Franck–Condon principle

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https://en.wikipedia.org/wiki/Franck-Condon_principle

Figure 1. Franck–Condon principle energy diagram. Since electronic transitions are very fast compared with nuclear motions, the vibrational states to and from which absorption and emission occur are those that correspond to a minimal change in the nuclear coordinates. As a result, both absorption and emission produce molecules in vibrationally excited states. The potential wells are shown favoring transitions with changes in ν.

The Franck–Condon principle is a rule in spectroscopy and quantum chemistry that explains the intensity of vibronic transitions. Vibronic transitions are the simultaneous changes in electronic and vibrational energy levels of a molecule due to the absorption or emission of a photon of the appropriate energy. The principle states that during an electronic transition, a change from one vibrational energy level to another will be more likely to happen if the two vibrational wave functions overlap more significantly.

Overview

Figure 2. Schematic representation of the absorption and fluorescence spectra corresponding to the energy diagram in Figure 1. The symmetry is due to the equal shape of the ground and excited state potential wells. The narrow lines can usually only be observed in the spectra of dilute gases. The darker curves represent the inhomogeneous broadening of the same transitions as occurs in liquids and solids. Electronic transitions between the lowest vibrational levels of the electronic states (the 0–0 transition) have the same energy in both absorption and fluorescence.

 

Figure 3. Semiclassical pendulum analogy of the Franck–Condon principle. Vibronic transitions are allowed at the classical turning points because both the momentum and the nuclear coordinates correspond in the two represented energy levels. In this illustration, the 0–2 vibrational transitions are favored.

The Franck–Condon principle has a well-established semiclassical interpretation based on the original contributions of James Franck. Electronic transitions are relatively instantaneous compared with the time scale of nuclear motions, therefore if the molecule is to move to a new vibrational level during the electronic transition, this new vibrational level must be instantaneously compatible with the nuclear positions and momenta of the vibrational level of the molecule in the originating electronic state. In the semiclassical picture of vibrations (oscillations) of a simple harmonic oscillator, the necessary conditions can occur at the turning points, where the momentum is zero.

Classically, the Franck–Condon principle is the approximation that an electronic transition is most likely to occur without changes in the positions of the nuclei in the molecular entity and its environment. The resulting state is called a Franck–Condon state, and the transition involved, a vertical transition. The quantum mechanical formulation of this principle is that the intensity of a vibronic transition is proportional to the square of the overlap integral between the vibrational wavefunctions of the two states that are involved in the transition.

— IUPAC Compendium of Chemical Terminology, 2nd Edition (1997)

In the quantum mechanical picture, the vibrational levels and vibrational wavefunctions are those of quantum harmonic oscillators, or of more complex approximations to the potential energy of molecules, such as the Morse potential. Figure 1 illustrates the Franck–Condon principle for vibronic transitions in a molecule with Morse-like potential energy functions in both the ground and excited electronic states. In the low temperature approximation, the molecule starts out in the v = 0 vibrational level of the ground electronic state and upon absorbing a photon of the necessary energy, makes a transition to the excited electronic state. The electron configuration of the new state may result in a shift of the equilibrium position of the nuclei constituting the molecule. In the figure this shift in nuclear coordinates between the ground and the first excited state is labeled as q ₀₁. In the simplest case of a diatomic molecule the nuclear coordinates axis refers to the internuclear separation. The vibronic transition is indicated by a vertical arrow due to the assumption of constant nuclear coordinates during the transition. The probability that the molecule can end up in any particular vibrational level is proportional to the square of the (vertical) overlap of the vibrational wavefunctions of the original and final state (see Quantum mechanical formulation section below). In the electronic excited state molecules quickly relax to the lowest vibrational level of the lowest electronic excitation state (Kasha's rule), and from there can decay to the electronic ground state via photon emission. The Franck–Condon principle is applied equally to absorption and to fluorescence.

The applicability of the Franck–Condon principle in both absorption and fluorescence, along with Kasha's rule leads to an approximate mirror symmetry shown in Figure 2. The vibrational structure of molecules in a cold, sparse gas is most clearly visible due to the absence of inhomogeneous broadening of the individual transitions. Vibronic transitions are drawn in Figure 2 as narrow, equally spaced Lorentzian lineshapes. Equal spacing between vibrational levels is only the case for the parabolic potential of simple harmonic oscillators, in more realistic potentials, such as those shown in Figure 1, energy spacing decreases with increasing vibrational energy. Electronic transitions to and from the lowest vibrational states are often referred to as 0–0 (zero zero) transitions and have the same energy in both absorption and fluorescence.

Development of the principle

In a report published in 1926 in Transactions of the Faraday Society, James Franck was concerned with the mechanisms of photon-induced chemical reactions. The presumed mechanism was the excitation of a molecule by a photon, followed by a collision with another molecule during the short period of excitation. The question was whether it was possible for a molecule to break into photoproducts in a single step, the absorption of a photon, and without a collision. In order for a molecule to break apart, it must acquire from the photon a vibrational energy exceeding the dissociation energy, that is, the energy to break a chemical bond. However, as was known at the time, molecules will only absorb energy corresponding to allowed quantum transitions, and there are no vibrational levels above the dissociation energy level of the potential well. High-energy photon absorption leads to a transition to a higher electronic state instead of dissociation. In examining how much vibrational energy a molecule could acquire when it is excited to a higher electronic level, and whether this vibrational energy could be enough to immediately break apart the molecule, he drew three diagrams representing the possible changes in binding energy between the lowest electronic state and higher electronic states.

Diagram I. shows a great weakening of the binding on a transition from the normal state n to the excited states a and a'. Here we have D > D' and D' > D". At the same time the equilibrium position of the nuclei moves with the excitation to greater values of r. If we go from the equilibrium position (the minimum of potential energy) of the n curve vertically [emphasis added] upwards to the a curves in Diagram I. the particles will have a potential energy greater than D' and will fly apart. In this case we have a very great change in the oscillation energy on excitation by light...

— James Franck, 1926

James Franck recognized that changes in vibrational levels could be a consequence of the instantaneous nature of excitation to higher electronic energy levels and a new equilibrium position for the nuclear interaction potential. Edward Condon extended this insight beyond photoreactions in a 1926 Physical Review article titled "A Theory of Intensity Distribution in Band Systems". Here he formulates the semiclassical formulation in a manner quite similar to its modern form. The first joint reference to both Franck and Condon in regards to the new principle appears in the same 1926 issue of Physical Review in an article on the band structure of carbon monoxide by Raymond Birge.

