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doi:10.1039/BK9781837674244-00183

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

Lamb Formula

The Lamb formula is an expression for the diamagnetic contribution, σ_d, to the nuclear magnetic shielding constant of an atom:

σ_{d} = \frac{e^{2} μ_{0}}{3 m_{e}} \int_{0}^{\infty} r ρ (r) d r

where ρ(r) is the electron probability density at a distance r from the nucleus. For a hydrogenic atom of atomic number Z with an electron in a 1s orbital, the expression evaluates to

σ_{d} = \frac{Z e^{2} μ_{0}}{12 π m_{e} a_{0}}

where a₀ is the Bohr radius.

Lambda Transition

A lambda transition (a λ transition) is a phase transition that is not first-order according to the Ehrenfest classification but for which the heat capacity becomes infinite at the transition. It is so called because the shape of the graph of heat capacity against temperature resembles the Greek letter lambda, λ (Figure L.1). Examples include order–disorder transitions in alloys, the onset of ferromagnetism, and the fluid–superfluid transition of liquid helium.

Figure L.1

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The lambda transition.

Langevin Function

The Langevin function,

L (x)

⁠, is

L (x) = \frac{e^{x} + e^{- x}}{e^{x} - e^{- x}} - \frac{1}{x} = \frac{1}{3} x + \dots when x < < 1

See Figure L.2. The function appears in the expression for the mean electric dipole moment of a collection of polar molecules that are free to rotate in an electric field of strength

E

at a temperature T:

〈 μ 〉 = μ L (μ E / k T) \approx \frac{μ^{2} E}{3 k T} when μ^{2} E < < 3 k T

Figure L.2

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The Langevin function and its approximation.

The mean electric dipole moment contributes to the polarization of a medium.

Langmuir–Blodgett Film

A Langmuir–Blodgett film is a layer of molecules adsorbed on a solid substrate. Such a film can be formed by withdrawing a glass slide from a film-coated solution. The process may be repeated to give a bilayer. In certain cases, the second layer contributes to a bilayer that resembles a biological cell wall.

Laporte Selection Rule

The Laporte selection rule states that for atoms and for molecules with a centre of inversion, the only electric dipole transitions allowed are accompanied by a change of parity: g ⇄ u. The rule arises from the fact that the electric dipole moment has odd parity, so the transition dipole moment is zero except for states such that one is g and the other is u, for then g × u × u = g. The selection rule is broken in vibronic transitions, in which vibrations of the molecule remove the centre of inversion.

Larmor Frequency

The Larmor frequency, ν_L, of a nucleus in a magnetic field of flux density

B

ν_{L} = \frac{| γ_{N} | B}{2 π}

where γ_N is the magnetogyric ratio of the nucleus. At resonance, the applied radiofrequency field has the Larmor frequency. In the vector model of nuclear spin angular momentum, the vector representing the spin precesses at the Larmor frequency around the direction of the applied field (Figure L.3).

Figure L.3

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Precession.

Laser

The acronym laser stands for light amplification by the stimulated emission of radiation. The operation of a laser relies on achieving a population inversion, a greater number of molecules in an upper energy state than in some lower state, and then stimulating a radiative transition from the upper to the lower state (Figure L.4).

Figure L.4

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Transitions in a four-level laser.

The characteristics of laser radiation and their application in chemistry are:

Intense; and therefore especially useful for Raman spectroscopy.
Pulsed (in some cases); as the pulses may be of very short duration (down to about 1 as), phenomena can be monitored on a very short timescale.
Monochromatic; and therefore suitable for application in high-resolution spectroscopy.
Collimated; and therefore suitable both for high-resolution studies of the spatial distribution of molecules and also in Raman spectroscopy for the detection of forward-scattered radiation.
Coherent; see CARS.

