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doi:10.1039/BK9781837674244-00196

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

Magic Angle Spinning

Magic angle spinning (MAS) in solid-state NMR makes use of the fact that the energy of the interactions between parallel magnetic dipole and also the anisotropies in chemical shifts are proportional to 1−3 cos² θ, where θ is the angle that the magnetic moments make to the vector that joins them. The ‘magic angle’ is the angle at which 1− 3 cos² θ = 0, which is 54.74°. In the technique, the sample is spun at high speed at this angle to the applied field and the anisotropies average to zero.

Magnetic Moment

The magnetic moment, μ, of a body is the magnitude of its magnetic dipole moment, μ . An electron has an intrinsic magnetic moment due to its spin, s :

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

where γ_e is the magnetogyric ratio (or gyromagnetic ratio) of the electron, μ_B is the Bohr magneton, and g_e is the g-factor of the electron. Their values are as follows:

γ_{e} \approx 1.760 859 630 23 \times 10^{11} s^{- 1} T^{- 1}

μ_{B} \approx 9.274 010 0783 \times 10^{- 24} J T^{- 1}

g_{e} \approx 2.002 319 304 362 56

An electron may also possess a magnetic moment by virtue of its orbital angular momentum, l :

μ = - (μ_{B} / ℏ) l

A consequence of the spin magnetic moment being anomalously large (due to the presence of g_e ≈ 2) is that the magnetic moment of an atom in a state with total angular momentum quantum number J is not related simply to the total orbital angular momentum quantum number L and the total spin angular momentum S. Instead (with g_e approximated by the integer 2),

μ = - g_{J} (L, S) (μ_{B} / ℏ) J, g_{J} (L, S) = \frac{J (J + 1) + S (S + 1) - L (L + 1)}{2 J (J + 1)}

where g_J (L,S) is the Landé g-factor.

Nuclei with nonzero spin I (and the neutron) possess a magnetic moment:

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

where γ_N is the nuclear magnetogyric ratio, g_N is the nuclear g-factor, and μ_N is the nuclear magneton. The first two depend on the identity of the nucleus and its internal structure; the value of the last is

μ_{N} \approx 5.050 783 7461 \times 10^{- 27} J T^{- 1}

The energy of interaction between two magnetic dipoles, μ ₁ and μ ₂ separated by a vector r in a vacuum is

V = - \frac{μ_{0}}{4 π r^{3}} (μ_{1} \cdot μ_{2} - 3 \frac{μ_{1} \cdot rr \cdot μ_{2}}{r^{2}})

where μ₀ is the magnetic constant. If the magnetic moments are parallel and located at an angle θ, then

V = - \frac{μ_{0} μ_{1} μ_{2}}{4 π r^{3}} (1 - 3 \cos^{2} θ)

Magnetic Susceptibility

The magnetic susceptibility (or volume magnetic susceptibility), χ, is the constant of proportionality between the magnetic moment density,

M

⁠, and the strength of the applied field,

H

⁠:

M = χ H

With

M

and

H

both in A m⁻¹, χ is dimensionless and may be of either sign. Materials for which χ < 1 are classified as diamagnetic; those for which χ > 1 are classified as paramagnetic. The molar magnetic susceptibility, χ_m, is defined as

χ_{m} = χ V_{m}

where V_m is the molar volume of the sample; χ_m has the dimensions of molar volume (m³ mol⁻¹).

At a molecular level, an applied magnetic field induces a magnetic dipole moment, μ , with

μ = ξ B

where ξ is the magnetizability of the molecule (the magnetic analogue of electric polarizability) and

B

is the magnetic flux density. Field strength

H

and flux density

B

are related by

B

= μ₀(⁠

H

M

⁠). The units of magnetic flux density are tesla, T, and those of magnetizability are J T⁻². Note that 1 T = 1 Wb m⁻¹, with Wb denoting weber, and equivalently 1 T = 1 J A⁻¹ m⁻². The magnetic susceptibility of a sample arises from both the induced moment and any permanent magnetic moment due to electron spin, and

χ_{m} = N_{A} μ_{0} (ξ + \frac{μ^{2}}{3 k T})

When the spin magnetic moment dominates the induced moment

χ_{m} = \frac{N_{A} g_{e}^{2} μ_{0} μ_{B}}{3 k T} S (S + 1)