Figure 5. Figure 1 in Edward Condon's first publication on what is now the Franck–Condon principle [Condon 1926]. Condon chose to superimpose the potential curves to illustrate the method of estimating vibrational transitions.

Quantum mechanical formulation

Consider an electrical dipole transition from the initial vibrational state (υ) of the ground electronic level (ε), ${\displaystyle |\epsilon v\rangle }$, to some vibrational state (υ′) of an excited electronic state (ε′), ${\displaystyle |\epsilon 'v'\rangle }$ (see bra–ket notation). The molecular dipole operator μ is determined by the charge (−e) and locations (r_i) of the electrons as well as the charges (+Z_je) and locations (R_j) of the nuclei:

${\displaystyle {\boldsymbol {\mu }}={\boldsymbol {\mu }}_{e}+{\boldsymbol {\mu }}_{N}=-e\sum \limits _{i}{\boldsymbol {r}}_{i}+e\sum \limits _{j}Z_{j}{\boldsymbol {R}}_{j}.}$

The probability amplitude P for the transition between these two states is given by

${\displaystyle P=\left\langle \psi '\right|{\boldsymbol {\mu }}\left|\psi \right\rangle =\int {\psi '^{*}}{\boldsymbol {\mu }}\psi \,d\tau ,}$

where ${\displaystyle \psi }$ and ${\displaystyle \psi '}$ are, respectively, the overall wavefunctions of the initial and final state. The overall wavefunctions are the product of the individual vibrational (depending on spatial coordinates of the nuclei) and electronic space and spin wavefunctions:

${\displaystyle \psi =\psi _{e}\psi _{v}\psi _{s}.}$

This separation of the electronic and vibrational wavefunctions is an expression of the Born–Oppenheimer approximation and is the fundamental assumption of the Franck–Condon principle. Combining these equations leads to an expression for the probability amplitude in terms of separate electronic space, spin and vibrational contributions:

${\displaystyle P=\left\langle \psi _{e}'\psi _{v}'\psi _{s}'\right|{\boldsymbol {\mu }}\left|\psi _{e}\psi _{v}\psi _{s}\right\rangle =\int \psi _{e}'^{*}\psi _{v}'^{*}\psi _{s}'^{*}({\boldsymbol {\mu }}_{e}+{\boldsymbol {\mu }}_{N})\psi _{e}\psi _{v}\psi _{s}\,d\tau }$

${\displaystyle =\int \psi _{e}'^{*}\psi _{v}'^{*}\psi _{s}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\psi _{v}\psi _{s}\,d\tau +\int \psi _{e}'^{*}\psi _{v}'^{*}\psi _{s}'^{*}{\boldsymbol {\mu }}_{N}\psi _{e}\psi _{v}\psi _{s}\,d\tau }$

${\displaystyle =\underbrace {\int \psi _{v}'^{*}\psi _{v}\,d\tau _{n}} _{\displaystyle {{\text{Franck–Condon}} \atop {\text{factor}}}}\underbrace {\int \psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\,d\tau _{e}} _{\displaystyle {{\text{orbital}} \atop {\text{selection rule}}}}\underbrace {\int \psi _{s}'^{*}\psi _{s}\,d\tau _{s}} _{\displaystyle {{\text{spin}} \atop {\text{selection rule}}}}+\underbrace {\int \psi _{e}'^{*}\psi _{e}\,d\tau _{e}} _{\displaystyle 0}\int \psi _{v}'^{*}{\boldsymbol {\mu }}_{N}\psi _{v}\,d\tau _{v}\int \psi _{s}'^{*}\psi _{s}\,d\tau _{s}.}$

The spin-independent part of the initial integral is here approximated as a product of two integrals:

${\displaystyle \iint \psi _{v}'^{*}\psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\psi _{v}\,d\tau _{e}d\tau _{n}\approx \int \psi _{v}'^{*}\psi _{v}\,d\tau _{n}\int \psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\,d\tau _{e}.}$

This factorization would be exact if the integral ${\displaystyle \int \psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\,d\tau _{e}}$ over the spatial coordinates of the electrons would not depend on the nuclear coordinates. However, in the Born–Oppenheimer approximation ${\displaystyle \psi _{e}}$ and ${\displaystyle \psi '_{e}}$ do depend (parametrically) on the nuclear coordinates, so that the integral (a so-called transition dipole surface) is a function of nuclear coordinates. Since the dependence is usually rather smooth it is neglected (i.e., the assumption that the transition dipole surface is independent of nuclear coordinates, called the Condon approximation is often allowed).

The first integral after the plus sign is equal to zero because electronic wavefunctions of different states are orthogonal. Remaining is the product of three integrals. The first integral is the vibrational overlap integral, also called the Franck–Condon factor. The remaining two integrals contributing to the probability amplitude determine the electronic spatial and spin selection rules.

The Franck–Condon principle is a statement on allowed vibrational transitions between two different electronic states; other quantum mechanical selection rules may lower the probability of a transition or prohibit it altogether. Rotational selection rules have been neglected in the above derivation. Rotational contributions can be observed in the spectra of gases but are strongly suppressed in liquids and solids.

It should be clear that the quantum mechanical formulation of the Franck–Condon principle is the result of a series of approximations, principally the electrical dipole transition assumption and the Born–Oppenheimer approximation. Weaker magnetic dipole and electric quadrupole electronic transitions along with the incomplete validity of the factorization of the total wavefunction into nuclear, electronic spatial and spin wavefunctions means that the selection rules, including the Franck–Condon factor, are not strictly observed. For any given transition, the value of P is determined by all of the selection rules, however spin selection is the largest contributor, followed by electronic selection rules. The Franck–Condon factor only weakly modulates the intensity of transitions, i.e., it contributes with a factor on the order of 1 to the intensity of bands whose order of magnitude is determined by the other selection rules. The table below gives the range of extinction coefficients for the possible combinations of allowed and forbidden spin and orbital selection rules.

