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# Equilibrium chemistry

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### Equilibrium chemistry

Equilibrium chemistry is a concerned with systems in chemical equilibrium. The unifying principle is that the free energy of a system at equilibrium is the minimum possible, so that the slope of the free energy with respect to the reaction coordinate is zero.[1][2] This principle, applied to mixtures at equilibrium provides a definition of an equilibrium constant. Applications include acid-base, host-guest, metal-complex, solubility, partition, chromatography and redox equilibria.

## Thermodynamic equilibrium

A chemical system is said to be in equilibrium when the quantities of the chemical entities involved do not and cannot change in time without the application of an external influence. In this sense a system in chemical equilibrium is in a stable state. The system at chemical equilibrium will be at a constant temperature, pressure (or volume) and composition. It will be insulated from exchange of heat with the surroundings, that is, it is a closed system. A change of temperature, pressure (or volume) constitutes an external influence and the equilibrium quantities will change as a result of such a change. If there is a possibility that the composition might change, but the rate of change is negligibly slow, the system is said to be in a metastable state. The equation of chemical equilibrium can be expressed symbolically as

reactant(s) product(s)

The sign means "are in equilibrium with". This definition refers to macroscopic properties. Changes do occur at the microscopic level of atoms and molecules, but to such a minute extent that they are not measurable and in a balanced way so that the macroscopic quantities do not change. Chemical equilibrium is a dynamic state in which forward and backward reactions proceed at such rates that the macroscopic composition of the mixture is constant. Thus, equilibrium sign symbolizes the fact that reactions occur in both forward \rightharpoonup and backward \leftharpoondown directions.

A steady state, on the other hand, is not necessarily an equilibrium state in the chemical sense. For example, in a radioactive decay chain the concentrations of intermediate isotopes are constant because the rate of production is equal to the rate of decay. It is not a chemical equilibrium because the decay process occurs in one direction only.

Thermodynamic equilibrium is characterized by the free energy for the whole (closed) system being a minimum. For systems at constant volume the Helmholtz free energy is minimum and for systems at constant pressure the Gibbs free energy is minimum.[3] Thus a metastable state is one for which the free energy change between reactants and products is not minimal even though the composition does not change in time.[4]

The existence of this minimum is due to the free energy of mixing of reactants and products being always negative.[5] For ideal solutions the enthalpy of mixing is zero, so the minimum exists because the entropy of mixing is always positive.[6][7] The slope of the reaction free energy, δGr with respect to the reaction coordinate, ξ, is zero when the free energy is at its minimum value.

\delta G_r=\left(\frac{\partial G}{\partial \xi }\right)_{T,P}; \delta G_r(Eq)=0

## Equilibrium constant

Chemical potential is the partial molar free energy. The potential, μi, of the ith species in a chemical reaction is the partial derivative of the free energy with respect to the number of moles of that species, Ni

\mu_i=\left(\frac{\partial G}{\partial N_i}\right)_{T,P}

A general chemical equilibrium can be written as[note 1]

\sum_j n_j Reactant_j\rightleftharpoons \sum_k m_k Product_k

nj are the stoichiometric coefficients of the reactants in the equilibrium equation, and mj are the coefficients of the products. The value of δGr for these reactions is a function of the chemical potentials of all the species.

\delta G_r = \sum_k m_k \mu_k \, - \sum_j n_j \mu_j

The chemical potential, μi, of the ith species can be calculated in terms of its activity, ai.

\mu_i = \mu_i^\ominus + RT \ln a_i

μi is the standard chemical potential of the species, R is the gas constant and T is the temperature. Setting the sum for the reactants j to be equal to the sum for the products, k, so that δGr (Eq) = 0

\sum_j n_j(\mu_j^\ominus +RT\ln a_j)=\sum_k m_k(\mu_k^\ominus +RT\ln a_k)

Rearranging the terms,

\sum_k m_k\mu_k^\ominus-\sum_j n_j\mu_j^\ominus =-RT \left(\sum_k \ln {a_k}^{m_k}-\sum_j \ln {a_j}^{n_j}\right)
\Delta G^\ominus = -RT ln K.

This relates the standard Gibbs free energy change, ΔG to an equilibrium constant, K, the reaction quotient of activity values at equilibrium.

\Delta G^\ominus = \sum_k m_k\mu_k^\ominus-\sum_j n_j\mu_j^\ominus
\ln K= \sum_k \ln {a_k}^{m_k}-\sum_j \ln {a_j}^{n_j}; K=\frac{\prod_k {a_k}^{m_k}}{\prod_j {a_j}^{n_j}}

It follows that any equilibrium of this kind can be characterized either by the standard free energy change or by the equilibrium constant. In practice concentrations are more useful than activities. Activities can be calculated from concentrations if the activity coefficient are known, but this is rarely the case. Sometimes activity coefficients can be calculated using, for example, Pitzer equations or Specific ion interaction theory. Otherwise conditions must be adjusted so that activity coefficients do not vary much. For ionic solutions this is achieved by using a background ionic medium at a high concentration relative to the concentrations of the species in equilibrium.

