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doi:10.1039/BK9781837674244-00352

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

Walden’s Rule

Walden’s rule states that the product ηΛ_m, where η is the solvent viscosity and Λ_m the molar conductivity, is approximately constant for the same ions in different solvents.

Wavefunction

A wavefunction, ψ, is a mathematical function that contains all the dynamical information about a quantum mechanical system. It is obtained by solving the Schrödinger equation subject to the boundary conditions (and initial conditions if the system is evolving with time) characteristic of the system. It may also be regarded as an eigenfunction of the Hamiltonian operator for the system. A wavefunction is normalized (to 1) if

\int ψ^{⁎} ψ d τ = 1

where the integration is over all space. Wavefunctions are single-valued, continuous, not infinite over a finite region, and (except for certain potential energies) have continuous gradients. Many-particle wavefunctions are sometimes denoted Ψ( r ₁, r ₂,…) and are subject to the Pauli principle.

According to the Born interpretation, |ψ( r )|² (with ψ normalized) is the probability density of a particle being at r (Figure W.1). Other dynamical observables are extracted by applying the appropriate operator. If ψ is an eigenvalue of the operator

\hat{Ω}

corresponding to the observable Ω and the eigenvalue ω, then the outcome of a measurement of Ω will be ω. If it is not an eigenfunction, then the average value of a series of measurements of Ω will be the expectation value:

〈 Ω 〉 = {\int ψ}^{⁎} \hat{Ω} ψ d τ, ψ normalized

Figure W.1

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The Born interpretation of the wavefunction.

The ground-state wavefunction of a system has no nodes (locations where a function passes through zero). Regions of high curvature contribute high kinetic energy to the total energy (Figure W.2). Spatial wavefunctions may be complex or real: complex wavefunction correspond to a specific direction of travel; real wavefunctions do not. (See translational motion.) One-electron wavefunctions in atoms and molecules are known as orbitals.

Figure W.2

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Curvature and its contribution to the mean kinetic energy.

Wavelength

The wavelength, λ, of a harmonic wave, a wave of the form cos(2πx/λ − ωt), is its peak-to-peak distance. It is related to the frequency, ν = ω/2π, of the wave and its speed of propagation through the medium, c_medium, by

λ = \frac{c_{medium}}{ν}

For electromagnetic radiation in a vacuum, c_medium is the speed of light, c. Note that an electromagnetic wave retains its frequency when entering a medium but its wavelength changes on account of the change in its speed of propagation. See refractive index.

Wavenumber

The vacuum wavenumber,

\tilde{ν}

⁠, is the reciprocal of the vacuum wavelength:

\tilde{ν} = \frac{1}{λ} = \frac{ν}{c}

A wavenumber (which is commonly expressed in reciprocal centimetres, cm⁻¹) can be pictured as the number of wavelengths of the radiation per centimetre.

Work

In mechanics, work, w, is done when a body is moved against an opposing force, F. For an infinitesimal displacement, dx, the work done on the body is

d w = - F d x

More generally, if a body is moved along a path s and experiences a force F that might vary in strength and direction along the path, then the total work done on the body is

w = - \int_{Path} F \cdot d s

As a result of doing work, the potential energy of the body changes by w. For instance, to raise a body of mass m through a height h on the surface of the Earth, where the force of magnitude mg is directed downwards (F_z = −mg),

w = - \int_{0}^{h} (- m g) d z = + m g h

and the potential energy of the body increases by that amount.

In thermodynamics, work is a process equivalent to (in the sense that it can in principle be adapted to) achieving the raising of a weight in the surroundings. It is one of the ways in which energy may be transferred between a system and its surroundings and therefore bring about a change in the internal energy, U, of a system. The sign convention normally adopted in chemical thermodynamics (but not in engineering thermodynamics, where the opposite is commonly adopted), is that w < 0 if energy leaves the system as work.

Work is classified as expansion work or nonexpansion work. Expansion work is done when the system expands or contracts against an external pressure. If the external pressure acting on the system is p_ex, then when it undergoes a change of volume dV (which is positive for expansion and negative for compression, Figure W.3).

d w = - p_{ex} d V

Figure W.3

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Expansion work.

The expansion is reversible if p_ex = p, the pressure of the system, at all stages of the expansion. For the reversible, isothermal expansion of a perfect gas from V_i to V_f at a temperature T,

w = - n R T \ln \frac{V_{f}}{V_{i}}

Nonexpansion work (or additional work), work that does not involve expansion, includes driving an electric current through an external circuit. See Table W.1.

Table W.1

Varieties of work. ^a

Type of work	dw	Comments	Units ^b
Expansion	$- p_{ex} d V$	p_ex is the external pressure	Pa
Expansion	$- p_{ex} d V$	dV is the change in volume	$m^{3}$
Surface expansion	$γ d σ$	γ is the surface tension	$N m^{- 1}$
Surface expansion	$γ d σ$	dσ is the change in area	$m^{2}$
Extension	$f d l$	f is the tension	N
Extension	$f d l$	dl is the change in length	m
Electrical	$ϕ d Q$	ϕ is the electric potential	V
	$ϕ d Q$	dQ is the change in charge	C
	$Q d ϕ$	dϕ is the potential difference	V
	$Q d ϕ$	Q is the charge transferred	C

Type of work	dw	Comments	Units ^b
Expansion	$- p_{ex} d V$	p_ex is the external pressure	Pa
Expansion	$- p_{ex} d V$	dV is the change in volume	$m^{3}$
Surface expansion	$γ d σ$	γ is the surface tension	$N m^{- 1}$
Surface expansion	$γ d σ$	dσ is the change in area	$m^{2}$
Extension	$f d l$	f is the tension	N
Extension	$f d l$	dl is the change in length	m
Electrical	$ϕ d Q$	ϕ is the electric potential	V
	$ϕ d Q$	dQ is the change in charge	C
	$Q d ϕ$	dϕ is the potential difference	V
	$Q d ϕ$	Q is the charge transferred	C

In general, the work done on a system can be expressed in the form $d w = - | F | d z$ ⁠, where $| F |$ is the magnitude of a ‘generalized force’ and dz is a ‘generalized displacement’.

For work in joules (J). Note that 1 N m = 1 J and 1 V C = 1 J.

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The maximum work that can be done by a system at constant temperature is equal to the change in its Helmholtz energy, A:

w_{\max} = Δ A at constant T

The maximum nonexpansion work that a system can do at constant temperature and pressure is equal to the change in its Gibbs energy, G:

w_{nonexpansion, \max} = Δ G at constant T, p

In molecular terms, work is the transfer of energy that makes use of the uniform motion of atoms in the surroundings (Figure W.4); for instance, the uniform upward motion of the atoms of a weight.

Figure W.4

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The molecular interpretation of work.

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Walden’s Rule

Wavefunction

Wavelength

Wavenumber

Work

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