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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 I; where appropriate, the entries also describe subsidiary but related concepts. Refer to the Directory for a full list of all the concepts treated.

An ideal solution is a solution in which the vapour pressures, pJ, of both the solute and the solvent obey Raoult’s law, that
where $p J ⁎$ is the vapour pressure of the pure component J and xJ is its mole fraction in the solution (Figure I.1). An alternative, equivalent definition (because it implies Raoult’s law) is that the chemical potential of J in the solution is given by
Figure I.1

The vapour pressure of an ideal solution.

Figure I.1

The vapour pressure of an ideal solution.

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In an ideal solution, the interactions between the components are the same as in the pure materials and the solute is distributed randomly through the solvent. That is, the enthalpy of mixing of the component is zero and the entropy and Gibbs energy of mixing are
See Figure I.2 and I.3.
Figure I.2

The entropy of mixing of an ideal solution.

Figure I.2

The entropy of mixing of an ideal solution.

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Figure I.3

The Gibbs energy of mixing of an ideal solution.

Figure I.3

The Gibbs energy of mixing of an ideal solution.

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A regular solution is one for which the entropy of mixing has its ideal value (so the solute is distributed randomly) but the enthalpy of mixing is nonzero; see excess functions. Note that in an ideal system the interactions are all the same but in a perfect system (that is, a perfect gas) they are not only the same but also zero. This distinction between ideal and perfect is not always made.

An idealdilute solution is one in which the solvent obeys Raoult’s law and the solute obeys Henry’s law, that
where Ksolute is an empirical constant obtained by extrapolating the low concentration data to xsolute = 1. Henry’s law is used as the basis of the definition of activity coefficient of the solute and for the estimation of the solubilities of gases (Figure I.4).
Figure I.4

Henry’s and Raoult’s laws.

Figure I.4

Henry’s and Raoult’s laws.

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The identity operation, E, is the act of doing nothing. It is required in the formal structure of group theory. The largest character of the identity in a character table for a point group is the maximum degeneracy possible for a system (and atom, ion, or molecule) that belongs to that point group.

The impact parameter, b, is the initial perpendicular distance between the line of approach of a projectile molecule and the target molecule (Figure I.5).

Figure I.5

The definition of impact parameter.

Figure I.5

The definition of impact parameter.

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An n-fold improper rotation axis (or axis of improper rotation), Sn, is a combination of an n-fold axis and a horizontal mirror plane: Sn = σhCn. Note that S1 is equivalent to a mirror plane and that S2 is equivalent to a centre of inversion. Molecules that do not possess an Sn axis are chiral (Figure I.6).

Figure I.6

Improper rotation axes with n = 4 and 6.

Figure I.6

Improper rotation axes with n = 4 and 6.

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Incongruent melting is the decomposition of a compound as it melts. The corresponding feature in the phase diagram is indicated in Figure I.7 by the ringed area.

Figure I.7

A feature indicating incongruent melting.

Figure I.7

A feature indicating incongruent melting.

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Indexing reflections is the procedure in which a reflection in X-ray crystallography is ascribed to planes with the Miller indices {hkl}.

An inexact differential (or incomplete differential) is one for which its integral depends on the path of integration. They are sometimes denoted đx. Examples of inexact differentials are dw and dq for work and heat, respectively. An inexact differential can be converted into an exact differential by multiplication by an integrating factor. An example is the conversion of the inexact differential dqrev into the exact differential dS by multiplication by 1/T.

Infrared spectroscopy (IR spectroscopy) is the observation of the absorption of infrared radiation by the vibrational modes of a molecule. A molecule consisting of N atoms has 3N – 6 normal modes of vibration (3N – 5 if it is linear), where a normal mode is a synchronous, independent vibrational displacement of its atoms that leaves its centre of mass and its orientation unchanged. A normal mode is infrared active in the sense that its transition dipole moment is nonzero, and therefore interacts with the electromagnetic field, if it is accompanied by a changing electric dipole moment. The frequency of the vibration depends on a combination of force constants for the displacement (with the force constants for angular displacements typically smaller than those for changes of bond length) and on the effective mass of the mode, a measure of the quantity of matter moved in the course of the vibration. The independence of the normal modes depends on their motion being harmonic (that is, in accord with Hooke’s law). The single normal vibrational mode of a diatomic molecule AB with atomic masses mA and mB, and its vibrational frequency is

Of the four normal modes of an ABA linear molecule, two are degenerate (in the sense that they have the same frequency. The allowed transitions, those for which $Δ v = 1$, are the first harmonics. When the motion is anharmonic (when the restoring force deviates from Hooke’s law), then overtones with $Δ v > 1$ (the higher harmonics) are allowed with low intensity. In the gas phase, vibrational transitions are accompanied by rotational transitions, giving rise to a branch structure of the vibrational spectrum.