Marcus Theory

Marcus theory provides a theoretical framework for the discussion of electron transfer reactions. Its core concept is that an electron can migrate from donor to acceptor only when the donor and acceptor have distorted to the point at which their molecular potential energy curves intersect and tunnelling is thereby facilitated. The core equations are as follows:

Rate constant : k_{r} = κ (kT / h) e^{- β r} e^{- Δ^{‡} G^{⦵} / RT}

Activation Gibbs energy : Δ^{‡} G^{⦵} = {({Δ_{reorg} G}^{⦵} + {Δ_{r} G}^{⦵})}^{2} / 4 {Δ_{reorg} G}^{⦵}

C r o s s - r e l a t i o n : k_{r, DA} = {(k_{r, AA} k_{r, DD} K_{DA})}^{1 / 2} f

Here, k_r,DA is the rate constant of interest, that for the transfer D → A, k_r,AA and k_r,DD are the rate constants for the two self-exchange processes (D → D and A → A), and K_DA is the equilibrium constant for the overall reaction. The factor f is a combination of a variety of factors, including encounter rates, and is often set equal to 1 in approximate calculations.

Maxwell Construction

The Maxwell construction is a procedure for replacing the van der Waals loops in isotherms by horizontal lines. The line is placed such that the areas of the loops on each side of the line are the same (Figure M.1). The construction is based on the requirement that an integral of dU around the cycle ABCDEA must be zero because the internal energy U is a state function. However, because only expansion work is allowed, it follows that the integral $\oint p d V = 0$ ⁠; so the areas between the area shown shaded must be equal.

Figure M.1

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The Maxwell construction.

Maxwell Relations

The Maxwell relations are a set of four relations between the first derivatives of thermodynamic quantities:

\begin{matrix} {(\frac{\partial T}{\partial V})}_{S} = - {(\frac{\partial p}{\partial S})}_{V}, {(\frac{\partial T}{\partial p})}_{S} = {(\frac{\partial V}{\partial S})}_{p} \\ {(\frac{\partial p}{\partial T})}_{V} = {(\frac{\partial S}{\partial V})}_{T}, {(\frac{\partial V}{\partial T})}_{p} = - {(\frac{\partial S}{\partial p})}_{T} \end{matrix}

They are based on the mathematical requirement that if dX(x,y) = f(x,y)dx + g(x,y)dy is an exact differential, then

{(\frac{\partial g (x, y)}{\partial y})}_{x} = {(\frac{\partial h (x, y)}{\partial x})}_{y}

Maxwell Distribution

The Maxwell distribution (or Maxwell–Boltzmann distribution) is an expression for the proportion of molecules of molar mass M (and mass m) that have a speed in the range

v

v + d v

d P (v) = f (v) d v

⁠, with

f (v) = 4 π {(\frac{m}{2 π k T})}^{3 / 2} v^{2} e^{- m v^{2} / 2 k T} = 4 π {(\frac{M}{2 π R T})}^{3 / 2} v^{2} e^{- M v^{2} / 2 R T}

See Figure M.2. The following features may be calculated from the distribution:

v_{mean} = {(\frac{8 k T}{π m})}^{1 / 2}, v_{rms} = {(\frac{3 k T}{m})}^{1 / 2}, v_{mp} = {(\frac{2 k T}{m})}^{1 / 2}

where

v_{mean}

is the mean speed,

v_{r m s}

is the root-mean-square speed, and

v_{mp}

is the most probable speed of the molecules. In each case k/m could be replaced by R/M.

Figure M.2

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The Maxwell distribution.

Mean Activity Coefficient

There is no experimental way of assigning departures from ideality to individual types of ions in an ionic solution, so it is conventional to regard the nonideality as shared equally between them. Therefore, the (geometric) mean activity coefficient, γ_±, is used instead. For the compound M_pX_q, which is present in solution as p cations and q anions,

γ_{\pm} = {(γ_{+}^{p} γ_{-}^{q})}^{1 / s}, s = p + q

Then, the chemical potential of each type of ion is

μ_{J} = μ_{J}^{ideal} + R T \ln γ_{\pm}

where

μ_{J}^{ideal}

is the chemical potential of J in an ideal solution. The molar Gibbs energy of the ions in solution is then

G_{m} = p μ_{M} + q μ_{X}

Mean Bond Enthalpy

The mean bond enthalpy, ΔH(AB), is the AB bond dissociation enthalpy averaged over a range of similar compounds.

