Defining equation (physical chemistry)

In physical chemistry, there are numerous quantities associated with chemical compounds and reactions; notably in terms of amounts of substance, activity or concentration of a substance, and the rate of reaction. This article uses SI units.

Introduction

Theoretical chemistry requires quantities from core physics, such as time, volume, temperature, and pressure. But the highly quantitative nature of physical chemistry, in a more specialized way than core physics, uses molar amounts of substance rather than simply counting numbers; this leads to the specialized definitions in this article. Core physics itself rarely uses the mole, except in areas overlapping thermodynamics and chemistry.

Notes on nomenclature

Entity refers to the type of particle/s in question, such as atoms, molecules, complexes, radicals, ions, electrons etc.^[1]

Conventionally for concentrations and activities, square brackets [ ] are used around the chemical molecular formula. For an arbitrary atom, generic letters in upright non-bold typeface such as A, B, R, X or Y etc. are often used.

No standard symbols are used for the following quantities, as specifically applied to a substance:

• the mass of a substance m,
• the number of moles of the substance n,
• partial pressure of a gas in a gaseous mixture p (or P),
• some form of energy of a substance (for chemistry enthalpy H is common),
• entropy of a substance S
• the electronegativity of an atom or chemical bond χ.

Usually the symbol for the quantity with a subscript of some reference to the quantity is used, or the quantity is written with the reference to the chemical in round brackets. For example, the mass of water might be written in subscripts as m_H2O, m_water, m_aq, m_w (if clear from context) etc., or simply as m(H₂O). Another example could be the electronegativity of the fluorine-fluorine covalent bond, which might be written with subscripts χ_F-F, χ_FF or χ_F-F etc., or brackets χ(F-F), χ(FF) etc.

Neither is standard. For the purpose of this article, the nomenclature is as follows, closely (but not exactly) matching standard use.

For general equations with no specific reference to an entity, quantities are written as their symbols with an index to label the component of the mixture - i.e. q_i. The labeling is arbitrary in initial choice, but once chosen fixed for the calculation.

If any reference to an actual entity (say hydrogen ions H⁺) or any entity at all (say X) is made, the quantity symbol q is followed by curved ( ) brackets enclosing the molecular formula of X, i.e. q(X), or for a component i of a mixture q(X_i). No confusion should arise with the notation for a mathematical function.

Quantification

General basic quantities

Quantity (Common Name/s)	(Common) Symbol/s	SI Units	Dimension
Number of molecules	N	dimensionless	dimensionless
Mass	m	kg	[M]
Number of moles, amount of substance, amount	n	mol	[N]
Volume of mixture or solvent, unless otherwise stated	V	m^3	[L]^3

General derived quantities

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Relative atomic mass of an element	Ar, A, mram	$A_r \left ( {\rm X} \right ) = \frac{\langle m \left ( {\rm X} \right ) \rangle }{m \left ( ^{12}{\rm C} \right ) / 12}$ The average mass $\langle m \left ( {\rm X} \right ) \rangle$ is the average of the T masses mi(X) corresponding the T isotopes of X (i is a dummy index labelling each isotope): $\langle m \left ( {\rm X} \right ) \rangle = \frac{1}{T} \sum_i^T m \left ( {\rm X}_i \right )$	dimensionless	dimensionless
Relative formula mass of a compound, containing elements Xj	Mr, M, mrfm	$M_r \left ( {\rm Y} \right ) = \sum_j N \left ( {\rm X}_j \right ) A_r \left ( {\rm X}_j \right ) = \frac{\sum_j N \left ( {\rm X}_j \right ) \langle m \left ( {\rm X}_j \right ) \rangle }{m \left ( ^{12}{\rm C} \right ) / 12}$ j = index labelling each element, N = number of atoms of each element Xi.	dimensionless	dimensionless
Molar concentration, concentration, molarity of a component i in a mixture	ci, [Xi]	$c_i = \left [ {\rm X}_i \right ] = \frac{\mathrm{d} n_i}{\mathrm{d} V}$	mol dm^−3 = 10^−3 mol m^−3	[N] [L]^−3
Molality of a component i in a mixture	bi, b(Xi)	$b_i = \frac{n_i}{m_{\rm solv}}$ where solv = solvent (liquid solution).	mol dm^−3 kg = 10^−3 mol m^−3	[N] [L]^−3
Mole fraction of a component i in a mixture	xi, x(Xi)	$x_i = \frac{n_i}{n_{\rm mix}}$ where Mix = mixture.	dimensionless	dimensionless
Partial pressure of a gaseous component i in a gas mixture	pi, p(Xi)	$p \left( {\rm X}_i \right ) = x_i p\left( {\rm mix} \right )$ where mix = gaseous mixture.	Pa = N m^−2	[M][T][L]^−1
Density, mass concentration	ρi, γi, ρ(Xi)	$\rho = m_i/V\,\!$	kg m^−3	[M] [L]^3
Number density, number concentration	Ci, C(Xi)	$C_i = N_i/V\,\!$	m^− 3	[L]^− 3
Volume fraction, volume concentration	ϕi, ϕ(Xi)	$\phi_i = \frac{V_i}{V_{\rm mix}}$	dimensionless	dimensionless
Mixing ratio, mole ratio	ri, r(Xi)	$r_i = \frac{n_i}{n_{\rm mix}- n_i}$	dimensionless	dimensionless
Mass fraction	wi, w(Xi)	$w_i = m_i / m_{\rm mix} \,\!$ m(Xi) = mass of Xi	dimensionless	dimensionless
Mixing ratio, mass ratio	ζi, ζ(Xi)	$\zeta_i = \frac{m_i}{m_{\rm mix}- m_i}$ m(Xi) = mass of Xi	dimensionless	dimensionless