The vibrational spectra of large molecules can be very complex. The fingerprint region of complex structure is characteristic of the substance, and can be used to recognize it by reference to a library of spectra.

Inhibition in biochemistry is the reduction in efficiency of an enzyme. Inhibitors are classified in terms of their binding properties, or equivalently in terms of their effect on the values of the maximum velocity, $v max$, and the Michaelis constant, KM, of the uninhibited enzyme. They can be distinguished experimentally by making a Lineweaver−Burk plot (Figure I.8).

Figure I.8

Lineweaver−Burk plots for a variety of inhibition schemes.

Figure I.8

Lineweaver−Burk plots for a variety of inhibition schemes.

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Competitive inhibition: the inhibitor binds to E but not to ES. Inhibitor binding is often but not always at the active site, and the formation of EI prevents binding of the substrate (Figure I.9).

Figure I.9

Competitive inhibition.

Figure I.9

Competitive inhibition.

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The y-intercept of a Lineweaver–Burk plot is unchanged, but the slope increases by a factor 1 + [I]/Kd,EI. Thus, $K M inhibited$ increases as [I] increases, but $v max$ is unaffected.

Uncompetitive inhibition: the inhibitor binds to ES but not E, thereby reducing the concentration of the catalytically active ES through the formation of ESI (Figure I.10).

Figure I.10

Uncompetitive inhibition.

Figure I.10

Uncompetitive inhibition.

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In this case Kd,EI → ∞ because EI does not form. The slope of the Lineweaver–Burk plot is unchanged, but the y-intercept of the plot increases by a factor of 1 + [I]/Kd,ESI. Thus, as [I] increases uncompetitive inhibition reduces $K M inhibited$ and $v max inhibited$ by the same amount.

Mixed inhibition: the inhibitor binds to both E and ES (Figure I.11).

Figure I.11

Mixed inhibition.

Figure I.11

Mixed inhibition.

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Both the slope and intercept of the Lineweaver–Burk plot increase upon addition of the inhibitor.

Non-competitive inhibition: is a special case of mixed inhibition in which the inhibitor has no effect on the binding of the substrate to the active site, and equally the substrate has no effect on the binding of the inhibitor.

Here Kd,ESI = Kd,EI because the dissociation equilibria ESI ⇌ ES + I and EI ⇌ E + I are unaffected by the presence of S in ESI. In this case, the lines for the uninhibited and inhibited enzymes in the Lineweaver–Burk plot intersect on the horizontal axis, showing that KM is unaffected by the inhibitor.

Inhomogeneous broadening is the line broadening in NMR arising from imperfections in the magnetic field across the sample. It is expressed in terms of the effective transverse relaxation time,
where Δν1/2 is the observed width of the resonance line at half height.

A property is classified as intensive if it is independent of the amount of substance in the sample. More precisely, if the sample is imagined to be divided into smaller samples, then a property is intensive if its value for the original sample is the same as the values for each of the smaller samples. Properties that are not intensive are extensive. Examples of intensive properties are temperature, pressure, mass density, and all molar quantities. In some cases (temperature and pressure, for instance) the properties are intrinsically independent of the size of the sample. In others (density and molar quantities, for instance), the property is a ratio of two extensive properties.

Interference is the interaction between the amplitudes of waves (including wavefunctions) that results in a wave with altered amplitude. In constructive interference the waves have the same phase in the region where they overlap and the total amplitude is larger than either component alone. In destructive interference the phases of the components are opposite and the amplitude of one wave subtracts from the other to result in a diminished amplitude.

Intermolecular forces are the forces of interaction of attraction and repulsion that act between closed shell atoms and molecules. Some of the interactions are universal, in the sense that (apart from their strength) they act regardless of the identity of the species. These universal interactions are electromagnetic in origin and most can be traced to the Coulombic interaction between permanent or transient charges. The distance dependence of their potential energies are as follows (where z is the charge number of an ion, μ is the dipole moment of a polar molecule, and α is the molecular polarizability):

1. Ion–ion interaction. The electrostatic interaction between charges; Vz1z2/r.

2. Ion–dipole interaction. The electrostatic interaction between the charge of an ion and the partial charges of a permanent electric dipole; Vz1μ2/r 2.

3. Dipole–dipole interaction. The electrostatic interaction between the partial charges of two polar molecules. Vμ1μ2/r 3 if the dipoles are locked in position; V ∝ (μ1μ2)2/Tr 6 if they are freely rotating at a temperature T.