If activity coefficients are unknown they may be subsumed into the equilibrium constant, which becomes a concentration quotient.[8] Each activity ai is assumed to be the product of a concentration, [Ai], and an activity coefficient, γi

a_i=[A_i]\gamma_i

This expression for activity is placed in the expression defining the equilibrium constant.[9]

K=\frac{\prod_k {a_k}^{m_k}}{\prod_j {a_j}^{n_j}} =\frac{\prod_k \left([A_k]\gamma_k\right)^{m_k}}{\prod_j \left([A_j]\gamma_j\right)^{n_j}} =\frac{\prod_k [A_k]^{m_k}}{\prod_j [A_j]^{n_j}}\times \frac{\prod_k {\gamma_k}^{m_k}}{\prod_j {\gamma_j}^{n_j}} =\frac{\prod_k [A_k]^{m_k}}{\prod_j [A_j]^{n_j}}\times \Gamma

By setting the quotient of activity coefficients, Γ, equal to one [note 2] the equilibrium constant is defined as a quotient of concentrations.

K=\frac{\prod_k [A_k]^{m_k}}{\prod_j [A_j]^{n_j}}

In more familiar notation, for a general equilibrium

\alpha A +\beta B ... \rightleftharpoons \sigma S+\tau T ...
K=\frac^2}{f_{N_2}{f_{H_2}}^3}

This reaction is strongly exothermic, so the equilibrium constant decreases with temperature. However, a temperature of around 400 °C is required in order to achieve a reasonable rate of reaction with currently available catalysts. Formation of ammonia is also favoured by high pressure, as the volume decreases when the reaction takes place. It is interesting to note that the same reaction, nitrogen fixation, occurs at ambient temperatures in nature, when the catalyst is an enzyme such as nitrogenase. Much energy is needed initially to break the N-N triple bond even though the overall reaction is exothermic.

Gas-phase equilibria occur during combustion and were studied as early as 1943 in connection with the development of the V2 rocket engine.[12]

The calculation of composition for a gaseous equilibrium at constant pressure is often carried out using ΔG values, rather than equilibrium constants.[13][14]

### Multiple equilibria

Two or more equilibria can exist at the same time. When this is so, equilibrium constants can be ascribed to individual equilibria, but they are not always unique. For example, three equilibrium constants can be defined for a dibasic acid, H2A.[15][note 3]

A^{2-} + H^+ \rightleftharpoons HA^-; K_1=\frac}
HA^- + H^+ \rightleftharpoons H_2A; K_2=\frac
A^{2-} + 2H^+ \rightleftharpoons H_2A; \beta_2=\frac}

The three constants are not independent of each other and it is easy to see that β2= K1K2. The constants K1 and K2 are stepwise constants and β is an example of an overall constant.

### Speciation

Speciation diagram for a solution of citric acid as a function of pH.

The concentrations of species in equilibrium are usually calculated under the assumption that activity coefficients are either known or can be ignored. In this case, each equilibrium constant for the formation of a complex in a set of multiple equilibria can be defined as follows

\alpha A +\beta B \ldots \rightleftharpoons A_\alpha B_\beta\ldots; K_{\alpha \beta \ldots}=\frac

The definition can easily be extended to include any number of reagents. It includes hydroxide complexes because the concentration of the hydroxide ions is related to the concentration of hydrogen ions by the self-ionization of water

[OH-] = KW [H+]-1

Stability constants defined in this way, are association constants. This can lead to some confusion as pKa values are dissociation constants. In general purpose computer programs it is customary to define all constants as association constants. The relationship between the two types of constant is given in association and dissociation constants.

In biochemistry, an oxygen molecule can bind to an iron (II) atom in a heme prosthetic group in hemoglobin. The equilibrium is usually written, denoting hemoglobin by Hb, as

Hb + O2 HbO2

but this representation is incomplete as the Bohr effect shows that the equilibrium concentrations are pH-dependent. A better representation would be

[HbH]+ + O2 HbO2 + H+

as this shows that when hydrogen ion concentration increases the equilibrium is shifted to the left in accordance with Le Chatelier's principle. Hydrogen ion concentration can be increased by the presence of carbon dioxide, which behaves as a weak acid.

H2O + CO2 HCO3- + H+

The iron atom can also bind to other molecules such as carbon monoxide. Cigarette smoke contains some carbon monoxide so the equilibrium

HbO2 + CO Hb(CO) + O2

is established in the blood of cigarette smokers.

