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doi:10.1039/BK9781837674244-00129

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

g and u

The labels g (gerade, even) and u (ungerade, odd) denote the parity of a molecular orbital, its behaviour under inversion (Figure G.1). If the orbital does not change sign under inversion, it is denoted g; if it does change sign, it is denoted u. The classification is applicable only if the molecule has a centre of inversion (for example, a homonuclear diatomic molecule but not a heteronuclear diatomic molecule). The overall parity of a many-electron molecule is obtained by multiplying the parities of the wavefunctions of all the electrons and using the rules g × g = g, g × u = u, and u × u = g. The parity classification is used in the specification of certain selection rules.

Figure G.1

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The parity classification of orbitals.

g-Factor

The magnetic moment, μ , of a free electron is related to its spin angular momentum, s , by

μ = γ_{e} s, γ_{e} = - \frac{g_{e} e}{2 m_{e}}

where γ_e is the magnetogyric ratio of the electron and g_e is its g-factor. In terms of the Bohr magneton, μ_B,

μ = - g_{e} (μ_{B} / ℏ) s, μ_{B} = \frac{e ℏ}{2 m_{e}}

According to the Dirac equation (the relativistic version of the Schrödinger equation), g_e = 2. The experimental value is g_e = 2.002 319 304 352 56. (Note that some sources report g_e with a negative sign and define γ_e as g_ee/2m_e.) The difference from the integer 2 is due to the influence of zero-point fluctuations of the electromagnetic field (the ‘vacuum’). When the electron is part of a molecule its g-value is written simply g and differs slightly from the free-spin g-factor due to its interaction with the orbital angular momentum of its own and other electrons in the molecule. The nuclear g-factor, γ_N, is the analogous relation between the magnetic moment of a nucleus and its spin angular momentum, I :

μ = γ_{N} I, γ_{N} = \frac{g_{N} e}{2 m_{p}} = g_{N} μ_{N} / ℏ, μ_{N} = \frac{e ℏ}{2 m_{p}}

Here, μ_N is the nuclear magneton and m_p is the mass of a proton. The nuclear g-factor depends on largely unknown details of nuclear structure and is an empirical quantity.

γ

The heat capacity ratio γ (or adiabatic constant) is defined as

γ = \frac{C_{p, m}}{C_{V, m}}

From the equipartition theorem (disregarding vibrational contributions) and the relation C_p,m − C_V,m = R for a perfect gas, its values are as follows:

	Atoms	Linear molecules	Nonlinear molecules
γ	5/3	7/5	4/3

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The heat capacity ratio appears in the expression for the pressure−volume relation in a reversible adiabatic expansion:

p_{i} V_{i}^{γ} = p_{f} V_{f}^{γ}

It also appears in the expression for the speed of sound (which propagates as a sequence of adiabatic expansions and compressions) through a gas of molecules of molar mass M:

c_{s} = {(\frac{γ R T}{M})}^{1 / 2}

For other uses of the symbol γ see the discussion of the magnetogyric ratio in the entry on g-factor and the discussions of activities and surface tension.

Galvani Potential Difference

The Galvani potential difference, Δϕ, is the difference in electric potential between points in the metal electrode and the bulk solution.

Gas Constant

The (molar) gas constant, R, is defined as R = N_Ak, where k is Boltzmann’s constant and N_A is Avogadro’s constant, both of which currently have defined, exact values which imply that R ≈ 8.314 462 618 J K⁻¹ mol⁻¹. Its name arises from its original appearance in the perfect gas equation of state, pV = nRT. However, it appears in expressions unrelated to gases by virtue of its relation to the more fundamental Boltzmann’s constant.

Gas Laws

The gas laws are statements that summarize the observation of the response of gases to changes in the conditions and are now collected into the perfect gas equation of state, pV = nRT. The individual components of this expression are

Boyle ’ s law : p V = constant, p \propto 1 / V at constant T

Charles ’ law : p \propto T at constant V; V \propto T at constant p

Avogadro ’ s principle : p \propto n at constant V, T

Avogadro’s contribution is called a principle rather than a law because it is based on the molecular model rather than being a direct summary of experience.

Gaussian Function

A Gaussian function is a bell-shaped function of the form e^{−ax ²} as illustrated in Figure G.2. Gaussian functions are encountered as components of the wavefunctions of the harmonic oscillator, as the shapes of certain spectral lines, and in expressions related to diffusion.

Figure G.2

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The Gaussian function (the function is symmetrical on either side of x = 0).

