 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
M

Published:17 May 2024
Concepts in Physical Chemistry, Royal Society of Chemistry, 2nd edn, 2024, pp. 196217.
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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 solidstate 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 cos^{2} 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 cos^{2} 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
Magnetic Susceptibility
Marcus Theory
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 $\u222e p d V =0$ ; so the areas between the area shown shaded must be equal.
Maxwell Relations
Maxwell Distribution
Mean Activity Coefficient
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 longrange order in one direction and shortrange 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
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.
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 Xray 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}.
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
Molality
Molar Concentration
Molar Mass
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(^{12}C)/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 ×10^{23} 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
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
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 buildingup principle. The molecular orbital energy level diagrams for secondperiod 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, selfconsistent field, and density functional procedures).
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.
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 timedependent 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
1. Diatomic molecules  
$I=\mu R2\mu =mAmBm$  
2. Triatomic linear rotors  
$I=mAR2+mCR\u20322\u2212(mAR\u2212mCR\u2032)2m$  
$I=2mAR2$  
3. Symmetric rotors  
$I=2mAf1(\theta )R2$  
$I\u22a5=mAf1(\theta )R2+mAm(mB+mA)f2(\theta )R2+mCm{(3mA+mB)R\u2032+6mAR[13f2(\theta )]1/2}R\u2032$  
$I=2mAf1(\theta )R2$  
$I\u22a5=mAf1(\theta )R2+mAmBmf2(\theta )R2$  
$I=4mAR2I\u22a5=2mAR2+2mCR\u20322$  
4. Spherical rotors  
$I=83mAR2$  
$I=4mAR2$ 
1. Diatomic molecules  
$I=\mu R2\mu =mAmBm$  
2. Triatomic linear rotors  
$I=mAR2+mCR\u20322\u2212(mAR\u2212mCR\u2032)2m$  
$I=2mAR2$  
3. Symmetric rotors  
$I=2mAf1(\theta )R2$  
$I\u22a5=mAf1(\theta )R2+mAm(mB+mA)f2(\theta )R2+mCm{(3mA+mB)R\u2032+6mAR[13f2(\theta )]1/2}R\u2032$  
$I=2mAf1(\theta )R2$  
$I\u22a5=mAf1(\theta )R2+mAmBmf2(\theta )R2$  
$I=4mAR2I\u22a5=2mAR2+2mCR\u20322$  
4. Spherical rotors  
$I=83mAR2$  
$I=4mAR2$ 
f_{1}(θ) = 1 − cos θ, f_{2}(θ) = 1 + 2cos θ; in each case, m is the total mass of the molecule.
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):
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
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 \u2212 \Delta 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 \u2212 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
Complicated analytical expressions for the wavefunctions are available. Two wavefunctions are shown in Figure M.11.
Moving Boundary Method
Multiplicity
The multiplicity of a spectroscopic term ^{2S+1}X is the value of 2S +1 and, provided S is not greater than L, is the number of levels (distinguished by J in ^{2S+1}X_{J}) of the term. A term with S = 0, denoted ^{1}X, is called a singlet term. A term with S = 1, denoted ^{3}X, is called a triplet term and (provided L > 0) has three levels.