4. Dipole–induced dipole interaction. In this interaction, the partial charges of a polar molecule induces a dipole moment in a neighbouring molecule (which might already be polar) and the initial and induced dipoles interact. V ∝ (μ1α2)2/r 6.

5. Induced-dipole–induced-dipole interaction (or London interaction, dispersion interaction). A fluctuation in the electron distribution in one molecule gives rise to a transient electric dipole, which induces an electric dipole in a neighbouring molecule, and the two transient dipoles interact. V ∝ (α1α2)2/r 6.

6. Repulsion. Once the two molecules come into contact and their wavefunctions start to overlap, they repel each other. This repulsion increases sharply with decreasing distance.

All the interactions may be present, depending on the charge and polarity of the molecules involved. The interactions that are proportional to 1/r 6 are classified as van der Waals interactions, although the term is often applied to all nonbonded interactions. Note that the force is proportional to dV/dr, so a potential energy proportional to 1/r n is associated with a force proportional to 1/r n+1. At large molecular separations, the potential energy of the dispersion interaction becomes proportional to 1/r 7. This weakening is due to the time it takes for information to pass between the molecules.

A specific interaction, one that does depend on the identity of the molecules, is hydrogen bonding; see that entry.

Internal conversion is the process by which a molecule in one electronic state converts non-radiatively into another electronic state of the same multiplicity (Figure I.12). The conversion occurs in accord with the Franck–Condon principle and takes place at the intersection of the two molecular potential energy curves.

Figure I.12

The process of internal conversion.

Figure I.12

The process of internal conversion.

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The internal energy, U, of a system is the total energy of the system (strictly, in the frame moving with the observer, disregarding its motion through space). Its thermodynamic definition is based on the First law and the statement that for a closed system
where dw is the (infinitesimal) energy supplied as work done on the system and dq is the (infinitesimal) energy supplied as heat to the system. This expression summarizes the observation that work and heat are equivalent ways of transferring energy between a system and its surroundings and the statement of the First law that the internal energy of an isolated system (dw = 0, dq = 0) is constant. A more subtle definition is that the difference in internal energy between two specific states is equal to the work needed to take the system from one state to another along an adiabatic path:
As work can be identified with a mechanical process (the raising of a weight), this definition links thermodynamics with mechanics.
The internal energy is a state function. For a closed system that does no other kind of work, a change in internal energy can be identified with the energy transferred as heat at constant volume:
The temperature dependence of the internal energy is summarized by the isochoric heat capacity:
A change in internal energy may also be expressed in terms of changes in volume and entropy through
from which various relations may be inferred, such as
This relation is a purely thermodynamic definition of temperature (and combines concepts from the Zeroth law, T; the First law, U; and the Second law, S).
The internal energy of a system relative to its value at T = 0 is related to the canonical partition function Q by
For a system composed of N independent molecules with partition function q,
Thus, the internal energy can be interpreted as the population-weighted average over the energies of the available states of a system.

Intersystem crossing (ISC) is the radiationless transition of a molecule from one electronic state into another with a different multiplicity (Figure I.13). An example is the singlet-to-triplet crossing that occurs as a step in the mechanism of phosphorescence.

Figure I.13

The process of intersystem crossing.

Figure I.13

The process of intersystem crossing.

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Intersystem crossing is brought about by spin−orbit coupling and entails the transfer of orbital angular momentum to spin angular momentum (the change from S = 0 to S = 1 in singlet-to-triplet ISC). The mechanism of this transfer is the magnetic field due to orbital motion of the electron acting on its own spin and thereby reversing its orientation relative to a second spin (↑↓ → ↑↑, Figure I.14). The opportunity for ISC, which conforms to the Franck–Condon principle, arises where the two molecular potential energy curves intersect. The intersection of curves effectively opens the door to the crossing and the spin–orbit coupling pushes the molecules through. As heavy atoms have strong spin–orbit couplings, ISC is most likely to occur in molecules that contain them.

Figure I.14

The vector model of intersystem crossing from a singlet to a triplet state.

Figure I.14

The vector model of intersystem crossing from a singlet to a triplet state.

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An ionic bond is an electrostatic attraction between oppositely charged ions. Except in the case of the formation of ion pairs in the gas phase, it is best regarded as a global property of an aggregation of ions rather than a localized interaction between one cation and one anion. The formation of an ionic bond involves transfer of an electron from one atom to another, so its formation is facilitated by a low ionization energy of the element that is to form the cation. Low ionization energies are associated with metallic elements, so ionic bonds are typically characteristic of a metal in combination with a nonmetal. There are major exceptions, such as the formation of compounds containing NH4 +. The energetics of formation of an ionic bond take into account the ionization energy of one element, the electron affinity of the other, and the global Coulombic interaction energy of the ions they form; the net result of the first two alone only rarely accounts for the lower energy of an ionic solid relative to the free atoms.