Kinetics and equilibria

The defining formulae for the equilibrium constants K_c (all reactions) and K_p (gaseous reactions) apply to the general chemical reaction:

$\nu_1 {\rm X}_1 + \nu_2 {\rm X}_2 + \cdots + \nu_r {\rm X}_r \rightleftharpoons \eta_1 {\rm Y}_1 + \eta_2 {\rm Y}_2 + \cdots + \eta_p {\rm Y}_p \,,$

and the defining equation for the rate constant k applies to the simpler synthesis reaction (one product only):

$\nu_1 {\rm X}_1 + \nu_2 {\rm X}_2 + \cdots + \nu_r {\rm X}_r \rightarrow \eta {\rm Y} \,,$

where:

• i = dummy index labelling component i of reactant mixture,
• j = dummy index labelling component i of product mixture,
• X_i = component i of the reactant mixture,
• Y_j = reactant component j of the product mixture,
• r (as an index) = number of reactant components,
• p (as an index) = number of product components,
• ν_i = stoichiometry number for component i in product mixture,
• η_j = stoichiometry number for component j in product mixture,
• σ_i = order of reaction for component i in reactant mixture.

The dummy indices on the substances X and Y label the components (arbitrary but fixed for calculation); they are not the numbers of each component molecules as in usual chemistry notation.

The units for the chemical constants are unusual since they can vary depending on the stoichiometry of the reaction, and the number of reactant and product components. The general units for equilibrium constants can be determined by usual methods of dimensional analysis. For the generality of the kinetics and equilibria units below, let the indices for the units be;

$S_1 = \sum_{j=1}^p \eta_j - \sum_{i=1}^r \nu_i \,,\quad\, S_2 = 1-\sum_{i=1}^{r} \sigma_i\,. \,\!$

  Click here to see their derivation

For the constant K_c;

Substitute the concentration units into the equation and simplify:,

$\begin{align} K_c = \frac{\prod_{j=1}^p \left [ Y_j \right ]^{y_j}}{\prod_{i=1}^r \left [ X_i \right ]^{x_i} } \,\!\end{align}$

Failed to parse (syntax error): \begin{align} [K_c] & = \frac{\prod_{j=1}^p [ {\rm mol \; dm^{{-3}} ]^{{y_j}}{\prod_{i=1}^r [ {\rm mol \; dm^{{-3}} ]^{x_i} } \\ & = \frac{[ {\rm mol \; dm^{{-3}} ]^{y_1} [ {\rm mol \; dm^{{-3}} ]^{y_2} \cdots [ {\rm mol \; dm^{{-3}} ]^{{y_p}}{[ {\rm mol \; dm^{{-3}} ]^{x_1} [ {\rm mol \; dm^{{-3}} ]^{x_2} \cdots [ {\rm mol \; dm^{{-3}} ]^{x_r} } \\ & = \frac{[ {\rm mol \; dm^{{-3}} ]^{\sum_{j=1}^p y_j}}{[ {\rm mol \; dm^{{-3}} ]^{\sum_{i=1}^r x_i} } \\ & = [ {\rm mol \; dm^{{-3}} ]^{\sum_{j=1}^p y_j - \sum_{i=1}^r x_i} \end{align}

The procedure is exactly identical for K_p.