Melting Point

The melting temperature is the temperature at which the solid and liquid phases of a substance are in equilibrium at a given pressure; the normal melting point is the melting temperature when that pressure is 1 atm. The standard melting temperature is the melting temperature when the pressure is 1 bar. The freezing temperature is the same as the melting temperature in all but certain exotic materials. The melting temperature of a biopolymer is the temperature at which 50 per cent of its natural form has undergone degradation. Polydisperse polymers typically melt over a range of temperatures.

Mesophase

A mesophase is a phase intermediate in character between a solid and a liquid, specifically with long-range order in one direction and short-range order in other directions. See liquid crystal.

Metastable Phase

A metastable phase is a phase that is thermodynamically unstable but prevented by kinetic reasons from undergoing a transition to a stable phase. Examples are superheated and supercooled liquids.

Michaelis–Menten Mechanism

The original formulation of the Michaelis–Menten mechanism of enzyme action proposed that an enzyme E establishes a pre-equilibrium concentration of the bound substrate, ES, with the substrate, S, and then goes on to form a product at the rate

v = d [P] / d t

with rate constant k_r:

E + S ⇌ ES \to E + P, v = k_{r} {[E]}_{0}, k_{r} = \frac{k_{r, 2} [S]}{K c^{⦵} + [S]}, K = \frac{[ES] c^{⦵}}{[E] [S]}

Here, [E]₀ is the total concentration of enzyme: [E]₀ = [E] + [ES]. A more flexible formulation treats the forward and reverse reactions involving the complex individually, with forward and reverse rate constants k_r,1 and

{k'}_{r, 1}

⁠, respectively, and applies the steady-state approximation to establish its concentration. Then

E + S ⇄ ES \to E + P, v = k_{r} {[E]}_{0}, k_{r} = \frac{k_{r, 2} [S]}{K_{M} + [S]}, K_{M} = \frac{k_{r, 1} + {k'}_{r, 1}}{k_{r, 2}}

In this expression, K_M is the Michaelis constant; note that it has dimensions of molar concentration.

The maximum velocity,

v_{\max}

⁠, of the reaction occurs when [S] ≫ K_M, and then

v_{\max} = k_{r, 2} {[E]}_{0}

The rate constant k_r,2 is called the maximum turnover number. It follows from this relation and the general rate law given above that

\frac{1}{v} = \frac{1}{v_{\max}} + \frac{K_{M}}{v_{\max}} \times \frac{1}{[S]}

Therefore, a Lineweaver–Burk plot (Figure M.3) of

1 / v

against 1/[S] should give a straight line with slope

K_{M} / v_{\max}

and intercept with the vertical axis (where 1/[S] = 0) at

1 / v_{\max}

and extrapolated intercept with the horizontal axis at −1/K_M.

Figure M.3

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A Lineweaver–Burk plot.

Miller Indices

The Miller indices (hkl) are used to label a plane in a crystal (Figure M.4); sets of parallel planes are denoted {hkl}. They are constructed as follows.

Write the intersections of the plane with the three crystallographic axes as multiples of the unit cell dimensions a, b, and c.
Take reciprocals of these multiples.
Clear fractions.

Figure M.4

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Three lattice planes and their Miller indices.

A negative index is denoted by an overbar, as in (12̄3). The distance, d_hkl, between neighbouring {hkl} planes in an orthorhombic (right-angled) lattice is given by

\frac{1}{d_{h k l}^{2}} = \frac{h^{2}}{a^{2}} + \frac{k^{2}}{b^{2}} + \frac{l^{2}}{c^{2}}

The separation of the planes in which the indices are all increased by a factor n is reduced by that factor. Structure factors, F_hkl, in X-ray crystallography are labelled with Miller indices.

Mirror Plane

A mirror plane, σ, is a symmetry element corresponding to the operation of reflection (Figure M.5). If the plane is horizontal to the principal axis of symmetry it is denoted σ_h; if the principal axis lies in the plane, then it is classified as vertical and denoted σ_v; a dihedral plane (see that entry) is denoted σ_d.

Figure M.5

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A mirror plane.

mmHg (the unit)

The outmoded unit 1 mmHg (millimetre of mercury) was used to report the pressure of a gas. It is defined as the pressure exerted at the base of a column of mercury of height 1 mm when the density is 13.5951 g cm⁻³ and the acceleration of free fall, g, has its standard value. It follows from this definition that 1 atm ≈760 mmHg. The millimetre of mercury has been largely superseded by 1 Torr, with 1 atm = 760 Torr (exactly). In all except the most precise work the units 1 mmHg and 1 Torr can be used synonymously, but in fact 1 mmHg ≈ 1.000 000 142 Torr.