The (molar) lattice energy, Elattice,m, of an ionic compound, the difference in energy between the solid and the gas of infinitely separated ions, is inferred experimentally from a Born–Haber cycle and estimated from the BornMayer equation:
In this expression, zA and zB are the charge numbers of the ions, d is the sum of their ionic radii, d* is a constant typically taken to be 34.5 pm, which aims to take into account repulsive interactions, and A is the Madelung constant, a constant characteristic of the arrangement of ions in the solid (Table I.1).
Table I.1

Structural type A
Caesium chloride  1.763
Fluorite  2.519
Rock salt  1.748
Rutile  2.408
Sphalerite (zinc blende)  1.638
Wurtzite  1.641
Structural type A
Caesium chloride  1.763
Fluorite  2.519
Rock salt  1.748
Rutile  2.408
Sphalerite (zinc blende)  1.638
Wurtzite  1.641

The ionic radius is the notional radius of an ion in a solid. Its value depends on a largely arbitrary apportioning of the internuclear separation of the two ions, and a typical scale is based on ascribing the value 140 pm to the ionic radius of the O2− ion. Other scales are sometimes more appropriate, such as for halides. Ionic radii vary periodically, with values typically increasing down a group and decreasing from left to right across a period. The d-block elements show some subtle variations. Cations are smaller than their parent atoms and anions are larger. The lanthanide contraction is a marked decrease in atomic and ionic radius following the lanthanoids.

The (dimensionless) ionic strength, I, of a solution that contains irons of charge number zi at a molality bi is
where $b ⦵ =1molkg$ −1. Note that the sum extends over all the ions in the solution, including any spectator ions that may be present. The ionic strength apars in a number of expressions relating to the thermodynamic and kinetic properties of ions in solution, such as the Debye–Hückel limiting law and the kinetic salt effect. A characteristic of some properties of ionic solutions is that they approach a limiting value for I = 0 as I 1/2.
The ionization energy, I, of an element is the minimum energy required to remove an electron from one of its atoms in the gas phase:
Successive (and increasing) ionization energies are designated first, second, and third ionization energies. In some applications it is sometimes appropriate to consider the ionization energy from a specific orbital: see Koopmans’ theorem. First ionization energies increase from left right across a period and decrease down a group. The standard enthalpy of ionization, ΔionH , is related to the molar ionization energy by

An isenthalpic process is a process taking place at constant enthalpy. An example is the isothermal reversible expansion of a perfect gas (which is isenthalpic but endothermic). See Joule–Thomson effect.

An isobaric process is a process taking place at constant pressure. For an isobaric process in a closed (constant composition) system, ΔH = qp.

An isochoric process is a process taking place at constant volume. For an isochoric process in a closed (constant composition) system, ΔU = qV.

The isoelectric point of a macromolecule is the pH of a solution in which its net electric charge is zero.

The isolation method is a technique for examining the influence of an individual reactant on the rate of a chemical reaction and for determining the order of the reaction with respect to that reactant. In the procedure, each reactant except the one of interest is present in a large excess so that its concentration is effectively constant throughout the course of the reaction. The concentration of the remaining reactant is varied and the rate law for that component is inferred in the normal way.

Isomorphous replacement is a technique used in X-ray crystallography in which a modification to the diffraction pattern is obtained by the substitution of atoms of one element for those of another without significantly changing the structure of the crystal. The replacement can result in a greatly simplified diffraction pattern, especially if the replacement atoms dominate the diffraction, and so be a guide to the determination of the normal structure of the crystal.

An isopleth is a line of constant composition in a phase diagram (Figure I.15).

Figure I.15

Isopleths in a phase diagram.

Figure I.15

Isopleths in a phase diagram.

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An isosteric process is a process that occurs at constant surface coverage.

An isothermal process is a process that occurs at constant temperature. An isotherm is a line in a graphical representation of a process occurring at constant temperature; for example, the variation of the pressure of a gas with volume at constant temperature (Figure I.16).

Figure I.16

An isotherm and an adiabat of a perfect gas.

Figure I.16

An isotherm and an adiabat of a perfect gas.

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An adsorption isotherm is an expression for the variation of the extent of surface coverage with pressure. The work done by a gas expanding isothermally and reversibly is equal to the area below its isotherm and enclosed between the initial and final volumes (Figure I.17).

Figure I.17

The relation between reversible expansion work and area under an isotherm.

Figure I.17

The relation between reversible expansion work and area under an isotherm.

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