For the constant k

$\begin{align} k & = \frac{{\rm d}[Y]/{\rm d}t}{\prod_{i=1}^{r}[X_i]^{\sigma_i}} \\ [k] & = \frac{[ {\rm mol \; dm^{-3}} ]s^{-1}}{\prod_{i=1}^{r}[ {\rm mol \; dm^{-3}} ]^{\sigma_i}} \\ & = \frac{[ {\rm mol \; dm^{-3}} ]s^{-1}}{[ {\rm mol \; dm^{-3}} ]^{\sum_{i=1}^{r} \sigma_i}} \\ & = [ {\rm mol \; dm^{-3}} ]^{1-\sum_{i=1}^{r} \sigma_i} s^{-1} \end{align}$

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Reaction progress variable, extent of reaction	ξ	$\xi$	dimensionless	dimensionless
Stoichiometric coefficient of a component i in a mixture, in reaction j (many reactions could occur at once)	νi	$\nu_{ij} = \frac{{\rm d}N_i}{{\rm d}\xi_j} \,$ where Ni = number of molecules of component i.	dimensionless	dimensionless
Chemical affinity	A	$A = - \left ( \frac{\partial G }{\partial \xi} \right )_{P,T}$	J	[M][L]^2[T]^−2
Reaction rate with respect to component i	r, R	$R_i = \frac{1}{\nu_i} \frac{\mathrm{d} \left [ {\rm X}_i \right ]}{\mathrm{d} t}$	mol dm^−3 s^−1 = 10^−3 mol m^−3 s^−1	[N] [L]^−3 [T]^−1
Activity of a component i in a mixture	ai	$a_i = e^{\left ( \mu_i - \mu^{\ominus}_i \right )/RT}$	dimensionless	dimensionless
Mole fraction, molality, and molar concentration activity coefficients	γxi for mole fraction, γbi for molality, γci for molar concentration.	Three coefficients are used; $a_i = \gamma_{xi} x_i \,$ $a_i = \gamma_{bi} b_i/b^{\ominus} \,$ $a_i = \gamma_{ci} \left [ {\rm X}_i \right ]/\left [ {\rm X}_i \right ]^{\ominus} \,$	dimensionless	dimensionless
Rate constant	k	$k = \frac{{\rm d}\left [ {\rm Y} \right ]/{\rm d}t}{\prod_{i=1}^{r}\left [ {\rm X}_i \right ]^{\sigma_i}}$	(mol dm^−3)^(S2) s^−1	([N] [L]^−3)^(S2) [T]^−1
General equilibrium constant [2]	Kc	$K_c = \frac{\prod_{j=1}^p \left [ {\rm Y}_j \right ]^{\eta_j}}{\prod_{i=1}^r \left [ {\rm X}_i \right ]^{\nu_i} } \,\!$	(mol dm^−3)^(S1)	([N] [L]^−3)^(S1)
General thermodynamic activity constant [3]	K0	$K_0 = \frac{\prod_{j=1}^p a \left ( {\rm Y}_j \right )^{\eta_j}}{\prod_{i=1}^r a \left ( {\rm X}_i \right )^{\nu_i} } \,\!$ a(Xi) and a(Yj) are activities of Xi and Yj respectively.	(mol dm^−3)^(S1)	([N] [L]^−3)^(S1)
Equilibrium constant for gaseous reactions, using Partial pressures	Kp	$K_p = \frac{\prod_{j=1}^p p\left ( {\rm Y}_j \right )^{\eta_j}}{\prod_{i=1}^r p\left ( {\rm X}_i \right )^{\nu_i} } \,\!$	Pa^(S1)	([M] [L]^−1 [T]^−2)^(S1)
Logarithm of any equilibrium constant	pKc	${\rm p} K_c = -\log_{10} K_c = \sum_{j=1}^p \eta_j \log_{10} \left [ {\rm Y}_j \right ] - \sum_{i=1}^r \nu_i \log_{10} \left [ {\rm X}_i \right ] \,\!$	dimensionless	dimensionless
Logarithm of dissociation constant	pK	${\rm p} K = -\log_{10} K \,\!$	dimensionless	dimensionless
Logarithm of hydrogen ion (H^+) activity, pH	pH	${\rm pH} = -\log_{10} [ {\rm H^{+}}] \,\!$	dimensionless	dimensionless
Logarithm of hydroxide ion (OH^−) activity, pOH	pOH	${\rm pOH} = -\log_{10} [ {\rm OH^{-}}] \,\!$	dimensionless	dimensionless

Electrochemistry

Notation for half-reaction standard electrode potentials is as follows. The redox reaction

$\mathrm{A} + \mathrm{BX} \rightleftharpoons \mathrm{B} + \mathrm{AX}$

split into:

a reduction reaction: $\mathrm{B}^{+} + \mathrm{e}^{-} \rightleftharpoons \mathrm{B}$

and an oxidation reaction: $\mathrm{A}^{+} + \mathrm{e}^{-} \rightleftharpoons \mathrm{A}$

(written this way by convention) the electrode potential for the half reactions are written as $E^\ominus\left( \mathrm{A}^{+} \vert \mathrm{A} \right)$ and $E^\ominus\left( \mathrm{B}^{+} \vert \mathrm{B} \right)$ respectively.