Lattice Enthalpy

The (standard) lattice enthalpy, $Δ H_{L}^{⦵}$ ⁠, is the change in standard molar enthalpy when a solid converts to a vapour, as in the process MX(s) →M⁺(g) + X⁻(g) for an ionic solid or A(s) → A(g) for a molecular or covalent solid. It is typically inferred from a Born–Haber cycle; for ionic solids, it is estimated from the Born–Mayer equation. High lattice enthalpies are characteristic of solids with strong internal binding forces, such as ionic solids composed of small, highly charged ions. The term is often used as a synonym for lattice energy, but the two are identical only in the limit T → 0.

Le Chatelier’s Principle

Le Chatelier’s principle states that, a system at equilibrium responds to a change in conditions by tending to minimize their effect. For instance, when the temperature is raised, the composition of a chemical reaction at equilibrium tends to shift in the endothermic direction. When compressed, it tends to shift in the direction that reduces the number of gas-phase molecules. The thermodynamic basis of the effect of temperature is the variation of the equilibrium constant as expressed by the van ’t Hoff equation. The basis of the pressure-dependence is the independence of pressure of the equilibrium constant in association with the manner in which the thermodynamic equilibrium constant depends on the partial pressure of the participants in the reaction. That is, although dK/dp = 0, the numerator and denominator of K may both change but their ratio remains constant.

Lennard Jones Potential

The Lennard Jones potential is a model intermolecular potential energy that expresses the potential energy of two closed-shell molecules at a separation r in terms of two parameters, one (C₆) representing the effect of attraction and the other (C_n, n > 6) of repulsion:

V (r) = \frac{C_{n}}{r^{n}} - \frac{C_{6}}{r^{6}}

The higher power of the repulsive contribution represents its shorter range. The form of the attractive contribution is inspired by the character of van der Waals interactions, which typically vary as the sixth power of the separation. One common form of this expression is the Lennard Jones (6,12)-potential (Figure L.5), which is parametrized as follows:

V (r) = 4 ε {{(\frac{r_{0}}{r})}^{12} - {(\frac{r_{0}}{r})}^{6}}

Figure L.5

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The Lennard Jones (6,12)-potential.

The depth of the well is ε and is located at r = 2^1/6r₀.

Level

An energy level is an allowed energy of a quantum system; it might be degenerate in the sense that several individual states may possess the same energy. In atomic spectroscopy a level is specifically a group of states of an atom that have the same total angular momentum quantum number j (for a hydrogenic atom) or J (for a many-electron atom). Thus, a ²P term has two levels, with j = 1/2 and 3/2, denoted ²P_1/2 and ²P_3/2, with a difference in energy due to spin–orbit coupling (Figure L.6). The individual states of an atomic level are distinguished by the quantum number m_j or M_J, written as a right superscript, as in

{}^{2}P_{3 / 2}^{+ 1 / 2}

⁠. The energy due to spin–orbit coupling of a level with quantum numbers L, S, and J is

E_{L, S, J} = \frac{1}{2} h A {J (J + 1) - L (L + 1) - S (S + 1)}

where A is the spin–orbit coupling parameter.

Figure L.6

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The two levels of a ²P term.

Lever Rule

The lever rule (Figure L.7) expresses the relative abundances n_α and n_β of two components in terms of the distances along a tie line, l_α and l_β, from the coexistence curves in a phase diagram as

n_{α} l_{α} = n_{β} l_{β}

Figure L.7

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The lever rule.

Lifetime Broadening

Lifetime broadening is the broadening of a spectral line due to the finite lifetime of the excited state. Solution of the time-dependent Schrödinger equation shows that if the wavefunction decays as

e^{- t / 2 τ}

(so its square modulus decays with time constant τ), then its energy lies within a range of width

δ E \approx \frac{ℏ}{τ}

Any transition that involves this state therefore has an intrinsic width of this magnitude. In terms of wavenumbers:

δ \tilde{ν} / {cm}^{- 1} \approx \frac{5.3}{τ / ps}

Any process, such as collision, that shortens the lifetime of the state contributes to this broadening. Every excited state has a finite lifetime even in the absence of collisions on account of spontaneous emission, and the resulting intrinsic spectral linewidth is called the natural linewidth of the transition involving it. Lifetime broadening is often attributed to the uncertainty principle and known as uncertainty broadening, but there are technical reasons why the name and attribution are inappropriate (see uncertainty principle).

