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doi:10.1039/BK9781837674244-00339

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

Ultraviolet Catastrophe

The ultraviolet catastrophe is the prediction of classical physics that the energy density of black-body radiation becomes infinite at high frequencies or short wavelengths. This implication is expressed by the Rayleigh–Jeans law for the energy density, ρ(λ):

d E = ρ (λ) d λ, ρ (λ) = \frac{8 π k T}{λ^{4}}, ρ (λ) \to \infty as λ \to 0

The catastrophe was avoided by Planck’s introduction of quantization, which quenches the contribution of the high-frequency, short-wavelength oscillations of the electromagnetic field. See black-body radiation.

Uncertainty Principle

The (Heisenberg) uncertainty principle states that it is impossible to specify simultaneously with arbitrary precision both the linear momentum, p_q, and position, q, of a particle:

Δ q Δ p_{q} \geq \frac{1}{2} ℏ

The explanation of the principle in this form is based on the expression of the wavefunction of a localized particle as a superposition of waves, each one representing a state of linear momentum (Figure U.1). To achieve a tightly localized superposition, a wide range of wavelength is required, so the representation of a localized particle implies that its linear momentum is correspondingly indefinite.

Figure U.1

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The superposition of 2, 5, and 21 linear momentum states.

The more general form of the principle recognizes that two variables Ω₁ and Ω₂ are complementary, not have the same eigenfunctions, if the corresponding operators,

{\hat{Ω}}_{1} and {\hat{Ω}}_{2}

⁠, to not commute. Then

Δ Ω_{1} Δ Ω_{2} \geq \frac{1}{2} 〈 | [{\hat{Ω}}_{1}, {\hat{Ω}}_{2}] | 〉, Δ Ω = {〈 {\hat{Ω}}^{2} 〉 - {〈 \hat{Ω} 〉}^{2}}^{1 / 2}

The so-called energy–time uncertainty relation, ΔE ≈ ħ/τ, does not fit into this pattern as there is no operator for time in quantum mechanics; it is best regarded as a consequence of the Schrödinger equation and the fact that the energy of a state is ill-defined if the state has a finite lifetime.

Unit Cell

A unit cell is an imaginary parallelepiped (a parallel-sided figure) that replicates a crystal by translational displacements. It can be thought of as the fundamental unit from which the entire crystal can be constructed by purely translational displacements. A primitive unit cell is formed by drawing straight lines between neighbouring lattice points so that each cell has one lattice point in it at its origin. It is often more convenient to draw a nonprimitive unit cell that has a lattice points at its centre or on pairs of faces and chosen to have sides that have the shortest lengths and are most perpendicular to one another. See Bravais lattice. The lengths of the sides are denoted a, b, and c and the angles between them are denoted α, β, and γ (Figure U.2). The volume of such a cell is

V = a \cdot b \times c = a b c {(1 - \cos^{2} α - \cos^{2} β - \cos^{2} γ + 2 \cos α \cos β \cos γ)}^{1 / 2}

Figure U.2

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A generic unit cell.

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U

Ultraviolet Catastrophe

Uncertainty Principle

Unit Cell

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