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doi:10.1039/BK9781837674244-00325

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

Temperature

The phenomenological definition of temperature, T, is that it is a parameter that governs the direction of flow of energy as heat through a diathermic boundary: energy as heat flows from high to low temperature. The thermodynamic definition of temperature originates in the Zeroth law of thermodynamics as the parameter that determines whether systems in contact through diathermic walls are in thermal equilibrium. The molecular definition of temperature is that it is the parameter in the Boltzmann distribution that determines the relative occupation of the available energy levels of any substance. There is a further definition in terms of a combination of the properties introduced by the laws of thermodynamics: if the internal energy, U (as introduced by the First law) and entropy, S (as introduced by the Second law), of a system are regarded as fundamental, being measures of the quantity and quality of stored energy, respectively, then they define temperature (as introduced by the Zeroth law) through the relation

{(\frac{\partial U}{\partial S})}_{V} = T

The formal definition of the thermodynamic temperature scale is in terms of the efficiency, η, of a heat engine working between a hot source at a known temperature, T_h, and a cold sink at the temperature, T, of interest, for then

T = (1 - η) T_{h}, η = \frac{work done}{heat absorbed}

The heat absorbed in a process can be assessed, in principle at least, by comparing the work done between two states with the work done in an adiabatic change between the same two states. Therefore, this definition of a temperature scale in principle makes use only of mechanical concepts (work). Setting T_h equal to the triple point temperature of water, 273.16 K, and noting that the lowest attainable temperature is T = 0, was the original definition of the Kelvin scale of temperature; that definition has been superseded by one that makes use of the defined values of fundamental constants. The Celsius scale of temperature, in which temperatures are denoted θ (so that T is reserved for thermodynamic temperature) is defined in terms of the Kelvin scale by the exact relation

θ / ° C = T / K - 273.15

Note that because Δθ/°C = ΔT/K, the size of the degree Celsius (°C) is the same as the size of the kelvin (K). The Fahrenheit scale is defined in terms of the Celsius scale and therefore the Kelvin scale by the exact relations

θ / ° F = \frac{9}{5} θ / ° C + 32 = \frac{9}{5} T / K - 459.67

The Rankine scale is an absolute scale starting at T = 0 in which the size of the degree Rankine (°R) is the same as the degree Fahrenheit (°F), and therefore

T / ° R = \frac{9}{5} T / K

Temperature Jump Technique

In the temperature jump technique, a reaction mixture is subjected to a sharp rise in temperature and its return to the new equilibrium composition is monitored. The relaxation time is interpreted in terms of the rate constants (at the new temperature) of the forward and reverse reactions.

Term

A spectroscopic term is the initial or final state (or group of states) between which a transition occurs; in somewhat outmoded language, two terms are said to ‘combine’ to account for a spectroscopic line. A term is specified by its term symbol and often the configuration that gives rise to the term. In atoms, a term consists of several levels labelled J split by spin−orbit coupling, with each level consisting of 2J + 1 states labelled M_J. For an atom, a term symbol has the form

{}^{Multiplicity \to 2 S + 1}{L}_{J \leftarrow Level}^{M_{J} \leftarrow State}

(The M_J label is commonly omitted.) Here, {L} is an upright uppercase letter used to denote the total orbital angular momentum:

{L}

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The notation continues G, H,…. The values of S, L, and J are obtained for the configuration by coupling the angular momenta in accord with the Russell–Saunders coupling scheme.

For linear molecules, a term has the form

{}^{Multiplicity \to 2 S + 1}{Λ}_{g / u \leftarrow Parity}^{\pm \leftarrow Reflection symmetry}

where {Λ} is an upright Greek letter representing the component of the orbital angular momentum around the internuclear axis:

±1

±2

±3

{Λ}

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The overall parity of the term (which is relevant only if the molecule has a centre of inversion) is evaluated by using the rules g × g = g, g × u = u, and u × u = g for all the electrons outside a closed shell. The reflection label (±) is used only for Σ terms and indicates the overall symmetry under reflection in a plane that includes the internuclear axis. For nonlinear polyatomic molecules, the term symbol has the form

{}^{Multiplicity \to 2 S + 1}Γ_{g / u \leftarrow Parity}

where Γ is the symmetry species of the overall orbital electronic state; once again, the parity designation is relevant only if the molecule has a centre of inversion.

Theoretical Plate

A theoretical plate is a tie line in a binary liquid/vapour phase diagram linking the liquid and vapour phase compositions that are in equilibrium at any given overall composition of the mixture. The term is used in connection with fractional distillation, with the number of theoretical plates for a succession of vaporizations and condensations between an initial composition and a final composition (Figure T.1). That number is a measure of the efficiency of the separation process.

Figure T.1

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Counting theoretical plates.