Intensities of electronic transitions

	Range of extinction coefficient (ε) values (mol−1 cm−1)
Spin and orbitally allowed	103 to 105
Spin allowed but orbitally forbidden	100 to 103
Spin forbidden but orbitally allowed	10−5 to 100

Franck–Condon metaphors in spectroscopy

The Franck–Condon principle, in its canonical form, applies only to changes in the vibrational levels of a molecule in the course of a change in electronic levels by either absorption or emission of a photon. The physical intuition of this principle is anchored by the idea that the nuclear coordinates of the atoms constituting the molecule do not have time to change during the very brief amount of time involved in an electronic transition. However, this physical intuition can be, and is indeed, routinely extended to interactions between light-absorbing or emitting molecules (chromophores) and their environment. Franck–Condon metaphors are appropriate because molecules often interact strongly with surrounding molecules, particularly in liquids and solids, and these interactions modify the nuclear coordinates of the chromophore in ways closely analogous to the molecular vibrations considered by the Franck–Condon principle.

Figure 6. Energy diagram of an electronic transition with phonon coupling along the configurational coordinate q _i, a normal mode of the lattice. The upwards arrows represent absorption without phonons and with three phonons. The downwards arrows represent the symmetric process in emission.

Franck–Condon principle for phonons

The closest Franck–Condon analogy is due to the interaction of phonons (quanta of lattice vibrations) with the electronic transitions of chromophores embedded as impurities in the lattice. In this situation, transitions to higher electronic levels can take place when the energy of the photon corresponds to the purely electronic transition energy or to the purely electronic transition energy plus the energy of one or more lattice phonons. In the low-temperature approximation, emission is from the zero-phonon level of the excited state to the zero-phonon level of the ground state or to higher phonon levels of the ground state. Just like in the Franck–Condon principle, the probability of transitions involving phonons is determined by the overlap of the phonon wavefunctions at the initial and final energy levels. For the Franck–Condon principle applied to phonon transitions, the label of the horizontal axis of Figure 1 is replaced in Figure 6 with the configurational coordinate for a normal mode. The lattice mode ${\displaystyle q_{i}}$ potential energy in Figure 6 is represented as that of a harmonic oscillator, and the spacing between phonon levels (${\displaystyle \hbar \Omega _{i}}$) is determined by lattice parameters. Because the energy of single phonons is generally quite small, zero- or few-phonon transitions can only be observed at temperatures below about 40 kelvins.

See Zero-phonon line and phonon sideband for further details and references.

Franck–Condon principle in solvation

Figure 7. Energy diagram illustrating the Franck–Condon principle applied to the solvation of chromophores. The parabolic potential curves symbolize the interaction energy between the chromophores and the solvent. The Gaussian curves represent the distribution of this interaction energy.

Franck–Condon considerations can also be applied to the electronic transitions of chromophores dissolved in liquids. In this use of the Franck–Condon metaphor, the vibrational levels of the chromophores, as well as interactions of the chromophores with phonons in the liquid, continue to contribute to the structure of the absorption and emission spectra, but these effects are considered separately and independently.

Consider chromophores surrounded by solvent molecules. These surrounding molecules may interact with the chromophores, particularly if the solvent molecules are polar. This association between solvent and solute is referred to as solvation and is a stabilizing interaction, that is, the solvent molecules can move and rotate until the energy of the interaction is minimized. The interaction itself involves electrostatic and van der Waals forces and can also include hydrogen bonds. Franck–Condon principles can be applied when the interactions between the chromophore and the surrounding solvent molecules are different in the ground and in the excited electronic state. This change in interaction can originate, for example, due to different dipole moments in these two states. If the chromophore starts in its ground state and is close to equilibrium with the surrounding solvent molecules and then absorbs a photon that takes it to the excited state, its interaction with the solvent will be far from equilibrium in the excited state. This effect is analogous to the original Franck–Condon principle: the electronic transition is very fast compared with the motion of nuclei—the rearrangement of solvent molecules in the case of solvation. We now speak of a vertical transition, but now the horizontal coordinate is solvent-solute interaction space. This coordinate axis is often labeled as "Solvation Coordinate" and represents, somewhat abstractly, all of the relevant dimensions of motion of all of the interacting solvent molecules.

In the original Franck–Condon principle, after the electronic transition, the molecules which end up in higher vibrational states immediately begin to relax to the lowest vibrational state. In the case of solvation, the solvent molecules will immediately try to rearrange themselves in order to minimize the interaction energy. The rate of solvent relaxation depends on the viscosity of the solvent. Assuming the solvent relaxation time is short compared with the lifetime of the electronic excited state, emission will be from the lowest solvent energy state of the excited electronic state. For small-molecule solvents such as water or methanol at ambient temperature, solvent relaxation time is on the order of some tens of picoseconds whereas chromophore excited state lifetimes range from a few picoseconds to a few nanoseconds. Immediately after the transition to the ground electronic state, the solvent molecules must also rearrange themselves to accommodate the new electronic configuration of the chromophore. Figure 7 illustrates the Franck–Condon principle applied to solvation. When the solution is illuminated by light corresponding to the electronic transition energy, some of the chromophores will move to the excited state. Within this group of chromophores there will be a statistical distribution of solvent-chromophore interaction energies, represented in the figure by a Gaussian distribution function. The solvent-chromophore interaction is drawn as a parabolic potential in both electronic states. Since the electronic transition is essentially instantaneous on the time scale of solvent motion (vertical arrow), the collection of excited state chromophores is immediately far from equilibrium. The rearrangement of the solvent molecules according to the new potential energy curve is represented by the curved arrows in Figure 7. Note that while the electronic transitions are quantized, the chromophore-solvent interaction energy is treated as a classical continuum due to the large number of molecules involved. Although emission is depicted as taking place from the minimum of the excited state chromophore-solvent interaction potential, significant emission can take place before equilibrium is reached when the viscosity of the solvent is high or the lifetime of the excited state is short. The energy difference between absorbed and emitted photons depicted in Figure 7 is the solvation contribution to the Stokes shift.

Chemical equilibrium

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https://en.wikipedia.org/wiki/Chemical_equilibrium

In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the system. This state results when the forward reaction proceeds at the same rate as the reverse reaction. The reaction rates of the forward and backward reactions are generally not zero, but they are equal. Thus, there are no net changes in the concentrations of the reactants and products. Such a state is known as dynamic equilibrium.