Mobility

The mobility, u, of an ion of charge number z in solution is the constant of proportionality between its drift speed, s, and the strength of the applied electric field strength,

E

⁠:

s = u E

The drift speed is the final speed the ion reaches as result of the competition between the driving force of the electric field and the retarding force of friction:

s = \frac{| z | e E}{f}

where f is the frictional coefficient. It follows that

u = \frac{| z | e}{f}

If the frictional force is given by Stokes’ law (f = 6πηa, where a is the hydrodynamic radius of the ion and η the viscosity), then

u = \frac{| z | e}{6 π η a}

The mobility of an ion is related to its molar conductivity, λ, by

λ = | z | u F

Two further relations are the Einstein relation, which expresses the mobility in terms of the diffusion coefficient D:

u = \frac{| z | F D}{R T}

and the Stokes–Einstein equation, which expresses the ionic molar conductivity in terms of the diffusion coefficient:

λ = \frac{z^{2} F^{2} D}{R T}

Molality

The molality, b_J, is the amount of solute J of solute divided by the mass of the solvent: b_J = n_J/m_solvent. Molalities are commonly reported in moles of solute per kilogram of solvent. A convenient quantity is b ^⦵ = 1 mol kg⁻¹. Unlike molar concentration, c_J, molality is independent of temperature. The relations between molality and mole fraction are

b_{J} = \frac{x_{J}}{(1 - x_{J}) M_{solvent}}, x_{J} = \frac{b_{J} M_{solvent}}{1 + b_{J} M_{solvent}}

and the relations between molality and molar concentration are

b_{J} = \frac{c_{J}}{ρ - c_{J} M_{J}}, c_{J} = \frac{b_{J} ρ}{1 + b_{J} M_{J}}

where ρ is the mass density of the solution.

Molar Concentration

The molar concentration of a solute J, c_J or [J] (more formally, the amount of substance concentration, and often informally the ‘molarity’), is the amount of solute divided by the volume of the solution: c_J = n_J/V_solution. A convenient quantity is c ^⦵ = 1 mol dm⁻³. The symbol M is sometimes used adjectivally as an abbreviation for mol dm⁻³ (as in a 1 M solution) but the use of prefixes (as in a 1 mM solution) is deprecated. The relation between molar concentration and mole fraction is

c_{J} = \frac{x_{J} ρ}{(1 - x_{J}) M_{solvent} + x_{J} M_{J}} \approx {\begin{matrix} \frac{ρ}{(1 - x_{J}) M_{solvent}} x_{J} if x_{J} M_{J} ≪ (1 - x_{J}) M_{solvent} \\ \frac{ρ}{M_{solvent}} x_{J} as above, and x_{J} ≪ 1 \end{matrix}

where ρ is the mass density of the solution. For its relation to molality, see that entry.

Molar Mass

The molar mass of an element or compound, M, is the isotopic-composition-averaged mass of a formula unit multiplied by Avogadro’s constant: M(X) = m_XN_A. The traditional terms atomic weight, molecular weight, and formula weight are widely used to refer to the dimensionless quantity M(X)/(g mol⁻¹). Polymeric materials and some biological macromolecules are rarely exactly monodisperse (that is, have a single, precise molar mass), and three types of average molar mass are encountered, the number-average molar mass, M̄_n, the weight-average (or mass-average) molar mass, M̄_w, and the Z-average molar mass, M̄_Z:

\begin{array}{c} {\bar{M}}_{n} = \frac{1}{N} \sum_{i} N_{i} M_{i} = 〈 M 〉 \\ {\bar{M}}_{w} = \frac{1}{m} \sum_{i} m_{i} M_{i} = \frac{\sum_{i} N_{i} M_{i}^{2}}{\sum_{i} N_{i} M_{i}} = \frac{〈 M^{2} 〉}{〈 M 〉} \\ {\bar{M}}_{Z} = \frac{\sum_{i} N_{i} M_{i}^{3}}{\sum_{i} N_{i} M_{i}^{2}} = \frac{〈 M^{3} 〉}{〈 M^{2} 〉} \end{array}

where N_i and m_i are the number and mass of molecules with molar mass M_i, and N and m are the total number and mass of molecules in the sample. Different experimental techniques give different types of average: osmometry gives the number-average molar mass, light scattering the weight-average molar mass, and the sedimentation the Z-average molar mass. In practice, average molar masses are treated as empirical averages and not interpreted strictly in terms of mean square and mean cube molar mass. If all the molecules in the sample have the same molar mass, all three averages give the same value. The dispersity, Đ, of a sample is defined as

