Defining equation (physical chemistry)

For the detailed nature of defining equations, see Physical quantity. For a summary of thermodynamic quantities, see defining equation (physics).
Main article: 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:

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 mH2O, mwater, maq, mw (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 χF-F, χFF or χF-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. qi. 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(Xi). 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 m3 [L]3

General derived quantities

Main article: Concentration
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 Kc (all reactions) and Kp (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,
• Xi = component i of the reactant mixture,
• Yj = reactant component j of the product mixture,
• r (as an index) = number of reactant components,
• p (as an index) = number of product components,
• νi = stoichiometry number for component i in product mixture,
• ηj = stoichiometry number for component j in product mixture,
• σi = 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;

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