Historical introduction

The concept of chemical equilibrium was developed in 1803, after Berthollet found that some chemical reactions are reversible. For any reaction mixture to exist at equilibrium, the rates of the forward and backward (reverse) reactions must be equal. In the following chemical equation, arrows point both ways to indicate equilibrium. A and B are reactant chemical species, S and T are product species, and α, β, σ, and τ are the stoichiometric coefficients of the respective reactants and products:

α A + β B ⇌ σ S + τ T

The equilibrium concentration position of a reaction is said to lie "far to the right" if, at equilibrium, nearly all the reactants are consumed. Conversely the equilibrium position is said to be "far to the left" if hardly any product is formed from the reactants.

Guldberg and Waage (1865), building on Berthollet's ideas, proposed the law of mass action:

${\displaystyle {\begin{aligned}{\text{forward reaction rate}}&=k_{+}{\ce {A}}^{\alpha }{\ce {B}}^{\beta }\\{\text{backward reaction rate}}&=k_{-}{\ce {S}}^{\sigma }{\ce {T}}^{\tau }\end{aligned}}}$

where A, B, S and T are active masses and k₊ and k₋ are rate constants. Since at equilibrium forward and backward rates are equal:

${\displaystyle k_{+}\left\{{\ce {A}}\right\}^{\alpha }\left\{{\ce {B}}\right\}^{\beta }=k_{-}\left\{{\ce {S}}\right\}^{\sigma }\left\{{\ce {T}}\right\}^{\tau }}$

and the ratio of the rate constants is also a constant, now known as an equilibrium constant.

${\displaystyle K_{c}={\frac {k_{+}}{k_{-}}}={\frac {\{{\ce {S}}\}^{\sigma }\{{\ce {T}}\}^{\tau }}{\{{\ce {A}}\}^{\alpha }\{{\ce {B}}\}^{\beta }}}}$

By convention, the products form the numerator. However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed (see, for example, nucleophilic aliphatic substitution by S_N1 or reaction of hydrogen and bromine to form hydrogen bromide). Equality of forward and backward reaction rates, however, is a necessary condition for chemical equilibrium, though it is not sufficient to explain why equilibrium occurs.

Despite the limitations of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the van 't Hoff equation. Adding a catalyst will affect both the forward reaction and the reverse reaction in the same way and will not have an effect on the equilibrium constant. The catalyst will speed up both reactions thereby increasing the speed at which equilibrium is reached.^[2][6]

Although the macroscopic equilibrium concentrations are constant in time, reactions do occur at the molecular level. For example, in the case of acetic acid dissolved in water and forming acetate and hydronium ions,

CH₃CO₂H + H₂O ⇌ CH
₃CO⁻
₂ + H₃O⁺

a proton may hop from one molecule of acetic acid onto a water molecule and then onto an acetate anion to form another molecule of acetic acid and leaving the number of acetic acid molecules unchanged. This is an example of dynamic equilibrium. Equilibria, like the rest of thermodynamics, are statistical phenomena, averages of microscopic behavior.

Le Châtelier's principle (1884) predicts the behavior of an equilibrium system when changes to its reaction conditions occur. If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to partially reverse the change. For example, adding more S from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward (though the equilibrium constant will stay the same).

If mineral acid is added to the acetic acid mixture, increasing the concentration of hydronium ion, the amount of dissociation must decrease as the reaction is driven to the left in accordance with this principle. This can also be deduced from the equilibrium constant expression for the reaction:

${\displaystyle K={\frac {\{{\ce {CH3CO2-}}\}\{{\ce {H3O+}}\}}{{\ce {\{CH3CO2H\}}}}}}$

If {H₃O⁺} increases {CH₃CO₂H} must increase and CH
₃CO⁻
₂ must decrease. The H₂O is left out, as it is the solvent and its concentration remains high and nearly constant.

A quantitative version is given by the reaction quotient.

J. W. Gibbs suggested in 1873 that equilibrium is attained when the Gibbs free energy of the system is at its minimum value (assuming the reaction is carried out at a constant temperature and pressure). What this means is that the derivative of the Gibbs energy with respect to reaction coordinate (a measure of the extent of reaction that has occurred, ranging from zero for all reactants to a maximum for all products) vanishes, signaling a stationary point. This derivative is called the reaction Gibbs energy (or energy change) and corresponds to the difference between the chemical potentials of reactants and products at the composition of the reaction mixture. This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (or Helmholtz energy at constant volume reactions) is the "driving force" for the composition of the mixture to change until equilibrium is reached. The equilibrium constant can be related to the standard Gibbs free energy change for the reaction by the equation

${\displaystyle \Delta _{r}G^{\ominus }=-RT\ln K_{\mathrm {eq} }}$

where R is the universal gas constant and T the temperature.

When the reactants are dissolved in a medium of high ionic strength the quotient of activity coefficients may be taken to be constant. In that case the concentration quotient, K_c,

${\displaystyle K_{\ce {c}}={\frac {[{\ce {S}}]^{\sigma }[{\ce {T}}]^{\tau }}{[{\ce {A}}]^{\alpha }[{\ce {B}}]^{\beta }}}}$

where [A] is the concentration of A, etc., is independent of the analytical concentration of the reactants. For this reason, equilibrium constants for solutions are usually determined in media of high ionic strength. K_c varies with ionic strength, temperature and pressure (or volume). Likewise K_p for gases depends on partial pressure. These constants are easier to measure and encountered in high-school chemistry courses.

Thermodynamics

At constant temperature and pressure, one must consider the Gibbs free energy, G, while at constant temperature and volume, one must consider the Helmholtz free energy, A, for the reaction; and at constant internal energy and volume, one must consider the entropy, S, for the reaction.

The constant volume case is important in geochemistry and atmospheric chemistry where pressure variations are significant. Note that, if reactants and products were in standard state (completely pure), then there would be no reversibility and no equilibrium. Indeed, they would necessarily occupy disjoint volumes of space. The mixing of the products and reactants contributes a large entropy increase (known as entropy of mixing) to states containing equal mixture of products and reactants and gives rise to a distinctive minimum in the Gibbs energy as a function of the extent of reaction. The standard Gibbs energy change, together with the Gibbs energy of mixing, determine the equilibrium state.

