V

doi:10.1039/BK9781837674244-00342

Skip Nav Destination

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

Valence Bond Theory

Valence bond theory (VB theory) is a theory of the chemical bond that focuses on the linking of pairs of neighbouring atoms by the overlap of their orbitals and the pairing of spins of the electrons that occupy them. Thus, the bond between atoms A and B is described by the (unnormalized) wavefunction

ψ_{AB} (1, 2) = {ψ_{A} (1) ψ_{B} (2) + ψ_{A} (2) ψ_{B} (1)} {α (1) β (2) - α (2) β (1)}

The overall wavefunction of a polyatomic molecule is the product of such wavefunctions for all the bonds in the molecule as expressed by one of its canonical structures, a structure exhibiting the topological layout of bonds, such as one of the Kekulé structures of benzene. The basic theory is augmented by allowing resonance, which is the improvement of the description by the superposition of wavefunctions corresponding to different canonical structures, including ionic structures. The geometry of molecules is matched by allowing for the promotion of atoms and the hybridization of their atomic orbitals, and the resulting localized bonds are classified as σ and π. See promotion, hybrid orbitals, and resonance.

Valence Electron

A valence electron is an electron in the outermost occupied shell of an atom and is the focus of bond formation. It occupies one of the valence orbitals of an atom, an orbital in that shell.

van der Waals Equation

The van der Waals equation is a parametrized model equation of state of a real gas in which attractions and repulsions between the molecules are expressed by two parameters, a and b, respectively:

p = \frac{n R T}{V - n b} - a \frac{n^{2}}{V^{2}}

The reduced form of the equation is

p_{r} = \frac{8 T_{r}}{3 V_{r} - 1} - \frac{3}{V_{r}^{2}}; p_{r} = \frac{p}{p_{c}}, V_{r} = \frac{V_{m}}{V_{c}}, T_{r} = \frac{T}{T_{c}}

See Figure V.1. The critical constants are related to a and b by

p_{c} = \frac{a}{27 b^{2}}, V_{c} = 3 b, T_{c} = \frac{8 a}{27 b R}

Figure V.1

View large Download slide

van der Waals isotherms.

The reduced compression factor, Z_r, and the Boyle temperature, T_B, are

Z_{r} = \frac{3}{8}, T_{B} = \frac{a}{b R}

van der Waals Forces

van der Waals forces are the forces of attraction between closed-shell atoms and molecules, and specifically the interactions for which the potential energy is proportional to 1/R ⁶, where R is the internuclear separation. (The force itself is proportional to 1/R ⁷.) They are commonly classified as dipole–dipole, dipole–induced dipole, and induced dipole–induced dipole (London, dispersion) interactions. A van der Waals molecule is a loose cluster of closed-shell molecules or atoms held together by van der Waals forces.

van der Waals Loops

van der Waals loops are the unphysical oscillations of the isotherms predicted by the van der Waals equation of state below the critical temperature. They are replaced by straight lines by using the Maxwell construction.

van ’t Hoff Equation

There are two unrelated van ’t Hoff equations. One (which is also known as the van ’t Hoff isochore) relates the temperature dependence of the equilibrium constant to the standard reaction enthalpy:

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

It follows that the equilibrium constant increases with temperature if the reaction is endenthalpic

{(Δ}_{r} H^{⦵} > 0)

but decreases if it is exenthalpic

{(Δ}_{r} H^{⦵} < 0)

⁠. Provided the reaction enthalpy can be regarded as constant in the temperature range of interest, this equation implies that

\ln K (T_{2}) = \ln K (T_{1}) + \frac{Δ_{r} H^{⦵}}{R} (\frac{1}{T_{1}} - \frac{1}{T_{2}})

The kinetic explanation of this dependence notes that the activation energy of the reverse of an exenthalpic reaction is greater than that of the forward reaction (Figure V.2). Therefore, the rate of the reverse reaction increases with temperature more than the rate of the forward reaction increases. As a result, the equilibrium shifts towards reactants and K decreases.

Figure V.2

View large Download slide

The kinetic explanation of the shift in equilibrium with temperature.