Ligand Field Theory

Ligand field theory is an adaptation of molecular orbital theory used to describe the structure of d-metal complexes and to correlate their structural, spectroscopic, and magnetic properties. The molecular orbitals of the complex are constructed from the d orbitals of the central metal atom and symmetry-adapted linear combinations of orbitals supplied by the ligands (Figure L.8). It is a more sophisticated version of crystal field theory in which the ligands are modelled as point negative charges. A focus of ligand field theory is the separation of the frontier orbitals of the complex, which in an octahedral complex are the e_g and t_2g combinations and their energy difference, which is called the ligand filed splitting and denoted Δ_O. In a tetrahedral complex, the frontier orbitals are e and t₂, and the splitting is denoted Δ_T. A d^N complex contributes N electrons to the frontier orbitals.

Figure L.8

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The molecular orbitals of an octahedral complex.

The resulting configuration depends on a competition between the size of Δ_O and the cost in energy of doubly occupying a single orbital. When Δ_O is large, the lowest energy is achieved by pairing the spins and letting them all occupy the lower t_2g set of orbitals; this arrangement results in a low-spin complex (Figure L.9). When Δ_O is small, the lowest energy is obtained by distributing the electrons over the orbitals and allowing as many if them as possible to have parallel spins; this arrangement results in a high-spin complex.

Figure L.9

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Low- and high-spin d⁵ configurations.

Limiting Law

A limiting law is a scientific law that is applicable with increasing reliability as one variable of the system tends to zero. Examples of limiting laws are the Debye–Hückel limiting law (which achieves reliability as the ionic strength I → 0), the application of the perfect gas law to real gases as p → 0, and Henry’s law for the vapour pressure of solutes as c → 0. A limiting property is the limit of a property as the concentration approaches zero, as in the limiting molar conductivity and the limiting enthalpy of solution, the limits of these properties as the concentration approaches zero. Limiting values avoid the complication of solute−solute interactions.

Linear Combination of Atomic Orbitals

The linear combination of atomic orbitals (LCAO) method is a procedure for constructing molecular orbitals by expressing them as a weighted sum of atomic orbitals, χ_n, of the appropriate symmetry:

ψ = \sum_{n} c_{n} χ_{n}

The coefficients c_n are found by applying the variation principle. The probability that an electron described by this molecular orbital will be found in the orbital χ_n, is proportional to

c_{n}^{2}

⁠. The atomic orbitals that appear in the sum constitute the basis set.

Linear Momentum

In nonrelativistic classical mechanics the linear momentum, p, is the product of the mass, m, of the body and its velocity,

v

⁠:

p = m v

Its magnitude, p, is the product of mass and speed,

p = m v

⁠. In quantum mechanics linear momentum is represented by an operator, which in the position representation is

{\hat{p}}_{q} = \frac{ℏ}{i} \frac{\partial}{\partial q}, q = x, y, z

The (unnormalized) wavefunctions for linear momentum p = kħ to positive x (→) and negative x (←) are

ψ_{\to} (x) = e^{i k x}, ψ_{\leftarrow} (x) = e^{- i k x}

and correspond to a wave of wavelength λ = h/p, in accord with the de Broglie relation. Note that the wavefunctions for the opposite directions of travel are related by forming the complex conjugate:

ψ_{\to} (x) = ψ_{\leftarrow} (x) ⁎

⁠. Real wavefunctions therefore denote stationary particles. Complex wavefunctions can be represented by plotting the real and imaginary components on perpendicular axes (Figure L.10), in which case the handedness of the resulting helix represents the direction of travel, with a right-hand screw corresponding to motion to the right and a left-hand screw corresponding to motion to the left.

Figure L.10

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Complex wavefunctions and direction of travel.