Thermal Conduction

Thermal conduction is the transport of energy as heat down a temperature gradient. The rate of transport is given by the following expression for the energy flux, J:

J = - κ {(\frac{\partial T}{\partial x})}_{n}

where κ is the coefficient of thermal conductivity and the constraint is for constant composition. The kinetic theory of gases leads to the following expression for the coefficient when the molar concentration of a gas is [J], its molar isochoric heat capacity is C_V,m, the mean free path is λ, and the mean speed of its molecules is

v_{mean}

⁠:

κ = \frac{1}{3} λ v_{mean} C_{V, m} [J]

Note that because λ ∝ 1/p and [J] ∝ p, κ is independent of pressure: although more molecules are present to transport energy at high pressure, the distance over which they can do so is decreased. Specifically, with λ = kT/σp and [J] = p/RT = p/N_AkT,

κ = \frac{v_{mean} C_{V, m}}{3 σ N_{A}}

where σ is the collision cross-section of the molecules.

Thermal Motion

Thermal motion is chaotic molecular motion; the higher the temperature, the more energetic is the thermal motion.

Thermal Wavelength

The thermal wavelength, Λ, of a particle of mass m at a temperature T is

Λ = \frac{h}{{(2 π m k T)}^{1 / 2}}

The translational partition function, q, of a perfect gas in a container of volume V is expressed in terms of the thermal wavelength by

q = \frac{V}{Λ^{3}}

Many translational states are occupied when V ≫ Λ ³.

Thermochemistry

Thermochemistry is the branch of thermodynamics that deals with the energy released or required as heat by a chemical reaction. See enthalpy of reaction and calorimeter.

Thermodynamic Force

The thermodynamic force,

F_{m}

⁠, is an effective force acting as though there is an influence to achieve the equalization of chemical potential throughout the sample, although in fact that is simply a consequence of the spontaneous tendency of matter to disperse. Formally, it is defined as

F_{m} = - {(\frac{\partial μ}{\partial x})}_{T, p}

where μ is the chemical potential of the solute. If the solution is ideal with molar concentration c(x), then the thermodynamic force reflects the tendency of the solute to diffuse, and

F_{m} = - \frac{R T}{c} {(\frac{\partial c}{\partial x})}_{T, p}

The units of thermodynamic force, a molar quantity, are newtons per mole (N mol⁻¹).

Thermodynamics

Thermodynamics is the branch of science concerned with the transformations of energy. It is expressed in terms of four laws. See the entry for each law.

Third Law of Thermodynamics

The phenomenological (observation-based) statement of the Third law is:

It is impossible to reach T = 0 in a finite number of cyclic steps.

The most succinct statement in terms of the entropy (a property introduced by the Second law), which implies the phenomenological statement, is:

The entropy of a pure, perfectly crystalline substance is zero at T = 0.

The experimental foundation of the law is the Nernst heat theorem. Its importance lies in the implication that it permits the identification of the thermodynamic and statistical definitions of entropy.

Tie Line

A tie line in a phase diagram is a straight line connecting two phases that are in equilibrium for a given overall composition of the system (Figure T.2). See lever rule.

Figure T.2

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A tie line.

torr (the unit)

The torr (symbol: Torr) is a non-SI unit used to report pressure and defined such that 760 Torr = 1 atm exactly. It follows that 1 Torr ≈ 133.32 Pa. See mmHg.

Transition Dipole Moment

The (electric) transition dipole moment for two states i and f, μ _fi, is defined as

μ_{fi} = \int ψ_{f}^{⁎} μ ψ_{i} d τ

It is a measure of the dipolar character of the shift in charge that occurs during the transition. The transition is allowed if the transition moment is nonzero and forbidden if it is zero. The conditions under which the transition moment is nonzero are expressed by the selection rules. The transition dipole moment is a vector with three components: the nonvanishing components express the polarization of the electric vector in the electromagnetic radiation that is absorbed or generated in the transition.

Translational Motion

In nonrelativistic classical mechanics translational motion, the motion of a particle of mass m through space, is described by its velocity,

v

⁠, its magnitude, its speed,

v

⁠, and its linear momentum, p :

v = \frac{d r}{d t}, v = | v |, p = m v

Two further relations are the kinetic energy, E_k, and the acceleration, a , the rate of change of velocity, in the presence of a force, F :

E_{k} = \frac{1}{2} m v^{2} = \frac{p^{2}}{2 m}, a = \frac{d v}{d t}, F = m a

In quantum mechanics the (unnormalized) wavefunction for free translational motion is

ψ_{k} (r) = e^{i k \cdot r}, E = \frac{k^{2} ℏ^{2}}{2 m}, p = k ℏ

All values of the wavevector k are allowed if the system is unbounded. If confining bounding walls are present, only certain energies are permitted as a result of the imposition of boundary conditions. Note that the motion is in the direction of k , and taking the complex conjugate of the wavefunction corresponds to reversing the direction of motion (Figure T.3):

ψ_{k} {(r)}^{⁎} = e^{- i k \cdot r}, E = \frac{k^{2} ℏ^{2}}{2 m}, p = - k ℏ

Figure T.3

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Representation of the wavefunctions for free translational motion.