In this article only the constant pressure case is considered. The relation between the Gibbs free energy and the equilibrium constant can be found by considering chemical potentials.

At constant temperature and pressure in the absence of an applied voltage, the Gibbs free energy, G, for the reaction depends only on the extent of reaction: ξ (Greek letter xi), and can only decrease according to the second law of thermodynamics. It means that the derivative of G with respect to ξ must be negative if the reaction happens; at the equilibrium this derivative is equal to zero.

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=0~}$:     equilibrium

In order to meet the thermodynamic condition for equilibrium, the Gibbs energy must be stationary, meaning that the derivative of G with respect to the extent of reaction, ξ, must be zero. It can be shown that in this case, the sum of chemical potentials times the stoichiometric coefficients of the products is equal to the sum of those corresponding to the reactants. Therefore, the sum of the Gibbs energies of the reactants must be the equal to the sum of the Gibbs energies of the products.

${\displaystyle \alpha \mu _{\mathrm {A} }+\beta \mu _{\mathrm {B} }=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }\,}$

where μ is in this case a partial molar Gibbs energy, a chemical potential. The chemical potential of a reagent A is a function of the activity, {A} of that reagent.

${\displaystyle \mu _{\mathrm {A} }=\mu _{A}^{\ominus }+RT\ln\{\mathrm {A} \}\,}$

(where μ^o
_A is the standard chemical potential).

The definition of the Gibbs energy equation interacts with the fundamental thermodynamic relation to produce

${\displaystyle dG=Vdp-SdT+\sum _{i=1}^{k}\mu _{i}dN_{i}}$.

Inserting dN_i = ν_i dξ into the above equation gives a stoichiometric coefficient (${\displaystyle \nu _{i}~}$) and a differential that denotes the reaction occurring to an infinitesimal extent (dξ). At constant pressure and temperature the above equations can be written as

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\sum _{i=1}^{k}\mu _{i}\nu _{i}=\Delta _{\mathrm {r} }G_{T,p}}$ which is the "Gibbs free energy change for the reaction .

This results in:

${\displaystyle \Delta _{\mathrm {r} }G_{T,p}=\sigma \mu _{\mathrm {S} }+\tau \mu _{\mathrm {T} }-\alpha \mu _{\mathrm {A} }-\beta \mu _{\mathrm {B} }\,}$.

By substituting the chemical potentials:

${\displaystyle \Delta _{\mathrm {r} }G_{T,p}=(\sigma \mu _{\mathrm {S} }^{\ominus }+\tau \mu _{\mathrm {T} }^{\ominus })-(\alpha \mu _{\mathrm {A} }^{\ominus }+\beta \mu _{\mathrm {B} }^{\ominus })+(\sigma RT\ln\{\mathrm {S} \}+\tau RT\ln\{\mathrm {T} \})-(\alpha RT\ln\{\mathrm {A} \}+\beta RT\ln\{\mathrm {B} \})}$,

the relationship becomes:

${\displaystyle \Delta _{\mathrm {r} }G_{T,p}=\sum _{i=1}^{k}\mu _{i}^{\ominus }\nu _{i}+RT\ln {\frac {\{\mathrm {S} \}^{\sigma }\{\mathrm {T} \}^{\tau }}{\{\mathrm {A} \}^{\alpha }\{\mathrm {B} \}^{\beta }}}}$

${\displaystyle \sum _{i=1}^{k}\mu _{i}^{\ominus }\nu _{i}=\Delta _{\mathrm {r} }G^{\ominus }}$:

which is the standard Gibbs energy change for the reaction that can be calculated using thermodynamical tables. The reaction quotient is defined as:

${\displaystyle Q_{\mathrm {r} }={\frac {\{\mathrm {S} \}^{\sigma }\{\mathrm {T} \}^{\tau }}{\{\mathrm {A} \}^{\alpha }\{\mathrm {B} \}^{\beta }}}}$

Therefore,

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G_{T,p}=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln Q_{\mathrm {r} }}$

At equilibrium:

${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G_{T,p}=0}$

leading to:

${\displaystyle 0=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln K_{\mathrm {eq} }}$

and

${\displaystyle \Delta _{\mathrm {r} }G^{\ominus }=-RT\ln K_{\mathrm {eq} }}$

Obtaining the value of the standard Gibbs energy change, allows the calculation of the equilibrium constant.

Addition of reactants or products

For a reactional system at equilibrium: Q_r = K_eq; ξ = ξ_eq.

• If the activities of constituents are modified, the value of the reaction quotient changes and becomes different from the equilibrium constant: Q_r ≠ K_eq
  $${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=\Delta _{\mathrm {r} }G^{\ominus }+RT\ln Q_{\mathrm {r} }~}$$

  
  and
  $${\displaystyle \Delta _{\mathrm {r} }G^{\ominus }=-RT\ln K_{eq}~}$$

  
  then
  $${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}=RT\ln \left({\frac {Q_{\mathrm {r} }}{K_{\mathrm {eq} }}}\right)~}$$

  
• If activity of a reagent i increases
  $${\displaystyle Q_{\mathrm {r} }={\frac {\prod (a_{j})^{\nu _{j}}}{\prod (a_{i})^{\nu _{i}}}}~,}$$

  
  the reaction quotient decreases. Then
  $${\displaystyle Q_{\mathrm {r} }<K_{\mathrm {eq} }~}$$

  
  and
  $${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}<0~}$$

  
  The reaction will shift to the right (i.e. in the forward direction, and thus more products will form).
• If activity of a product j increases, then
  $${\displaystyle Q_{\mathrm {r} }>K_{\mathrm {eq} }~}$$

  
  and
  $${\displaystyle \left({\frac {dG}{d\xi }}\right)_{T,p}>0~}$$

  
  The reaction will shift to the left (i.e. in the reverse direction, and thus less products will form).

Note that activities and equilibrium constants are dimensionless numbers.

Treatment of activity

The expression for the equilibrium constant can be rewritten as the product of a concentration quotient, K_c and an activity coefficient quotient, Γ.

${\displaystyle K={\frac {[\mathrm {S} ]^{\sigma }[\mathrm {T} ]^{\tau }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}\times {\frac {{\gamma _{\mathrm {S} }}^{\sigma }{\gamma _{\mathrm {T} }}^{\tau }...}{{\gamma _{\mathrm {A} }}^{\alpha }{\gamma _{\mathrm {B} }}^{\beta }...}}=K_{\mathrm {c} }\Gamma }$

[A] is the concentration of reagent A, etc. It is possible in principle to obtain values of the activity coefficients, γ. For solutions, equations such as the Debye–Hückel equation or extensions such as Davies equation Specific ion interaction theory or Pitzer equations may be used. However this is not always possible. It is common practice to assume that Γ is a constant, and to use the concentration quotient in place of the thermodynamic equilibrium constant. It is also general practice to use the term equilibrium constant instead of the more accurate concentration quotient. This practice will be followed here.

For reactions in the gas phase partial pressure is used in place of concentration and fugacity coefficient in place of activity coefficient. In the real world, for example, when making ammonia in industry, fugacity coefficients must be taken into account. Fugacity, f, is the product of partial pressure and fugacity coefficient. The chemical potential of a species in the real gas phase is given by

${\displaystyle \mu =\mu ^{\ominus }+RT\ln \left({\frac {f}{\mathrm {bar} }}\right)=\mu ^{\ominus }+RT\ln \left({\frac {p}{\mathrm {bar} }}\right)+RT\ln \gamma }$

so the general expression defining an equilibrium constant is valid for both solution and gas phases.

Concentration quotients

In aqueous solution, equilibrium constants are usually determined in the presence of an "inert" electrolyte such as sodium nitrate, NaNO₃, or potassium perchlorate, KClO₄. The ionic strength of a solution is given by

${\displaystyle I={\frac {1}{2}}\sum _{i=1}^{N}c_{i}z_{i}^{2}}$

where c_i and z_i stand for the concentration and ionic charge of ion type i, and the sum is taken over all the N types of charged species in solution. When the concentration of dissolved salt is much higher than the analytical concentrations of the reagents, the ions originating from the dissolved salt determine the ionic strength, and the ionic strength is effectively constant. Since activity coefficients depend on ionic strength, the activity coefficients of the species are effectively independent of concentration. Thus, the assumption that Γ is constant is justified. The concentration quotient is a simple multiple of the equilibrium constant.

${\displaystyle K_{\mathrm {c} }={\frac {K}{\Gamma }}}$

However, K_c will vary with ionic strength. If it is measured at a series of different ionic strengths, the value can be extrapolated to zero ionic strength. The concentration quotient obtained in this manner is known, paradoxically, as a thermodynamic equilibrium constant.

Before using a published value of an equilibrium constant in conditions of ionic strength different from the conditions used in its determination, the value should be adjusted.

Metastable mixtures

A mixture may appear to have no tendency to change, though it is not at equilibrium. For example, a mixture of SO₂ and O₂ is metastable as there is a kinetic barrier to formation of the product, SO₃.

2 SO₂ + O₂ ⇌ 2 SO₃

The barrier can be overcome when a catalyst is also present in the mixture as in the contact process, but the catalyst does not affect the equilibrium concentrations.

Likewise, the formation of bicarbonate from carbon dioxide and water is very slow under normal conditions

CO₂ + 2 H₂O ⇌ HCO⁻
₃ + H₃O⁺

but almost instantaneous in the presence of the catalytic enzyme carbonic anhydrase.

Pure substances

When pure substances (liquids or solids) are involved in equilibria their activities do not appear in the equilibrium constant because their numerical values are considered one.

Applying the general formula for an equilibrium constant to the specific case of a dilute solution of acetic acid in water one obtains

CH₃CO₂H + H₂O ⇌ CH₃CO₂⁻ + H₃O⁺

${\displaystyle K_{\mathrm {c} }={\frac {\mathrm {[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]} }{\mathrm {[{CH_{3}CO_{2}H}][{H_{2}O}]} }}}$

For all but very concentrated solutions, the water can be considered a "pure" liquid, and therefore it has an activity of one. The equilibrium constant expression is therefore usually written as

${\displaystyle K={\frac {\mathrm {[{CH_{3}CO_{2}}^{-}][{H_{3}O}^{+}]} }{\mathrm {[{CH_{3}CO_{2}H}]} }}=K_{\mathrm {c} }}$.

A particular case is the self-ionization of water

2 H₂O ⇌ H₃O⁺ + OH⁻

Because water is the solvent, and has an activity of one, the self-ionization constant of water is defined as

${\displaystyle K_{\mathrm {w} }=\mathrm {[H^{+}][OH^{-}]} }$

It is perfectly legitimate to write [H⁺] for the hydronium ion concentration, since the state of solvation of the proton is constant (in dilute solutions) and so does not affect the equilibrium concentrations. K_w varies with variation in ionic strength and/or temperature.

The concentrations of H⁺ and OH⁻ are not independent quantities. Most commonly [OH⁻] is replaced by K_w[H⁺]⁻¹ in equilibrium constant expressions which would otherwise include hydroxide ion.

Solids also do not appear in the equilibrium constant expression, if they are considered to be pure and thus their activities taken to be one. An example is the Boudouard reaction:

2 CO ⇌ CO₂ + C

for which the equation (without solid carbon) is written as:

${\displaystyle K_{\mathrm {c} }={\frac {\mathrm {[CO_{2}]} }{\mathrm {[CO]^{2}} }}}$

Multiple equilibria

Consider the case of a dibasic acid H₂A. When dissolved in water, the mixture will contain H₂A, HA⁻ and A²⁻. This equilibrium can be split into two steps in each of which one proton is liberated.

${\displaystyle {\begin{array}{rl}{\ce {H2A <=> HA^- + H+}}:&K_{1}={\frac {{\ce {[HA-][H+]}}}{{\ce {[H2A]}}}}\\{\ce {HA- <=> A^2- + H+}}:&K_{2}={\frac {{\ce {[A^{2-}][H+]}}}{{\ce {[HA-]}}}}\end{array}}}$

K₁ and K₂ are examples of stepwise equilibrium constants. The overall equilibrium constant, β_D, is product of the stepwise constants.

${\displaystyle {\ce {{H2A}<=>{A^{2-}}+{2H+}}}}$:     ${\displaystyle \beta _{{\ce {D}}}={\frac {{\ce {[A^{2-}][H^+]^2}}}{{\ce {[H_2A]}}}}=K_{1}K_{2}}$

Note that these constants are dissociation constants because the products on the right hand side of the equilibrium expression are dissociation products. In many systems, it is preferable to use association constants.

${\displaystyle {\begin{array}{ll}{\ce {A^2- + H+ <=> HA-}}:&\beta _{1}={\frac {{\ce {[HA^-]}}}{{\ce {[A^{2-}][H+]}}}}\\{\ce {A^2- + 2H+ <=> H2A}}:&\beta _{2}={\frac {{\ce {[H2A]}}}{{\ce {[A^{2-}][H+]^2}}}}\end{array}}}$

β₁ and β₂ are examples of association constants. Clearly β₁ = 1/K₂ and β₂ = 1/β_D; log β₁ = pK₂ and log β₂ = pK₂ + pK₁ For multiple equilibrium systems, also see: theory of Response reactions.

Effect of temperature

The effect of changing temperature on an equilibrium constant is given by the van 't Hoff equation

${\displaystyle {\frac {d\ln K}{dT}}={\frac {\Delta H_{\mathrm {m} }^{\ominus }}{RT^{2}}}}$

Thus, for exothermic reactions (ΔH is negative), K decreases with an increase in temperature, but, for endothermic reactions, (ΔH is positive) K increases with an increase in temperature. An alternative formulation is

${\displaystyle {\frac {d\ln K}{d(T^{-1})}}=-{\frac {\Delta H_{\mathrm {m} }^{\ominus }}{R}}}$

At first sight this appears to offer a means of obtaining the standard molar enthalpy of the reaction by studying the variation of K with temperature. In practice, however, the method is unreliable because error propagation almost always gives very large errors on the values calculated in this way.

Effect of electric and magnetic fields

The effect of electric field on equilibrium has been studied by Manfred Eigen among others.

Types of equilibrium

Haber–Bosch process

1. N₂ (g) ⇌ N₂ (adsorbed)
2. N₂ (adsorbed) ⇌ 2 N (adsorbed)
3. H₂ (g) ⇌ H₂ (adsorbed)
4. H₂ (adsorbed) ⇌ 2 H (adsorbed)
5. N (adsorbed) + 3 H(adsorbed) ⇌ NH₃ (adsorbed)
6. NH₃ (adsorbed) ⇌ NH₃ (g)

Equilibrium can be broadly classified as heterogeneous and homogeneous equilibrium. Homogeneous equilibrium consists of reactants and products belonging in the same phase whereas heterogeneous equilibrium comes into play for reactants and products in different phases.

• In the gas phase: rocket engines
• The industrial synthesis such as ammonia in the Haber–Bosch process (depicted right) takes place through a succession of equilibrium steps including adsorption processes
• Atmospheric chemistry
• Seawater and other natural waters: chemical oceanography
• Distribution between two phases
  • log D distribution coefficient: important for pharmaceuticals where lipophilicity is a significant property of a drug
  • Liquid–liquid extraction, Ion exchange, Chromatography
  • Solubility product
  • Uptake and release of oxygen by hemoglobin in blood
• Acid–base equilibria: acid dissociation constant, hydrolysis, buffer solutions, indicators, acid–base homeostasis
• Metal–ligand complexation: sequestering agents, chelation therapy, MRI contrast reagents, Schlenk equilibrium
• Adduct formation: host–guest chemistry, supramolecular chemistry, molecular recognition, dinitrogen tetroxide
• In certain oscillating reactions, the approach to equilibrium is not asymptotically but in the form of a damped oscillation.
• The related Nernst equation in electrochemistry gives the difference in electrode potential as a function of redox concentrations.
• When molecules on each side of the equilibrium are able to further react irreversibly in secondary reactions, the final product ratio is determined according to the Curtin–Hammett principle.

In these applications, terms such as stability constant, formation constant, binding constant, affinity constant, association constant and dissociation constant are used. In biochemistry, it is common to give units for binding constants, which serve to define the concentration units used when the constant's value was determined.

Composition of a mixture

When the only equilibrium is that of the formation of a 1:1 adduct as the composition of a mixture, there are many ways that the composition of a mixture can be calculated. For example, see ICE table for a traditional method of calculating the pH of a solution of a weak acid.

There are three approaches to the general calculation of the composition of a mixture at equilibrium.

1. The most basic approach is to manipulate the various equilibrium constants until the desired concentrations are expressed in terms of measured equilibrium constants (equivalent to measuring chemical potentials) and initial conditions.
2. Minimize the Gibbs energy of the system.
3. Satisfy the equation of mass balance. The equations of mass balance are simply statements that demonstrate that the total concentration of each reactant must be constant by the law of conservation of mass.

Mass-balance equations

In general, the calculations are rather complicated or complex. For instance, in the case of a dibasic acid, H₂A dissolved in water the two reactants can be specified as the conjugate base, A²⁻, and the proton, H⁺. The following equations of mass-balance could apply equally well to a base such as 1,2-diaminoethane, in which case the base itself is designated as the reactant A:

${\displaystyle T_{\mathrm {A} }=\mathrm {[A]+[HA]+[H_{2}A]} \,}$

${\displaystyle T_{\mathrm {H} }=\mathrm {[H]+[HA]+2[H_{2}A]-[OH]} \,}$

with T_A the total concentration of species A. Note that it is customary to omit the ionic charges when writing and using these equations.

When the equilibrium constants are known and the total concentrations are specified there are two equations in two unknown "free concentrations" [A] and [H]. This follows from the fact that [HA] = β₁[A][H], [H₂A] = β₂[A][H]² and [OH] = K_w[H]⁻¹

${\displaystyle T_{\mathrm {A} }=\mathrm {[A]} +\beta _{1}\mathrm {[A][H]} +\beta _{2}\mathrm {[A][H]} ^{2}\,}$

${\displaystyle T_{\mathrm {H} }=\mathrm {[H]} +\beta _{1}\mathrm {[A][H]} +2\beta _{2}\mathrm {[A][H]} ^{2}-K_{w}[\mathrm {H} ]^{-1}\,}$

so the concentrations of the "complexes" are calculated from the free concentrations and the equilibrium constants. General expressions applicable to all systems with two reagents, A and B would be

${\displaystyle T_{\mathrm {A} }=[\mathrm {A} ]+\sum _{i}p_{i}\beta _{i}[\mathrm {A} ]^{p_{i}}[\mathrm {B} ]^{q_{i}}}$

${\displaystyle T_{\mathrm {B} }=[\mathrm {B} ]+\sum _{i}q_{i}\beta _{i}[\mathrm {A} ]^{p_{i}}[\mathrm {B} ]^{q_{i}}}$

It is easy to see how this can be extended to three or more reagents.