Gibbs–Duhem Equation

The Gibbs–Duhem equation is a relation between changes that occur in the partial molar quantities X̄_J of the components of a mixture at constant temperature and pressure:

\sum_{J} n_{J} d {\bar{X}}_{J} = 0, n \cdot d \bar{X} = 0

It follows that for a binary mixture of A and B

d {\bar{X}}_{B} = - \frac{n_{A}}{n_{B}} d {\bar{X}}_{A}

A consequence of the Gibbs–Duhem equation is the Gibbs–Duhem–Margules equation for the relation between the changes to vapour pressures (more exactly the vapour fugacity, f_J) that occur when the composition of a binary mixture is varied:

{(\frac{\partial \ln f_{A}}{\partial \ln x_{A}})}_{T, p} = {(\frac{\partial \ln f_{B}}{\partial \ln x_{B}})}_{T, p}

Gibbs Energy

The Gibbs energy (which is also known as the Gibbs free energy or simply the free energy), G, is defined in terms of the enthalpy, entropy, and temperature of a system as

G = H - T S

It is a state function. Its importance stems from the fact that at constant temperature and pressure

Δ G = - T Δ S_{total}

and therefore that a spontaneous process (which is always accompanied by an increase in total entropy, the sum of changes in the system and the surroundings) under these conditions is accompanied by a decrease in Gibbs energy (a property of the system alone). A change in Gibbs energy at constant pressure and temperature is equal to the nonexpansion work that a system can do:

w_{nonexpansion} = Δ G

One important application of this relation is to the prediction of a cell potential, E_cell:

Δ G = - n F E_{cell}

In general, a change in Gibbs energy may arise from changes in temperature, pressure, and composition, with

d G = V d p - S d T + \sum_{J} μ_{J} d n_{J}, d G = V d p - S d T + μ \cdot d n

See fundamental equation. The Gibbs energy depends on pressure as

{(\frac{\partial G}{\partial p})}_{T} = V

For an incompressible phase G depends on measurable changes in pressure as

G (p) = G^{⦵} + (p - p^{⦵}) V

where

G^{⦵}

is the standard Gibbs energy, the value of G at the pressure

p^{⦵} = 1

bar. The corresponding expression for a (compressible) perfect gas is

G (p) = G^{⦵} + nRT \ln (p / p^{⦵})

See Figure G.3. For a real gas, replace the pressure by the fugacity. The temperature dependence is given by

{(\frac{\partial G}{\partial T})}_{p} = - S

This relation implies the Gibbs–Helmholtz equation:

{(\frac{\partial (G / T)}{\partial T})}_{p} = - \frac{H}{T^{2}}

Figure G.3

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The variation of Gibbs energy with the pressure of a perfect gas.

Gibbs Energy of Formation

The Gibbs energy of formation of J, Δ_fG(J). is the Gibbs energy of reaction for the formation of J from its elements in their reference states expressed as a quantity per mole of J. The reference state of an element is its thermodynamically most stable form at the specified temperature. (The exception to this general rule is phosphorus, for which the reference state is white phosphorus.) The standard Gibbs energy of formation, Δ_fG ^⦵(J), adds to this definition that all components of the formation reaction are in their standard state (pure, 1 bar) at the specified temperature. To apportion values between cations and anions in solution, the standard Gibbs energy of formation of hydrogen ions in aqueous solution is defined as zero for all temperatures: Δ_fG ^⦵(H⁺, aq)=0. Compounds for which Δ_fG ^⦵(J) > 0 are classified as endergonic and are unstable with respect to their elements (Figure G.4). Compounds for which Δ_fG ^⦵(J) < 0 are classified as exergonic and are stable with respect to their elements. The principal application of standard Gibbs energies of formation is to the calculation of the standard Gibbs energy of reaction by taking the difference between the stoichiometrically weighted values for the products and reactants (see that entry).

Figure G.4

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The thermodynamic classification of compounds.

Gibbs Energy of Reaction

The Gibbs energy of reaction, Δ_fG, is formally defined in terms of the extent of reaction:

Δ_{r} G = {(\frac{\partial G}{\partial ξ})}_{T, p}

Although formally a differential property (with dimensions of a molar energy) the fact that dG can be expressed in terms of chemical potential and changes in amount of all the substances involved in the reaction, dn_J = ν_Jdξ, means that it can be written as the stoichiometrically weighted (products – reactants) chemical potentials evaluated at the current composition of the reaction mixture:

Δ_{r} G = \sum_{J} ν_{J} μ_{J}, Δ_{r} G = ν \cdot μ

For the standard reaction Gibbs energy, Δ_fG ^⦵, all reactants and products are taken to be in their standard states (pure, 1 bar), and

Δ_{r} G^{⦵} = \sum_{J} ν_{J} Δ_{f} G^{⦵} (J), Δ_{r} G^{⦵} = ν {\cdot Δ}_{f} G^{⦵}

The relation between Δ_fG and Δ_fG ^⦵ is

Δ_{r} G = Δ_{r} G^{⦵} + R T \ln Q, Q = \prod_{J} a_{J}^{ν_{J}}

where Q is the reaction quotient. At equilibrium, Δ_rG = 0 and Q has the value known as the equilibrium constant, K, for the reaction (Figure G.5). It follows that

Δ_{r} G^{⦵} = - R T \ln K, K = {(\prod_{J} a_{J}^{ν_{J}})}_{equilibrium}

Figure G.5

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The variation of Gibbs energy in the course of a reaction.