Transport Number

The transport number, t_±, of an ion in solution (+ for cations, − for anions) is the fraction of the total electric current, I, it carries through the solution:

t_{\pm} = \frac{I_{\pm}}{I}

The limiting transport number,

t_{\pm}^{○}

⁠, is the value of t_± in the limit of the concentration becoming zero. The limiting transport number of a salt M_pX_q is related to the mobilities, u_±, of the ions of a fully dissociated salt by

t_{\pm}^{○} = \frac{z_{\pm} ν_{\pm} u_{\pm}}{z_{+} ν_{+} u_{+} + z_{-} ν_{-} u_{-}}, ν_{+} = p, ν_{-} = q

It follows that the limiting transport numbers are related to the individual ion conductivities, λ_±, by

ν_{\pm} λ_{\pm} = t_{\pm}^{○} Λ_{m}^{○}

For a symmetrical electrolyte, in which the charge numbers of the ions are the same,

t_{\pm}^{○} = \frac{u_{\pm}}{u_{+} + u_{-}}

Transport Property

A transport property is the movement of a physical property through a nonuniform system. The flux of the property, J, is commonly proportional to the gradient of a related property, with

J = - coefficient \times gradient

Four examples are

Property transported	Gradient	Name	Coefficient
Matter	concentration	diffusion	D
Energy as heat	temperature	thermal conduction	κ
Momentum	velocity	viscosity	η
Charge	electric potential	electrical conduction

Property transported	Gradient	Name	Coefficient
Matter	concentration	diffusion	D
Energy as heat	temperature	thermal conduction	κ
Momentum	velocity	viscosity	η
Charge	electric potential	electrical conduction

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The kinetic theory of gases can be used to derive expressions for the corresponding coefficients in the first three cases and the individual entries should be consulted.

Triangular Phase Diagram

A triangular phase diagram is used to depict the phase equilibria of ternary (three-component) systems at constant pressure and temperature. Its interpretation is illustrated in Figure T.4. Note that for any point in the interior of the triangle it follows from the geometry of an equilateral triangle that x_A + x_B + x_C = 1 and that all points on a straight line terminating at the corner A have B and C in the same mole ratio, and likewise for the other two corners.

Figure T.4

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The interpretation of a triangular phase diagram.

Triple Point

A triple point in a phase diagram is where three phases are in mutual equilibrium (Figure T.5). For a system with a single component (C = 1), it is an invariant point (F = C − P + 2= 0; that is, the pressure and temperature are unchangeable). One triple point of water, the one involving ice-I (T₃ = 273.16 K, p₃ = 611.657 Pa), was used to define the size of the kelvin, but it is now an experimentally determined point.

Figure T.5

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A triple point in a phase diagram.

Trouton’s Rule

Trouton’s rule states that for most liquids, Δ_vapH ^⦵/T_b is a constant and equal to about 85 J K⁻¹ mol⁻¹. The exceptions include liquids in which there is a significant amount of structural ordering (by hydrogen bonding, for instance). The explanation of the rule is that Δ_vapH ^⦵/T_b is the standard entropy of vaporization, and there is a similar change in the degree of disorder whenever an unstructured liquid forms a vapour.

Tunnelling

Tunnelling is a quantum mechanical phenomenon in which a particle is found in or passing through regions from which it is excluded according to classical mechanics (where its kinetic energy would, formally, be negative). The effect arises from the fact that a wavefunction does not fall abruptly to zero at the walls of a container unless the potential energy there is infinite, but decays in the region. If the potential energy falls back to a low but nonzero level after a finite distance, the wavefunction will have a nonzero amplitude and therefore a nonzero probability of being found beyond the barrier (Figure T.6).

Figure T.6

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The components of reflected and transmitted wavefunctions.

For a rectangular potential energy barrier of height V and width L the probability, P, that a particle of mass m will be found on the far side of the barrier when it is incident on it from the left with energy E is

\begin{array}{l} E < V : P & = & {1 + \frac{e^{κ L} - e^{- κ L}}{16 ε (1 - ε)}}^{- 1}, κ = {2 m V (1 - ε)}^{1 / 2} / ℏ, ε = E / V \\ E > V : P & = & {1 + \frac{\sin^{2} k L}{14 ε (1 - ε)}}^{- 1}, k = {2 m V (ε - 1)}^{1 / 2} / ℏ, ε = E / V \end{array}

See Figure T.7.

When the barrier is high and wide in the sense that κL ≫ 1, then for E < V,

P \approx 16 ε (1 - ε) e^{- 2 κ L}

Figure T.7

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The dependence of the transmission probability on the relative incident energy for two values of (2mV)^1/2/ħ.

In this case, the tunnelling probability decreases exponentially with the height of the barrier and with the mass of the particle. Tunnelling is very important for electrons, moderately important for protons (which accounts for the rapid equilibration of proton transfer reactions), less important for deuterons, and not important in general for heavier particles. See scanning tunnelling microscopy for one application.

Two-dimensional NMR

In two-dimensional NMR, the spectrum is displayed along using two axes to facilitate its interpretation. For example, in correlation spectroscopy (COSY) the pulse sequence 90°(x)[delay t₁]90°(x)[acquisition t₂] is applied, followed by a double Fourier transform with respect to the two time delays t₁ and t₂.

Two-level System

A two-level system, a system of only two states separated by energy ε, has simple but instructive thermodynamic properties which may be expressed analytically by means of the molecular partition function. For N such systems at a temperature T: