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doi:10.1039/BK9781837674244-00228

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Concepts in Physical Chemistry

DOI:

https://doi.org/10.1039/9781837674244

Hardback ISBN:

978-1-83767-386-5

PDF ISBN:

978-1-83767-424-4

EPUB ISBN:

978-1-83767-425-1

No. of Pages:

394

Publication date:

17 May 2024

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

Oblate and Prolate

An oblate cylindrical body is disc shaped (I_‖ > I_⊥); a prolate cylindrical body is cigar shaped (I_‖ < I_⊥). See Figure O.1.

Figure O.1

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Oblate and prolate bodies.

Ohm’s Law

Ohm’s law states that the current, I, is proportional to the potential difference, Δϕ:

I = \frac{Δ ϕ}{R}

where R is the resistance of the conducting medium. With current in amperes (A) and potential difference in volts (V), the resistance is in ohms (Ω). The law is also commonly written V = IR, with V the potential difference.

Operator

An operator, $\hat{Ω}$ ⁠, is an instruction to modify (carry out an operation on) an entity. A mathematical operation is an operation on a function, such as differentiation or multiplication by x. The corresponding operators are d/dx and x × (the multiplication sign is commonly omitted). A symmetry operation is an operation on an entity (such as a rotation) that leaves it apparently unchanged.

A linear operator is a mathematical operator with the property that

\hat{Ω} (f + g) = \hat{Ω} f + \hat{Ω} g

Differentiation and multiplication are linear operators; exponentiation is not. An Hermitian operator is one for which

\int f^{⁎} \hat{Ω} g d τ = {(\int g^{⁎} \hat{Ω} f d τ)}^{⁎}

The eigenvalues of Hermitian operators are real. Eigenfunctions corresponding to different eigenvalues of an Hermitian operator are orthogonal. All observables in quantum mechanics correspond to linear, Hermitian operators.

The order in which operators are applied is in general significant, for it is not always the case that

\hat{Ω}

₁

\hat{Ω}

₂f =

\hat{Ω}

₂

\hat{Ω}

₁f. The commutator of two operators is defined as

[{\hat{Ω}}_{1}, {\hat{Ω}}_{2}] = {\hat{Ω}}_{1} {\hat{Ω}}_{2} - {\hat{Ω}}_{2} {\hat{Ω}}_{1}

Only operators that commute (that is, have a commutator equal to 0) have simultaneous eigenfunctions: see uncertainty principle. A fundamental proposition of quantum mechanics is that

[\hat{q}, {\hat{p}}_{q'}] = i ℏ δ_{q q'}, q, q' = x, y, z

Many conclusions can be inferred from the commutation properties of operators (such as almost the whole of the theory of angular momentum) while leaving the operators in an abstract form, as here. However, it is often appropriate to deal with a representation of the operators in actual mathematical form. Here are two representations of the fundamental commutation relation:

\begin{matrix} Position representation : \hat{q} = q \times, {\hat{p}}_{q} = \frac{ℏ}{i} \frac{\partial}{\partial q} \\ Momentum representation : {\hat{p}}_{q} = p_{q} \times, \hat{q} = - \frac{ℏ}{i} \frac{\partial}{\partial p_{q}} \end{matrix}

Operator representations of other dynamical variables may be built from these fundamental representations. See Hamiltonian.

Optical Activity

Optical activity is the rotation of the plane of polarization of light when it passes through a sample. The effect arises from the phase difference introduced as a result of circular birefringence, the different speeds of propagation of left- and right-circularly polarized radiation through the medium (Figure O.2). See birefringence. If the sample rotates the beam to the right as seen by the observer looking towards the oncoming beam, the sample is dextrorotatory and denoted (+). If the sample rotates the beams the left, the sample is laevorotatory and denoted (−). Optical activity is a property of chiral molecules, those lacking an S_n axis of improper rotation.

Figure O.2

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Circular birefringence and optical activity.

Orbital

An orbital is a one-electron wavefunction in an atom or molecule. See atomic orbital and molecular orbital.

Orbital Angular Momentum

In classical physics, orbital angular momentum, l , is defined as

l = r \times p

The three components are

l_{x} = y p_{z} - z p_{y}, l_{y} = z p_{x} - x p_{z}, l_{z} = x p_{y} - y p_{x}

and the square of the magnitude is

l^{2} = l_{x}^{2} + l_{y}^{2} + l_{z}^{2}

In quantum mechanics, these observables are treated as operators that obey the commutation relations

[{\hat{l}}_{x}, {\hat{l}}_{y}] = i ℏ {\hat{l}}_{z} and cyclic permutations; [{\hat{l}}^{2}, {\hat{l}}_{q}] = 0, q = x, y, z

It follows that only the magnitude and any one of the components may be specified. The eigenfunctions, subject to cyclic boundary conditions (and therefore for orbital motion but not spin), are the spherical harmonics,

Y_{l, m_{l}} (θ, ϕ)

⁠, with

{\hat{l}}^{2} Y_{l, m_{l}} (θ, ϕ) = l (l + 1) ℏ^{2} Y_{l, m_{l}} (θ, ϕ), {\hat{l}}_{z} Y_{l, m_{l}} (θ, ϕ) = m_{l} ℏ Y_{l, m_{l}} (θ, ϕ)

That is, the allowed values are:

\begin{array}{l} Magnitude : {l (l + 1)}^{1 / 2} ℏ, l = 0, 1, 2, \dots \\ z - Component : m_{l} ℏ, m_{l} = 0, \pm 1, \pm 2, \dots, \pm l \end{array}

In a hydrogenic atom, the value of l is constrained by another boundary condition and cannot exceed n − 1, where n is the principal quantum number. In general, angular momentum increases with the number of angular nodes in the wavefunction (the spherical harmonics).

The total orbital angular momentum of a collection of particles is specified by the quantum numbers L and M_L with their values found from the Clebsch–Gordan series.

Order of Reaction

If a rate law can be written in the form

v = k_{r} {[A]}^{a} {[B]}^{b} \dots

then it is classified as a-order in A, b-order in B, and so on, and its overall order is a + b +⋯. The order may be 0, integral, fractional, or negative. If all reactants and products except A are in great excess, they may be regarded as constant and the rate law is the pseudo-a order (commonly, when a = 1, pseudofirst-order) in A, with

v = k_{r, eff} {[A]}^{a}

with the constant concentrations of the other participants absorbed into the effective rate constant.

The order of a reaction is an empirical quantity, and only in special cases (elementary reactions) can it be inferred from the stoichiometry of the reaction. See first-order reactions and second-order reactions. Reactions with rate laws of the same order follow similar time evolutions (but at different rates) of the concentrations of the participants.

Orthogonality

Two wavefunctions ψ_a and ψ_b are orthogonal if

\int ψ_{a}^{⁎} ψ_{b} d τ = 0

where the integration is over all space (Figure O.3). It is a general feature of quantum mechanics that two eigenfunctions of an Hermitian operator are orthogonal if they correspond to different eigenvalues. In group theory, two functions are orthogonal if they span different irreducible representations of the relevant point group.

Figure O.3

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The orthogonality of two wavefunctions.

Oscillator Strength

The oscillator strength, f, of a transition i → f is related to the transition dipole moment μ_if and the frequency, ν, of the transition by

f = \frac{8 π^{2} m_{e} ν}{3 ℏ e} | μ_{fi} |^{2}

It is so defined that f = 1 for a transition in a three-dimensional harmonic oscillator, and therefore for an atom its value is an indication of how much the actual transition resembles such an oscillation. The oscillator strength is determined experimentally by measuring the integrated absorption coefficient,

A

⁠, for the transition and then using

f = \frac{4 m_{e} c ε_{0} \ln 10}{N_{A} e^{2}} A

Osmosis

Osmosis is the tendency of a solvent to flow into a more concentrated solution through a semipermeable membrane, a membrane that permits passage of the solvent molecules but not the solute. It is an example of a colligative property, one that depends on the number of solute molecules present but not their identity. The effect arises from the tendency of the solvent to equalize its chemical potential on each side of the membrane to compensate for the presence of the solute: the solute lowers the chemical potential of the solvent and a pressure, the osmotic pressure, Π, must be applied to the solution to restore it to the value characteristic of the pure solvent (Figure O.4). If the solution is ideal, then the osmotic pressure is given by the van ’t Hoff equation:

Π = R T [B]

where [B] is the molar concentration of the solute.