The other van ’t Hoff equation relates the osmotic pressure, Π, to the molar concentration, [J], of a solution:

Π = R T [J]

Vapour Pressure

The vapour pressure, p, is the pressure of the vapour in equilibrium with a condensed phase of the substance. The partial vapour pressure, p_J, is the vapour pressure of J when other gases are present. The term sublimation vapour pressure is sometimes used when the condensed phase is a solid. The temperature dependence of the vapour pressure is given by the Clausius–Clapeyron equation (an approximate version of the exact Clapeyron equation):

\frac{d \ln p}{d T} \approx \frac{Δ_{vap} H}{R T^{2}}

The vapour pressure also varies with the applied pressure, ΔP:

p (Δ P) = p (0) e^{V_{m} Δ P / R T}

The vapour pressure above a curved surface differs from that when the surface is flat, the dependence on the radius r of a presumed spherical surface being given by the Kelvin equation:

p (r) = p (\infty) e^{2 γ V_{m} / r R T}

(p(∞) corresponds to a flat surface.) The vapour pressure of a liquid in an ideal solution is expressed by Raoult’s law:

p (x_{solvent}) = x_{solvent} p ⁎

where p* is the vapour pressure of the pure solvent. For a nonideal solution, the mole fraction x_solvent is replaced by the solvent activity, a_solvent:

p (x_{solvent}) = a_{solvent} p ⁎

The boiling temperature, T_b, is the temperature at which the vapour pressure of a liquid becomes equal to the ambient pressure. The standard boiling point is the boiling temperature when the ambient pressure is 1 bar. The normal boiling point (the ‘boiling point’) is the boiling temperature when the ambient pressure is 1 atm.

Variation Principle

The variation principle states that if the energy is calculated from a trial wavefunction by varying its parameters, then that energy is never less than the true ground-state energy, E_0,true:

E_{calculated} = \frac{\int ψ_{trial}^{⁎} \hat{H} ψ_{trial} d τ}{\int ψ_{trial}^{⁎} ψ_{trial} d τ}, E_{calculated} \geq E_{0, true}

The implication is that the best form of a trial wavefunction, the form that most closely matches the true wavefunction of the system, is the one with values of the parameters that result in the lowest energy (Figure V.3). When the trial wavefunction is written as a linear combination with the coefficients the variable parameters, then the optimum form, the best values of the coefficients, is obtained by solving the secular equations.

Figure V.3

View large Download slide

The implication of the variation principle.

Vector Model

The vector model of angular momentum is a pictorial depiction of the state with quantum numbers j, m_j in which a vector of length {j(j + 1)}^1/2 and z-component m_j lies at a stationary but indeterminate azimuthal angle on a cone around the z-axis (Figure V.4). In the presence of a magnetic field, the vector precesses on the cone at the Larmor frequency. The vector model can be elaborated to depict the coupling of angular momenta, when the contributing vectors are drawn with definite phases to acquire the correct resultant vectors: see singlet and triplet states.

Figure V.4

View large Download slide

The vector model for j = 1/2, 1, and 2, and m_j ≥ 0.

Vertical Transition

A vertical transition is an electronic transition that, in accord with the Franck–Condon principle, occurs without change of the nuclear coordinates and therefore is depicted by a vertical line on a molecular potential-energy diagram.

Vibrational Motion

Vibrational motion is a repetitive oscillatory motion that, in molecules, leaves the centre of mass and the orientation of the molecule unchanged. Vibration is commonly expressed in terms of harmonic motion, in which a particle of mass m is subjected to a Hooke’s law force, that the restoring force is proportional to the displacement (Figure V.5). The classical behaviour, with initial conditions x(0) and ẋ(0), is then described as follows, where k_f is the force constant:

m a = m \frac{d^{2} x}{d t^{2}} = - k_{f} x, x (t) = x (0) \cos ω t + \frac{\dot{x} (0)}{ω} \sin ω t, ω = {(\frac{k_{f}}{m})}^{1 / 2}

Figure V.5

View large Download slide

The restoring force and potential energy of an harmonic oscillator.

The frequency of a pendulum of length l making small-amplitude swings is

ω = {(\frac{g}{l})}^{1 / 2}, ν = \frac{1}{2 π} {(\frac{g}{l})}^{1 / 2}

The quantum mechanical description of harmonic motion uses a parabolic potential

V = \frac{1}{2} k_{f} x^{2}

in the Schrödinger equation and, with the appropriate boundary conditions, establishes that the allowed energy levels (Figure V.6) are

E_{v} = (v + \frac{1}{2}) ℏ ω; v = 0, 1, 2, \dots, ω = {(\frac{k_{f}}{m})}^{1 / 2}

Figure V.6

View large Download slide

The potential energy and allowed energy levels of an harmonic oscillator.

See harmonic oscillator. A nonlinear molecule with N atoms has 3N − 6 normal modes of vibration with a restoring force which is in general a function of stretching and bonding contributions and an effective mass which depends on the quantity of matter that moves in the course of the vibration. When the displacements are so great that the potential can no longer be regarded as parabolic, the motion is anharmonic. See Morse potential.