Gibbs–Helmholtz Equation

The Gibbs–Helmholtz equation is

{(\frac{\partial (G / T)}{\partial T})}_{p} = - \frac{H}{T^{2}}

It also applies when a substance is in its standard state, in which case the partial derivative becomes a complete derivative because pressure is no longer a variable. It also applies to differences in standard Gibbs energies, therefore to the standard reaction Gibbs energy too, and through that to the equilibrium constant. In that case it becomes the van ’t Hoff equation (or van ’t Hoff isochore):

\frac{d \ln K}{d T} = \frac{{Δ_{r} H}^{⦵}}{{RT}^{2}}

Gibbs Surface Tension Equation

The Gibbs surface tension equation (or Gibbs isotherm) expresses the change in the surface tension, γ, of a liquid in terms of the changes in the chemical potentials of the components of a solution:

d γ = - \sum_{J} Γ_{J} d μ_{J}, d γ = - Γ \cdot d μ; Γ_{J} = \frac{n_{J} - (n_{J}^{α} + n_{J}^{β})}{σ}

In this expression, Γ_J is the surface excess, n_J is the total amount of J present in the system,

n_{J}^{α}

and

n_{J}^{β}

are the amounts of J in the two adjacent phases treated as uniform up to their interface, and σ is the area of the interface. If it is assumed that only a surfactant, S, accumulates in the interface, then the equation implies that

{(\frac{\partial γ}{\partial c})}_{T} = - \frac{R T Γ_{S}}{c}

where c is the molar concentration of S. This result implies that the surface tension decreases when a surfactant accumulates at the interface.

Glory Scattering

Glory scattering is the enhanced intensity of scattering in the forward direction; it arises when two paths interfere constructively (Figure G.6). One path is at large impact parameter and is undeflected by the target molecule. The other path is at low impact parameter and is deflected as it enters regions of attractive and repulsive regions of force but then continues in the forward direction.

Figure G.6

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Glory scattering.

Grotrian Diagram

A Grotrian diagram displays the energy of the states of an atom (or molecule) as a ladder of horizontal lines and depicts the observed transitions by lines connected the terms responsible for them. In some cases, the relative intensities of the transitions are depicted by the width of the lines.

Grotthuss Mechanism

In the Grotthuss mechanism of proton conduction in water, the migration of a hydrogen ion occurs by the coordinated adjustment of the locations of protons in a chain of neighbouring molecules rather than the actual motion of an identifiable proton through the liquid (Figure G.7). Its rate is determined by the ability of protons to tunnel through potential barriers and for the molecules to rotate in the chain.

Figure G.7

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The Grotthuss mechanism.

Ground State

The ground state of an atom or molecule is its state of lowest energy. It is a characteristic of quantum mechanics that the wavefunction of the ground state of a system has no nodes.

Group Theory

Group theory is the mathematical theory of symmetry. The h members of a set of elements g_i form a group of order h if they satisfy the following conditions:

The set includes the identity E, an element for which gE = Eg = g for all elements of the set.
The set includes the inverse g ⁻¹ for each member of the set, where gg ⁻¹ = g ⁻¹g = E.
The rule of combination is associative; that is, g_i(g_jg_k) = (g_ig_j)g_k.
All the elements of a set conform to the group property, that g_ig_j = g_k, a member of the set.

The link between this mathematical structure and symmetry is that the symmetry operations of an object fulfil the conditions for them to form a group. A point group consists of all symmetry operations that leave a single point unchanged; a space group extends that concept to include translational symmetry.

The theory is rendered quantitative in terms of numbers by representing the effect of symmetry operations by a set of matrices that obey the same multiplication properties as the symmetry operations. Each matrix is a representative of the corresponding symmetry operation and the entire set of matrices for a given basis is a matrix representation of the group. The representation is irreducible if a transformation of the basis cannot be found that simultaneously reduces all the matrices to block diagonal form. A matrix representation is characterized by the character of each representative, the sum of its diagonal elements. These characters are collected in character tables.

Group theory, particularly the information in character tables, is used to classify molecules according to the point group to which they belong, to establish selection rules, to classify orbitals, and to construct molecular orbitals from the appropriate symmetry-adapted combinations of atomic orbitals.

Log in

G

g and u

g-Factor

γ

Galvani Potential Difference

Gas Constant

Gas Laws

Gaussian Function

Gibbs–Duhem Equation

Gibbs Energy

Gibbs Energy of Formation

Gibbs Energy of Reaction

Gibbs–Helmholtz Equation

Gibbs Surface Tension Equation

Glory Scattering

Grotrian Diagram

Grotthuss Mechanism

Ground State

Group Theory

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