Đ = \frac{{\bar{M}}_{w}}{{\bar{M}}_{n}}

A synthetic sample is classified as monodisperse if Đ < 1.1. Samples for which the dispersity is larger are classified as polydisperse.

Note that the unit 1 dalton, 1 Da, is sometimes used to report molar mass. That is incorrect: 1 Da = m_u, where m_u = m(¹²C)/12 ≈1.660 539 066 60 × 10⁻²⁷ kg is the atomic mass constant, a measure of atomic or molecular mass, not molar mass.

Molar Quantity

A molar quantity, X_m, is defined as X/n, where n is the amount of substance in the sample. Molar quantities are intensive. An exception to this notational convention is the molar mass, which is commonly denoted simply M. Two other exceptions to the definition are molar conductivity, Λ_m, in which the divisor is the molar concentration of the electrolyte, and the molar concentration, c_J or [J], itself. The latter is more properly but hardly ever called the amount of substance concentration.

mole (the unit)

The unit 1 mole (1 mol) is used to report the amount of substance n_J of specified entities J in a sample, with 1 mol = 6.022 140 76 ×10²³ entities exactly. In practice, a desired amount of entities is measured out by noting the mass of the sample, m, and then using n_J = m/M_J, where M_J is the molar mass of J.

Mole Fraction

The mole fraction, x_J, of the component J of a mixture is its amount expressed as a fraction of the total amount, n, of all the components:

x_{J} = \frac{n_{J}}{n}, n = \sum_{J} n_{J}

It follows from the definition that

\sum_{J} x_{J} = 1

For its relation to molality, see that entry.

Molecular Beam

A molecular beam is a narrow stream of molecules with a narrow spread of speeds and in some cases in specific internal state states or orientation. Molecular beam studies of nonreactive collisions are used to explore the details of molecular interactions with a view to determining the intermolecular potential energy landscape. Molecular beam studies of reactive collisions are used to study the details of collisions that lead to the exchange of atoms. Surfaces, particularly those of catalysts, are also open to investigation by using molecular beams.

Molecular Orbital Theory

In molecular orbital theory (MO theory) the electrons are considered to occupy wavefunctions that spread over the entire molecule. These one-electron delocalized wavefunctions are the molecular orbitals of the theory. Two principal approximations are used to construct them. The first is the Born–Oppenheimer approximation, in which the nuclei are regarded as stationary in some preselected arrangement. The second is to use an LCAO procedure to express all the molecular orbitals as linear combinations of a basis set of atomic orbitals, χ_i:

ψ = \sum_{i} c_{i} χ_{i}

From a basis of N atomic orbitals, N molecular orbitals can be constructed and their energies calculated by manipulating the Schrōdinger equation. Electrons are then introduced into the available molecular orbitals by using the building-up principle. The molecular orbital energy level diagrams for second-period homonuclear diatomic molecules obtained in this way are shown in Figure M.6. The type of calculation involved varies from the very primitive (see FEMO, Hückel method, and semiempirical procedures) to the very sophisticated (see ab initio, self-consistent field, and density functional procedures).

Figure M.6

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Molecular orbital energy levels of homonuclear diatomic molecules.

A characteristic feature of MO theory is that it leads to electron distributions that can be interpreted as contributing to bonding and antibonding effects between neighbouring pairs of atoms (and for the entire molecule for diatomic molecules):

Bonding: A molecular orbital is bonding between neighbours if there is no internuclear node between them. There is constructive interference between the atomic orbitals on the two atoms and an accumulation of electron density in the internuclear region. An electron that occupies a bonding molecular orbital contributes to the lowering of energy of the entire molecule. Note that in a polyatomic molecule there may be different regions where a molecular orbital may be bonding but in others antibonding.
Nonbonding: A molecular orbital is nonbonding between to atoms if at least one of the atoms contributes no atomic orbital to the LCAO.
Antibonding: A molecular orbital is antibonding between neighbours if there is an internuclear node between them. There is destructive interference between the atomic orbitals on the two atoms and a reduction of electron density in the internuclear region. An electron that occupies an antibonding molecular orbital contributes to the raising of energy of the entire molecule. Note that in a polyatomic molecule there may be different regions where a molecular orbital may be bonding but in others antibonding.