Figure O.4

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The equilibria involved in the calculation of osmotic pressure.

To allow for departures from ideality, this expression is extended to

Π = R T [B] (1 + B [B] + \dots)

where B is the second osmotic virial coefficient. The procedure of osmometry is the application of the measurement of osmotic pressure to the determination of molar mass, particularly of macromolecules. In this application (which has largely been replaced by mass spectrometry), the osmotic virial equation is expressed in terms of the mass concentration of the solute, c_B = [B]M, where M is the molar mass, and written

\frac{Π}{c_{B}} = \frac{R T}{M} + \frac{R T B}{M^{2}} c_{B}

Then a plot of Π/c_B against c_B should be a straight line with an intercept RT/M at c_B = 0.

A more formal approach to the osmotic pressure of nonideal solutions notes that

Π = - R T \ln a_{A}

where a_A is the activity of the solvent in the solution. Then the osmotic activity coefficient, ϕ, is introduced, where

ϕ = - r \ln a_{A}, r = \frac{x_{A}}{x_{B}}

Then

Π = R T [B] ϕ

Overlap Integral

The overlap integral, S, between two atomic orbitals on neighbouring atoms is defined as

S = \int ψ_{A}^{⁎} ψ_{B} d τ

where the integration is over all space. Overlap integrals are encountered in the theory of the chemical bond, where a significant overlap between atomic orbitals on different atoms is an indication of their contribution to bonding (and antibonding). Analytical expressions for the overlap of various types of orbitals are available. For hydrogenic orbitals centred on identical nuclei a distance R apart:

\begin{matrix} S (1 s, 1 s; R) = {1 + \frac{Z R}{a_{0}} + \frac{1}{3} {(\frac{Z R}{a_{0}})}^{2}} e^{- Z R / a_{0}} \\ S (2 s, 2 s; R) = {1 + \frac{1}{2} (\frac{Z R}{a_{0}}) + \frac{1}{12} {(\frac{Z R}{a_{0}})}^{2} + \frac{1}{240} {(\frac{Z R}{a_{0}})}^{4}} e^{- Z R / 2 a_{0}} \\ S (2 p, 2 p; σ orientation; R) = - {1 + \frac{1}{2} (\frac{Z R}{a_{0}}) + \frac{1}{20} {(\frac{Z R}{a_{0}})}^{2} - \frac{1}{60} {(\frac{Z R}{a_{0}})}^{3} - \frac{1}{240} {(\frac{Z R}{a_{0}})}^{4}} e^{- Z R / 2 a_{0}} \\ S (2 p, 2 p; π orientation; R) = {1 + \frac{1}{2} (\frac{Z R}{a_{0}}) + \frac{1}{10} {(\frac{Z R}{a_{0}})}^{2} + \frac{1}{120} {(\frac{Z R}{a_{0}})}^{3}} e^{- Z R / 2 a_{0}} \end{matrix}

These expressions are plotted in Figure O.5.

Figure O.5

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Overlap integrals for hydrogenic orbitals.

Overpotential

The overpotential, η, is the difference between the potential of the electrode and its zero-current potential. The current density, j, at the electrode varies with overpotential in a manner expressed by the Butler–Volmer equation. Processes such as gas evolution and metal deposition proceed at an appreciable rate only if the overpotential exceeds a critical value, which is typically about 0.1 V for many reactions but depends strongly on the identity of the electrode and the type of reaction occurring at it.

Overtone

An overtone in the infrared spectrum of a molecule is a vibrational transition with $Δ v > 1$ ⁠. The first overtone is the second harmonic. Overtones are observed only in the presence of anharmonicity.

Oxidation

Oxidation is the is the loss of electrons from a species, as in the half-reaction

Re d (s t a t e) \to O x^{+} (s t a t e) + e^{-}

Oxidation is accompanied, and can be recognized, by an increase in oxidation number of an element. Oxidation is often accompanied by atom transfer. It is the opposite of reduction.

Log in

O

Oblate and Prolate

Ohm’s Law

Operator

Optical Activity

Orbital

Orbital Angular Momentum

Order of Reaction

Orthogonality

Oscillator Strength

Osmosis

Overlap Integral

Overpotential

Overtone

Oxidation

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