomness of the distribution of hydrogen bonds between neighbouring molecules (that is, whether an OHO unit is O–H⋯O or O⋯H–O).
Resonance is the strong interaction between two systems with the same natural frequency, such as two pendulums of the same length hanging from a common, torsionally flexible axle. One important case is the resonance absorption of radiation when its frequency matches that corresponding to the excitation of a molecule (Figure R.11). All spectroscopy depends on resonance of this kind, but the term is commonly reserved for magnetic resonance (NMR and EPR), where the energy levels of the sample are modified (by the application of a magnetic field) until they match the frequency of a monochromatic electromagnetic field, rather than vice versa.
The term resonance is also used in valence bond theory to refer to the superposition of wavefunctions corresponding to different electron distributions in the same nuclear framework, as in ionic–covalent resonance:
$\psi = c cov \psi A \u2212 B + c ion \psi A + B \u2212 $
Resonance, in this context, results in a variety of effects, one being the lowering of the calculated energy. That lowering is greatest when the contributing structures have the same energy (hence the term ‘resonance’), as in the resonance between the two Kekulé structures of benzene. Resonance also distributes double bond character over a molecule and thereby gives a more accurate description of the molecule and consequently a better approximation to its actual energy. Figure R.12 shows how the greatest effect on the energies of states is observed when the two contributing ‘canonical’ structures have similar energies.
A reversible process is a process that can be reversed by an infinitesimal change in an externally controllable variable. Systems that are reversible in this thermodynamic sense are in dynamic equilibrium. Thus, a system is in mechanical equilibrium with its surroundings if its volume responds in opposite directions to opposite infinitesimal changes in the applied pressure. It is in thermal equilibrium if the flow of energy as heat responds in opposite directions to opposite infinitesimal changes in temperature.
Reversible changes do maximum work. At constant temperature, the maximum work that a system can do, w_{max}, is equal to the change in Helmholtz energy, A:
$ w max =\Delta A$
At constant temperature and pressure, the maximum nonexpansion work that a system can do is equal to the change in its Gibbs energy, G:
$ w max , nonexp =\Delta G$
The work done by the reversible, isothermal expansion of a perfect gas is
$w=\u2212nRTln V f V i $
(The negative sign indicates that energy leaves the system as work when it expands.) The pressure and volume of a perfect gas undergoing a reversible adiabatic change are related by
$p V \gamma =constant,\gamma = C p , m C V , m $
Rock-salt Structure
The rock-salt structure is illustrated in Figure R.13 in two equivalent space-filling representations of the unit cell and one image that shows the locations of the ions. It consists of two interpenetrating face-centred lattices of cations and anions. The coordination of the lattice is (6,6).
The classical angular momentum, J, and kinetic energy, E_{k}, of rotation of a solid body of moment of inertia I about an axis are
$J=I\omega , E k = 1 2 I \omega 2 $
where ω is its angular velocity (typically in radians per second). If the molecule is free to rotate around all three axes, its total kinetic energy is
$ E k = 1 2 \u2211 q I q q \omega q 2 = \u2211 q J q 2 2 I q ,q=x,y,z$
In quantum mechanics, the rotational state of a molecule is specified by three quantum numbers J, K, and M_{J} (Figure R.14):
J specifies the magnitude of the angular momentum of the molecule, with J = 0, 1, 2,….
K specifies the component of angular momentum on the principal axis of the molecule, with K = 0, ±1,…, ±J, except for a linear molecules, when K ≡ 0 only.
M_{J} specifies the component of angular momentum on a fixed external axis, with M_{J} = 0, ±1,…, ±J.
The energy depends on M_{J} only if an electric field is present and the molecule is polar. In the absence of an electric field, the energies of the rotors are as follows:
$ Linear rotor : E J , M J = h c B \u02dc J ( J + 1 ) , B \u02dc = \u210f 2 4 \pi c I Symmetric rotor : E J , K , M J = h c B \u02dc J ( J + 1 ) + h c ( A \u02dc \u2212 B \u02dc ) K 2 , A \u02dc = \u210f 2 4 \pi c I \u2225 , B \u02dc = \u210f 2 4 \pi c I \u22a5 Spherical rotor : E J , K , M J = h c B \u02dc J ( J + 1 ) , B \u02dc = \u210f 2 4 \pi c I $
The quantities Ã and B̃ are the rotational constants of the molecule (here expressed as wavenumbers). The energy levels of a linear rotor are shown in Figure R.15.
It follows from the preceding expressions for the energy that each level of a linear rotor is (2J + 1)-fold degenerate; for a symmetrical rotor each level other than the one with K = 0 are 2(2J + 1)-fold degenerate; and for spherical rotors each level is (2J + 1)^{2}-fold degenerate. The Pauli principle disallows some states: see nuclear statistics.