The net bonding character of electrons is distributed over the entire molecule, not localized (as in valence bond theory). Broadly speaking, the energy of molecular orbitals built from a given basis increases as the number of internuclear nodes increases. The lowest energy orbital, the orbital with the strongest bonding character, has no internuclear nodes; the highest energy orbital, the most antibonding orbital, typically has an internuclear node between all neighbouring atoms.

Molecular Potential Energy Curve

A molecular potential energy curve shows how the energy of a diatomic molecule, disregarding the kinetic energy arising from nuclear motion, varies as its internuclear distance is changed (Figure M.7). The minimum of the curve corresponds to the equilibrium bond length and is closely related to the bond dissociation energy. The second derivative of the curve evaluated at the minimum gives the force constant for the bond and is used in the discussion of molecular vibration. The existence of the curve depends on the validity of the Born–Oppenheimer approximation. The corresponding depiction of a polyatomic molecule is a molecular potential energy surface.

Figure M.7

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Typical molecular potential energy curves.

Molecular Reaction Dynamics

Molecular reaction dynamics is the determination of the trajectory that interacting molecules take through a potential energy surface, particularly those corresponding to reactive collisions. In more quantum mechanical terms, it is the determination of the time-dependent wavefunctions for the collection of electrons in interacting molecules. Such studies, and their augmentation by experimental studies using molecular beams, give highly detailed information about the evolution of reactant molecules into product molecules in the course of a reactive collision.

Molecular Solid

A molecular solid is a solid composed of discrete molecules that cohere by nonbonding molecular interactions.

Moment of Inertia

The moment of inertia, I, of a body composed of point masses m_i at a perpendicular distance r_i from the axis of rotation is

I = \sum_{i} m_{i} r_{i}^{2}

Whereas mass is a measure of the resistance of a body to an applied force, F, in the sense that the rate of change of velocity is

d v / d t = F / m

⁠, for rotational motion the moment of inertia is a measure of the resistance of a body to an applied torque,

T

⁠, in the sense that

d ω / d t = T / I

⁠. The parallel axes theorem asserts that if I is the moment of inertia around an axis passing through the centre of mass of a body of mass m, then the moment of inertia, I′, around an axis that is parallel to the first axis but a distance d from it is

I' = I + m d^{2}

Explicit expressions for a variety of molecular shapes are listed in Table M.1.

Table M.1

Moments of inertia of molecules.^a

1. Diatomic molecules

I = μ R^{2} μ = \frac{m_{A} m_{B}}{m}

2. Triatomic linear rotors

I = m_{A} R^{2} + m_{C} {R^{'}}^{2} - \frac{{(m_{A} R - m_{C} R^{'})}^{2}}{m}

I = 2 m_{A} R^{2}

3. Symmetric rotors

I_{| |} = 2 m_{A} f_{1} (θ) R^{2}

\begin{array}{l} I_{⊥} = m_{A} f_{1} (θ) R^{2} + \frac{m_{A}}{m} (m_{B} + m_{A}) f_{2} (θ) R^{2} \\ + \frac{m_{C}}{m} {(3 m_{A} + m_{B}) R' + 6 m_{A} R {[\frac{1}{3} f_{2} (θ)]}^{1 / 2}} R' \end{array}

I_{| |} = 2 m_{A} f_{1} (θ) R^{2}

I_{⊥} = m_{A} f_{1} (θ) R^{2} + \frac{m_{A} m_{B}}{m} f_{2} (θ) R^{2}

\begin{matrix} I_{| |} = 4 m_{A} R^{2} \\ I_{⊥} = 2 m_{A} R^{2} + 2 m_{C} {R^{'}}^{2} \end{matrix}

4. Spherical rotors

I = \frac{8}{3} m_{A} R^{2}

I = 4 m_{A} R^{2}

f₁(θ) = 1 − cos θ, f₂(θ) = 1 + 2cos θ; in each case, m is the total mass of the molecule.