Vibrational Temperature

The vibrational temperature, θ ^V, of a harmonic oscillator is its characteristic vibrational frequency, ν, expressed as a temperature:

θ^{V} = \frac{h ν}{k}

Many vibrational states are occupied when T ≫ θ ^V and the vibrational partition function is then

q \approx \frac{T}{θ^{V}} = \frac{k T}{h ν}

Virial Theorem

The virial theorem states that if the potential energy has the form E_p= ax ^b, then the mean potential and kinetic energies are related by

2 〈 E_{k} 〉 = b 〈 E_{p} 〉

For a harmonic oscillator, b = 2, so 〈E_k〉 = 〈E_p〉; for a hydrogenic atom, b = −1, so $〈 E_{k} 〉 = - \frac{1}{2} 〈 E_{p} 〉$ ⁠, which implies that $E = \frac{1}{2} 〈 E_{p} 〉$ ⁠.

Virtual Orbital

A virtual orbital of an atom is an atomic orbital that is not occupied in the ground state of an atom but which is a member of the basis used to construct molecular orbitals as linear combinations of atomic orbitals.

Virtual Transition

When perturbation theory is used to model the distortion of a molecule by a perturbation, H ⁽¹⁾, it expresses the distortion by mixing into the ground state

ψ_{0}^{(0)}

wavefunctions corresponding to excited states of the unperturbed molecule

ψ_{n}^{(0)}

⁠:

ψ_{0} = ψ_{0}^{(0)} + ψ_{0}^{(1)}, ψ_{0}^{(1)} = \sum_{n \neq 0} c_{n} ψ_{n}^{(0)}, c_{n} = \frac{\int ψ_{n}^{(0) ⁎} H^{(1)} ψ_{0}^{(0)} d τ}{E_{n}^{(0)} - E_{0}^{(0)}}

It is sometimes said that the molecule has made a virtual transition to one of those excited states because if it is inspected there is a probability equal to |c_n|² that it would be found in the excited state $ψ_{n}^{(0)}$ ⁠.

Viscosity

Viscosity is a measure of a fluid’s resistance to flow. More formally, the coefficient of viscosity, η, is the constant of proportionality between the flux of momentum and the gradient of a perpendicular component of velocity:

J_{x} = - η \frac{d v_{z}}{d x}

The physical inspiration for this definition is that if the flow of the fluid is Newtonian (that is, can be regarded as a series of layers moving past each other at different rates), then the migration of a molecule from a slowly moving layer to a faster layer slows that layer and hence retards the flow of the liquid (Figure V.7). The SI units of η are kg m⁻¹ s⁻¹, but it is commonly reported in poise (P), with 1 P = 10⁻¹ kg m⁻¹ s⁻¹. See also diffusion.

Figure V.7

View large Download slide

The transfer of linear momentum between layers.

The kinetic theory of gases can be used to derive an approximate expression for the coefficient of viscosity. For a gas composed of molecules of molar mass M, mass m, mean free path λ, and collision cross-section σ:

η = \frac{1}{3} M λ v_{mean} [J] = \frac{m v_{mean}}{3 σ}

It follows that the viscosity of a perfect gas is independent of the pressure and increases as the temperature is increased. The independence of pressure arises because the increase of the number of molecules cancels a decrease in mean free path, so although there are more transporting molecules, they carry the momentum for a shorter distance. The increase with temperature (as T ^1/2) is due to the greater transport of momentum at higher temperatures.

The viscosity of a liquid decreases with increasing temperature, the temperature dependence being approximately

η = η_{0} e^{E_{a} / R T}

In this case, the migration of molecules is an activated process because a molecule must escape from the attraction of its neighbours.

The rheological (flow) properties of solutions of macromolecules are sometimes started in terms of their intrinsic viscosity, [η], where [η] is the coefficient in

η = η^{⁎} (1 + [η] c + \dots)

In this expression η* is the viscosity of the pure solvent and c is its mass concentration. Thus

[η] = \lim_{c \to 0} \frac{η / η^{⁎} - 1}{c}

One application of the intrinsic viscosity is to the estimation of the (viscosity-average) molar mass, M̅_v, by using the Mark–Houwink equation:

[η] = K {\bar{M}}_{v}^{a}

where K and a are empirical constants.

Log in

V

Valence Bond Theory

Valence Electron

van der Waals Equation

van der Waals Forces

van der Waals Loops

van ’t Hoff Equation

Vapour Pressure

Variation Principle

Vector Model

Vertical Transition

Vibrational Motion

Vibrational Temperature

Virial Theorem

Virtual Orbital

Virtual Transition

Viscosity

This Feature Is Available To Subscribers Only