Rotational Spectroscopy
Rotational spectroscopy is the detection and analysis of the rotational transitions of molecules. Observations take place in the microwave region of the electromagnetic spectrum and are applicable only to polar molecules. The selection rules are as follows:
$\Delta J=\xb11,\Delta K=0,\Delta M J =0,\xb11$
Note that ΔJ = +1 corresponds to absorption and ΔJ = −1 corresponds to emission. It follows that the absorptions J + 1←J occur at the following wavenumbers:
$ \nu \u02dc =2 B \u02dc (J+1),J=0,1,2,\u2026$
The net intensity of the transition depends on the difference in populations of the two levels and the electric dipole transition moment:
where μ is the permanent electric dipole moment of the molecule. Therefore, the absorption spectrum consists of a series of lines separated by 2B̃ and the maximum intensity of absorption occurs close to
$ J max = ( k T 2 h c B \u02dc ) 1 / 2 \u2212 1 2 $
Once the rotational constant has been determined, it can be used to infer bond lengths and bond angles, although in some cases a single value needs to be augmented by data on isotopomers.
The electric dipole moment of a molecule can be determined by making use of the Stark effect, the splitting of the lines into components by the application of an electric field. The field removes the degeneracy of the states with different values of M_{J}^{2} and gives a splitting proportional to μ^{2}.
When the molecule cannot be treated as a rigid rotor and undergoes centrifugal distortion the rotational energy levels are closer than expected and the transitions lie at
$ \nu \u02dc =2 B \u02dc (J+1)\u22124 D \u02dc ( J + 1 ) 2 ,J=0,1,2,\u2026$
where D̃ is the centrifugal distortion constant (a wavenumber; don’t confuse it with the dissociation energy).
Rotational Temperature
The rotational temperature, θ^{R}, is the rotational constant of a molecule expressed as an equivalent temperature through
$ \theta R = h c B \u02dc k $
For a collection of rotors at thermal equilibrium at a temperature T, many rotational states are occupied if T ≫ θ^{R}, and for the calculation of thermodynamic properties the system can be regarded as a collection of classical rotors.
Rotor
A (rigid) rotor is a model of rotating molecule in which it is supposed that the bond lengths and bond angles remain the same whatever the rate of molecular rotation. Rotors are classified as follows:
An asymmetric rotor has three different moments of inertia; it has no axis of symmetry of order higher than 2.
A symmetric rotor has two equal moments of inertia and one different from the other two; it has an axis of symmetry of order 3 or more.
A spherical rotor has three equal moments of inertia; it belongs to the cubic or icosahedral point group.
A linear rotor has one moment of inertia equal to 0; the other two moments of inertia are equal.
For the energy levels of these rotors, see rotation and moment of inertia.
RRK Theory
In the Rice–Ramsperger–Kassel theory (RRK theory) of rate constants it is supposed that collision results in the formation of an energized molecule, A*, which then undergoes a redistribution of the excitation energy, E_{0}*, and becomes the activated complex, A^{‡}. It is supposed that the initial excitation energy is spread as n quanta of magnitude hν over s vibrational modes. In the activated complex, m quanta are localized in the vibrational mode corresponding to the reaction coordinate. It then follows that the rate constant for the overall reaction, k_{r}, is related to the rate constant k^{‡} for the unimolecular decay of A^{‡} by k_{2} = fk^{‡}, where
$f= ( n \u2212 m n ) s \u2212 1 $
In the more sophisticated Rice–Ramsperger–Kassel–Marcus theory (RRKM theory) the total energy of A* is classified as active or inactive; the inactive energy remains in the same quantum state throughout the reaction so does not participate in bond scission. The limiting high pressure, first-order rate constant obtained from the theory has the form
$ k r = k T h \xd7 q \u2260 q \xd7 e \u2212 E 0 \u204e / k T $
where q and q^{‡} (for typographical reasons, here q^{≠}) are the molecular partition functions for the initial and activated molecules. This expression is essentially the same as that obtained from transition-state theory.
Russell–Saunders Coupling
In Russell–Saunders coupling the orbital angular momenta are coupled into a resultant with quantum number L and the spin angular momenta are coupled into a resultant with quantum number S. Finally, these two resultant angular momenta are coupled into a total angular momentum with quantum number J (Figure R.16). At each stage the result is obtained by using the Clebsch–Gordan series. This coupling scheme is appropriate for light atoms, those with weak spin–orbit coupling. The outcome is represented by a term symbol of structure ^{2S+1}{L}_{J}, with {L} denoting S, P, etc.