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The magnitude of the angular momentum around an axis with moment of inertia I and angular velocity ω is

J = I ω

and its kinetic energy is

E_{k} = \frac{1}{2} I ω^{2} = \frac{J^{2}}{2 I}

If a molecule has a moment of inertia I_qq and angular velocity ω_q around each of three mutually perpendicular axes then its total rotational (kinetic) energy is

E = \frac{1}{2} \sum_{q} I_{q q} ω_{q}^{2} = \sum_{q} \frac{J_{q}^{2}}{2 I_{q q}}

The quantum mechanical expression for the rotational energy of a rigid rotor depends on its symmetry. Rigid rotors are classified according to their moments of inertia, which in this context are denoted I_a, I_b, and I_c (Figure M.8):

Figure M.8

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The classification of rotors.

Linear rotor, I_a = 0, I_b = I_c = I

E_{J, M_{J}} = h c \tilde{B} J (J + 1), \tilde{B} = \frac{ℏ}{4 π c I}, J = 0, 1, 2, \dots, M_{J} = 0, \pm 1, \dots, \pm J

Symmetrical rotor, I_c = I_b (=I_⊥) > I_a (=I_∥), prolate; I_a = I_b (=I_⊥) < I_c (=I_∥), oblate

E_{J, K, M_{J}} = h c \tilde{B} J (J + 1) + h c (\tilde{A} - \tilde{B}) K^{2}, \tilde{B} = \frac{ℏ}{4 π c I_{⊥}}, \tilde{A} = \frac{ℏ}{4 π c I_{∥}}, J = 0, 1, 2, \dots, K, M_{J} = 0, \pm 1, \dots, \pm J

Spherical rotor, I_a = I_b = I_c = I

E_{J, K, M_{J}} = h c \tilde{B} J (J + 1), \tilde{B} = \frac{ℏ}{4 π c I}, J = 0, 1, 2, \dots, K, M_{J} = 0, \pm 1, \dots, \pm J

Asymmetrical rotor, I_a, I_b, I_c all different

No simple closed, expression

In group theoretical language, a spherical rotor belongs to a cubic or icosahedral point group, and a symmetric rotor belongs to a group having a rotational axis with n > 2.

Monolayer

A monolayer is a single layer of molecules on a liquid or solid surface. For the extent of surface coverage see adsorption. A monolayer on a liquid surface is characterised by the surface pressure, π, the difference between the surface tension of the solvent and the solution:

π = γ - γ ⁎

and by the collapse pressure the highest lateral pressure that can be sustained by the surface film (Figure M.9).

Figure M.9

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The surface pressure of two types of surfactant.

Monte Carlo Method

The Monte Carlo method for producing statistical averages over assemblies of particles allows each particle in system to move through an arbitrary but small distance. Then the total potential energy is calculated by using an appropriate intermolecular potential. Whether this new configuration is accepted and included in the computation of an average property is then judged as follows. If the calculated potential energy is not greater than before the change, then the new configuration is accepted. If the potential is energy is greater than before the change, then it is accepted or rejected with the probability of acceptance proportion to the value of $e^{- Δ V_{N} / k T}$ ⁠, where ΔV_N is the change in potential energy of the N particles in the system. This procedure ensures that the probability of occurrence of any configuration is proportional to the Boltzmann factor, $e^{- V_{N} / k T}$ ⁠. The thermodynamic or structural properties of the system are then be calculated by averaging the properties over the configurations generated in this way.

Morse Potential

The Morse potential is two-parameter analytical expression for the form of the molecular potential energy curve as the displacement, x, from equilibrium that mirrors its principal features (Figure M.10):

V (x) = h c {\tilde{D}}_{e} {1 - e^{- a x}}^{2}, a = {(\frac{m_{eff} c}{2 h {\tilde{D}}_{e}})}^{1 / 2} \frac{\tilde{ν}}{2 π}

Figure M.10

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The Morse potential and its energy levels.

The parameter D̃_e, a wavenumber, is the depth of the minimum of the curve and m_eff is the effective mass of the oscillator. The solutions of Schrödinger equation with this potential energy are

E_{v} = (v + \frac{1}{2}) h c \tilde{ν} - {(v + \frac{1}{2})}^{2} h c x_{e} \tilde{ν}, x_{e} = \frac{a^{2} ℏ}{4 π m_{eff} c \tilde{ν}} = \frac{\tilde{ν}}{4 {\tilde{D}}_{e}}

The parameters appearing in this expression and the Mores potential itself are found by fitting these energies to the observed spectrum. The quantum number

v

takes the finite series of values 0, 1, 2,…, with

v < \frac{2 {\tilde{D}}_{e}}{\tilde{ν}} - \frac{1}{2}

Complicated analytical expressions for the wavefunctions are available. Two wavefunctions are shown in Figure M.11.