For the case of a metal-metal half electrode, letting M represent the metal and z be its valency, the half reaction takes the form of a reduction reaction:

$\mathrm{M}^{+z} + z \mathrm{e}^{-} \rightleftharpoons \mathrm{M} .$

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Standard EMF of an electrode	$E^\ominus, E^\ominus\left( \mathrm{X} \right)$	$\Delta E^\ominus \left( \mathrm{X} \right ) = E^{\ominus} \left( \mathrm{X} \right ) - E^{\ominus} \left( \mathrm{Def} \right )$ where Def is the standard electrode of definition, defined to have zero potential. The chosen one is hydrogen: $E^{\ominus} \left( \mathrm{H}^{+} \right ) = E^{\ominus} \left( \mathrm{H}^{+} \vert \mathrm{H} \right ) = 0$	V	[M][L]^2[I][T]^−1
Standard EMF of an electrochemical cell	$E_\mathrm{cell}^\ominus , \Delta E^\ominus$	$E_\mathrm{cell}^\ominus = E^\ominus\left( \mathrm{Cat} \right ) - E^\ominus\left( \mathrm{An} \right )$ where Cat is the cathode substance and An is the anode substance.	V	[M][L]^2[I][T]^−1
Ionic strength	I	Two definitions are used, one using molarity concentration, $I = \frac{1}{2}\sum_{i = 1}^{N} z_i^{2} \left [ {\rm X}_i \right ]$ and one using molality,[4] $I = \frac{1}{2}\sum_{i = 1}^{N} z_i^{2} b_i$ The sum is taken over all ions in the solution.	mol dm^−3 or mol dm^−3 kg^−1	[N] [L]^−3 [M]^−1
Electrochemical potential (of component i in a mixture)	$\bar{\mu}_i \,\!$	$\bar{\mu}_i = \mu - z e N_A \phi \,\!$ φ = local electrostatic potential (see below also) zi = valency (charge) of the ion i	J	[M][L]^2[T]^−2

Quantum chemistry

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Electronegativity	χ	Pauling (difference between atoms A and B): $\chi_{\rm A} - \chi_{\rm B} = ({\rm eV})^{-1/2} \sqrt{E_{\rm d}({\rm AB}) - [E_{\rm d}({\rm AA}) + E_{\rm d}({\rm BB})]/2}$ Mulliken (absolute): $\chi = 10^{-3}\left [ 187 \left ( E_{I} + E_{EA} \right ) + 170 \right ] \,$ Energies (in eV) Ed = Bond dissociation EI = Ionization EEA = Electron affinity	dimensionless	dimensionless

References

1. http://goldbook.iupac.org/index.html
2. Quantitative Chemical Analysis (4th Edition), I.M. Kolthoff, E.B. Sandell, E.J. Meehan, S. Bruckenstein, The Macmillan Co. (USA) 1969, Library of Congress Catalogue Number 69 10291
3. Quantitative Chemical Analysis (4th Edition), I.M. Kolthoff, E.B. Sandell, E.J. Meehan, S. Bruckenstein, The Macmillan Co. (USA) 1969, Library of Congress Catalogue Number 69 10291
4. Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7

Sources

• Physical chemistry, P.W. Atkins, Oxford University Press, 1978, ISBN 0-19-855148-7
• Chemistry, Matter and the Universe, R.E. Dickerson, I. Geis, W.A. Benjamin Inc. (USA), 1976, ISBN 0-8053-2369-4
• http://goldbook.iupac.org/index.html
• Chemical thermodynamics, D.J.G. Ives, University Cchemistry Series, Macdonald Technical and Scientific co. ISBN 0-356-03736-3.
• Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Principles of Chemistry, Addison-Wesley, 1974, ISBN 0-201-05229-6
• Statistical Physics (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, ISBN 978-0-471-91533-1

Further reading

• Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-1
• Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISTRY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0
• Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009, ISBN 978-1-4200-7368-3
• Properties of matter, B.H. Flowers, E. Mendoza, Manchester Physics Series, J. Wiley and Sons, 1970, ISBN 978-0-471-26498-9