Linear Rotor

A linear rotor is a body with one moment of inertia that is zero and two equal moments of inertia perpendicular to the axis of the body. Rigid linear molecules are linear rotors. Their rotational energy levels are specified by the quantum numbers J and M_J and are

E_{J, M_{J}} = h B J (J + 1); B = \frac{ℏ}{4 π I}, J = 0, 1, 2, \dots, M_{J} = 0, \pm 1, \dots, \pm J

See Figure L.11.

Figure L.11

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The energy levels of a linear rotor.

The partition function of a linear rotor at high temperature, when many rotational states are occupied, is

q = \frac{k T}{σ h B}

where σ is the symmetry number of the molecule. The mean rotational energy is kT and the rotational contribution to the molar isochoric (constant-volume) heat capacity is R.

Linear Superposition

A linear superposition, F, of functions f_n is

F = \sum_{n} c_{n} f_{n}, F = c \cdot f

A fundamental aspect of the structure of of quantum mechanics is that if several outcomes of an observation are possible, then the wavefunction of the system ψ is a linear superposition of the eigenfunctions of the operator corresponding to the observable:

ψ_{superposition} = \sum_{n} c_{n} ψ_{n}, ψ_{superposition} = c \cdot ψ

The probability that a single observation of the property Ω, represented by its operator

\hat{Ω}

⁠, will give the eigenvalue ω_n of

\hat{Ω}

corresponding to the eigenfunction ψ_n is |c_n|². The formation of a linear superposition of the wavefunctions corresponding to possible outcomes represents a central difference between classical and quantum mechanics. In classical mechanics, the probabilities of outcomes are added; in quantum mechanics the amplitudes of the outcomes, the wavefunctions, are added and then the probability of a specific outcome is calculated by evaluating the square of the amplitude. The latter procedure admits the possibility of interference between outcomes, such as the interference between the paths of a particle as it travels between two points (Figure L.12).

Figure L.12

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Classical (left) and quantum mechanical (right) composite processes.

Liquid Crystal

A liquid crystal is a mesophase that shows structural characteristics of liquids in some directions and of solids in others. The three classes of liquid crystal are cholesteric, nematic, and smectic. See those entries.

Liquid Junction Potential

A liquid junction potential, E_LJ, is the contribution to the potential of a galvanic cell that arises from the presence of an interface between two electrolyte solutions. It can be at least partially eliminated by joining the two solutions with a salt bridge, a suspension of ions in a gel.

Lorentzian Line

A Lorentzian line is a spectroscopic line that has a shape given by the following expression (Figure L.13):

S (ω) = \frac{1}{1 + {(ω - ω_{0})}^{2} T_{2}^{2}}

Figure L.13

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Lorentzian lineshapes.

The centre of the line is at ω₀ and its half width at half height is 1/T₂. In magnetic resonance, the parameter T₂ is called the transverse relaxation time. The Fourier transform of the line is

I (t) = \frac{π^{1 / 2}}{2^{3 / 2} T_{2}} \cos (ω_{0} t) e^{- t / T_{2}}

Lyman Series

The Lyman series consists of lines in the spectrum of atomic hydrogen arising from the transitions n₂ → n₁ = 1 and therefore with wavenumbers

\tilde{ν} = {\tilde{R}}_{H} (1 - \frac{1}{n_{2}^{2}}), n_{2} = 2, 3, \dots

where R̃_H is the Rydberg constant for hydrogen. All the lines lie in the ultraviolet region of the spectrum and terminate at the wavenumber corresponding to the ionization of the ground-state atom.

Log in

L

Lamb Formula

Lambda Transition

Langevin Function

Langmuir–Blodgett Film

Laporte Selection Rule

Larmor Frequency

Laser

Lattice Enthalpy

Le Chatelier’s Principle

Lennard Jones Potential

Level

Lever Rule

Lifetime Broadening

Ligand Field Theory

Limiting Law

Linear Combination of Atomic Orbitals

Linear Momentum

Linear Rotor

Linear Superposition

Liquid Crystal

Liquid Junction Potential

Lorentzian Line

Lyman Series

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