Figure M.11

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Two wavefunctions of a Morse oscillator.

Moving Boundary Method

The moving boundary method is a technique for the determination of transport numbers in which the motion of a boundary between two ionic solutions having a common ion is observed as current flows. If MX is the salt of interest, a solution of another salt NX is selected that gives a denser solution. The MX solution is called the leading solution and the NX solution is the indicator solution. The leading solution lies on top of the indicator solution. The boundary between them sweeps out a volume V in a time Δt when a current I is passed. The transport number t₊ is given by

t_{+} = \frac{z c F}{I Δ t}

where z is the charge number of the M cations and c is their molar concentration.

Multiplicity

The multiplicity of a spectroscopic term ^2S+1X is the value of 2S +1 and, provided S is not greater than L, is the number of levels (distinguished by J in ^2S+1X_J) of the term. A term with S = 0, denoted ¹X, is called a singlet term. A term with S = 1, denoted ³X, is called a triplet term and (provided L > 0) has three levels.

Multipole

An electric multipole is an array of point charges with an n-pole moment but no lower moment (Figure M.12). Thus, a single point charge is a monopole, two equal and opposite charges is a dipole, and four point charges with zero overall charge and no dipole moment is a quadrupole. Multipoles may also be formed with more than n charges provided that they conform to these definitions. Thus, two dipoles may be arranged to have no net dipole moment but a nonzero quadrupole moment (CO₂ is an example of a molecule with no net charge, no dipole moment, but a nonzero quadrupole moment). A point multipole is a multipole in which the separation of the charges is much smaller than the distance at which their influence is measured. The potential energy of the interaction of a stationary point n-pole and a stationary point m-pole separated by r depends on that distance as

E_{p} \propto \frac{1}{r^{n + m - 1}}

Figure M.12

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Examples of electric multipoles.

1. Diatomic molecules
	$I = μ R^{2} μ = \frac{m_{A} m_{B}}{m}$
2. Triatomic linear rotors
	$I = m_{A} R^{2} + m_{C} {R^{'}}^{2} - \frac{{(m_{A} R - m_{C} R^{'})}^{2}}{m}$
	$I = 2 m_{A} R^{2}$
3. Symmetric rotors
	$I_{\| \|} = 2 m_{A} f_{1} (θ) R^{2}$
	$\begin{array}{l} I_{⊥} = m_{A} f_{1} (θ) R^{2} + \frac{m_{A}}{m} (m_{B} + m_{A}) f_{2} (θ) R^{2} \\ + \frac{m_{C}}{m} {(3 m_{A} + m_{B}) R' + 6 m_{A} R {[\frac{1}{3} f_{2} (θ)]}^{1 / 2}} R' \end{array}$
	$I_{\| \|} = 2 m_{A} f_{1} (θ) R^{2}$
	$I_{⊥} = m_{A} f_{1} (θ) R^{2} + \frac{m_{A} m_{B}}{m} f_{2} (θ) R^{2}$
	$\begin{matrix} I_{\| \|} = 4 m_{A} R^{2} \\ I_{⊥} = 2 m_{A} R^{2} + 2 m_{C} {R^{'}}^{2} \end{matrix}$
4. Spherical rotors
	$I = \frac{8}{3} m_{A} R^{2}$
	$I = 4 m_{A} R^{2}$

Log in

M

Magic Angle Spinning

Magnetic Moment

Magnetic Susceptibility

Marcus Theory

Maxwell Construction

Maxwell Relations

Maxwell Distribution

Mean Activity Coefficient

Mean Bond Enthalpy

Melting Point

Mesophase

Metastable Phase

Michaelis–Menten Mechanism

Miller Indices

Mirror Plane

mmHg (the unit)

Mobility

Molality

Molar Concentration

Molar Mass

Molar Quantity

mole (the unit)

Mole Fraction

Molecular Beam

Molecular Orbital Theory

Molecular Potential Energy Curve

Molecular Reaction Dynamics

Molecular Solid

Moment of Inertia

Monolayer

Monte Carlo Method

Morse Potential

Moving Boundary Method

Multiplicity

Multipole

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