Polybasic acids

Species concentrations during hydrolysis of the aluminium.

The composition of solutions containing reactants A and H is easy to calculate as a function of p[H]. When [H] is known, the free concentration [A] is calculated from the mass-balance equation in A.

The diagram alongside, shows an example of the hydrolysis of the aluminium Lewis acid Al³⁺_(aq) shows the species concentrations for a 5 × 10⁻⁶ M solution of an aluminium salt as a function of pH. Each concentration is shown as a percentage of the total aluminium.

Solution and precipitation

The diagram above illustrates the point that a precipitate that is not one of the main species in the solution equilibrium may be formed. At pH just below 5.5 the main species present in a 5 μM solution of Al³⁺ are aluminium hydroxides Al(OH)²⁺, AlOH⁺
₂ and Al
₁₃(OH)⁷⁺
₃₂, but on raising the pH Al(OH)₃ precipitates from the solution. This occurs because Al(OH)₃ has a very large lattice energy. As the pH rises more and more Al(OH)₃ comes out of solution. This is an example of Le Châtelier's principle in action: Increasing the concentration of the hydroxide ion causes more aluminium hydroxide to precipitate, which removes hydroxide from the solution. When the hydroxide concentration becomes sufficiently high the soluble aluminate, Al(OH)⁻
₄, is formed.

Another common instance where precipitation occurs is when a metal cation interacts with an anionic ligand to form an electrically neutral complex. If the complex is hydrophobic, it will precipitate out of water. This occurs with the nickel ion Ni²⁺ and dimethylglyoxime, (dmgH₂): in this case the lattice energy of the solid is not particularly large, but it greatly exceeds the energy of solvation of the molecule Ni(dmgH)₂.

Minimization of Gibbs energy

At equilibrium, at a specified temperature and pressure, and with no external forces, the Gibbs free energy G is at a minimum:

${\displaystyle dG=\sum _{j=1}^{m}\mu _{j}\,dN_{j}=0}$

where μ_j is the chemical potential of molecular species j, and N_j is the amount of molecular species j. It may be expressed in terms of thermodynamic activity as:

${\displaystyle \mu _{j}=\mu _{j}^{\ominus }+RT\ln {A_{j}}}$

where ${\displaystyle \mu _{j}^{\ominus }}$ is the chemical potential in the standard state, R is the gas constant T is the absolute temperature, and A_j is the activity.

For a closed system, no particles may enter or leave, although they may combine in various ways. The total number of atoms of each element will remain constant. This means that the minimization above must be subjected to the constraints:

${\displaystyle \sum _{j=1}^{m}a_{ij}N_{j}=b_{i}^{0}}$

where a_ij is the number of atoms of element i in molecule j and b⁰
_i is the total number of atoms of element i, which is a constant, since the system is closed. If there are a total of k types of atoms in the system, then there will be k such equations. If ions are involved, an additional row is added to the a_ij matrix specifying the respective charge on each molecule which will sum to zero.

This is a standard problem in optimisation, known as constrained minimisation. The most common method of solving it is using the method of Lagrange multipliers (although other methods may be used).

Define:

${\displaystyle {\mathcal {G}}=G+\sum _{i=1}^{k}\lambda _{i}\left(\sum _{j=1}^{m}a_{ij}N_{j}-b_{i}^{0}\right)=0}$

where the λ_i are the Lagrange multipliers, one for each element. This allows each of the N_j and λ_j to be treated independently, and it can be shown using the tools of multivariate calculus that the equilibrium condition is given by

${\displaystyle 0={\frac {\partial {\mathcal {G}}}{\partial N_{j}}}=\mu _{j}+\sum _{i=1}^{k}\lambda _{i}a_{ij}}$

${\displaystyle 0={\frac {\partial {\mathcal {G}}}{\partial \lambda _{i}}}=\sum _{j=1}^{m}a_{ij}N_{j}-b_{i}^{0}}$

(For proof see Lagrange multipliers.) This is a set of (m + k) equations in (m + k) unknowns (the N_j and the λ_i) and may, therefore, be solved for the equilibrium concentrations N_j as long as the chemical activities are known as functions of the concentrations at the given temperature and pressure. (In the ideal case, activities are proportional to concentrations.) (See Thermodynamic databases for pure substances.) Note that the second equation is just the initial constraints for minimization.

This method of calculating equilibrium chemical concentrations is useful for systems with a large number of different molecules. The use of k atomic element conservation equations for the mass constraint is straightforward, and replaces the use of the stoichiometric coefficient equations. The results are consistent with those specified by chemical equations. For example, if equilibrium is specified by a single chemical equation:

${\displaystyle \sum _{j=0}^{m}\nu _{j}R_{j}=0}$

where ν_j is the stoichiometric coefficient for the j th molecule (negative for reactants, positive for products) and R_j is the symbol for the j th molecule, a properly balanced equation will obey:

${\displaystyle \sum _{j=1}^{m}a_{ij}\nu _{j}=0}$

Multiplying the first equilibrium condition by ν_j and using the above equation yields:

${\displaystyle 0=\sum _{j=1}^{m}\nu _{j}\mu _{j}+\sum _{j=1}^{m}\sum _{i=1}^{k}\nu _{j}\lambda _{i}a_{ij}=\sum _{j=1}^{m}\nu _{j}\mu _{j}}$

As above, defining ΔG

${\displaystyle \Delta G=\sum _{j=1}^{m}\nu _{j}\mu _{j}=\sum _{j=1}^{m}\nu _{j}(\mu _{j}^{\ominus }+RT\ln(\{R_{j}\}))=\Delta G^{\ominus }+RT\ln \left(\prod _{j=1}^{m}\{R_{j}\}^{\nu _{j}}\right)=\Delta G^{\ominus }+RT\ln(K_{c})}$

where K_c is the equilibrium constant, and ΔG will be zero at equilibrium.

Analogous procedures exist for the minimization of other thermodynamic potentials.