Chelation therapy is based on the principle of using chelating ligands with a high binding selectivity for a particular metal to remove that metal from the human body.

Complexes with polyamino carboxylic acids find a wide range of applications. EDTA in particular is used extensively.

## Redox equilibria

A reduction-oxidation (redox) equilibrium can be handled in exactly the same way as any other chemical equilibrium. For example

Fe^{2+} + Ce^{4+} \rightleftharpoons Fe^{3+} + Ce^{3+}; K=\frac[Ce^{3+}]}[Ce^{4+}]}

However, in the case of redox reactions it is convenient to split the overall reaction into two half-reactions. In this example

Fe^{3+} + e^- \rightleftharpoons Fe^{2+}
Ce^{4+} + e^- \rightleftharpoons Ce^{3+}

The standard free energy change, which is related to the equilibrium constant by

\Delta G^\ominus=-RT \ln K\,

can be split into two components,

\Delta G^\ominus=\Delta G^\ominus_{Fe}+\Delta G^\ominus_{Ce}

The concentration of free electrons is effectively zero as the electrons are transferred directly from the reductant to the oxidant. The standard electrode potential, E0 for the each half-reaction is related to the standard free energy change by[30]

\Delta G^\ominus_{Fe} = -nFE^0_{Fe};\Delta G^\ominus_{Ce} = -nFE^0_{Ce}

where n is the number of electrons transferred and F is the Faraday constant. Now, the free energy for an actual reaction is given by

\Delta G=\Delta G^\ominus +RT \ln Q

where R is the gas constant and Q a reaction quotient. Strictly speaking Q is a quotient of activities, but it is common practice to use concentrations instead of activities. Therefore

E_{Fe}=E_{Fe}^0 + \frac{RT}{nF} \ln \frac}}

For any half-reaction, the redox potential of an actual mixture is given by the generalized expression[note 6]

E=E^0 + \frac{RT}{nF} \ln \frac}}

This is an example of the Nernst equation. The potential is known as a reduction potential. Standard electrode potentials are available in a table of values. Using these values, the actual electrode potential for a redox couple can be calculated as a function of the ratio of concentrations.

The equilibrium potential for a general redox half-reaction (See #Equilibrium constant above for an explanation of the symbols)

\alpha A +\beta B ... +ne^- \rightleftharpoons \sigma S+\tau T ...

is given by[31]

E=E^\ominus + \frac{RT}{nF}\ln\frac ^\sigma {\{T\}}^\tau ... } ^\alpha {\{B\}}^\beta ...}

Use of this expression allows the effect of a species not involved in the redox reaction, such as the hydrogen ion in a half-reaction such as

MnO4- + 8H+ +5e- Mn2+ + 4H2O

to be taken into account.

The equilibrium constant for a full redox reaction can be obtained from the standard redox potentials of the constituent half-reactions. At equilibrium the potential for the two half-reactions must be equal to each other and, of course, the number of electrons exchanged must be the same in the two half reactions.[32]

Redox equilibria play an important role in the electron transport chain. The various cytochromes in the chain have different standard redox potentials, each one adapted for a specific redox reaction. This allows, for example, atmospheric oxygen to be reduced in photosynthesis. A distinct family of cytochromes, the cytochrome P450 oxidases, are involved in steroidogenesis and detoxification.

## Solubility

When a solute forms a saturated solution in a solvent, the concentration of the solute, at a given temperature, is determined by the equilibrium constant at that temperature.[33]

ln K=-RT \ln \left(\frac{\sum_k {a_k}^{m_k} (solution)}{a (solid)}\right)

The activity of a pure substance in the solid state is one, by definition, so the expression simplifies to

ln K=-RT \ln \left(\sum_k {a_k}^{m_k} (solution)\right)

If the solute does not dissociate the summation is replaced by a single term, but if dissociation occurs, as with ionic substances

K_{SP}=\prod_k

For example, with Na2SO4 m1=2 and m2=1 so the solubility product is written as

K_{SP}=[Na^+]^2[SO_4^{2-}]

Concentrations, indicated by [..], are usually used in place of activities, but activity must be taken into account of the presence of another salt with no ions in common, the so-called salt effect. When another salt is present that has an ion in common, the common-ion effect comes into play, reducing the solubility of the primary solute.[34]

## Partition

When a solution of a substance in one solvent is brought into equilibrium with a second solvent that is immiscible with the first solvent, the dissolved substance may be partitioned between the two solvents. The ratio of concentrations in the two solvents is known as a partition coefficient or distribution coefficient.[note 7] The partition coefficient is defined as the ratio of the analytical concentrations of the solute in the two phases. By convention the value is reported in logarithmic form.

\log p = \log \frac

The partition coefficient is defined at a specified temperature and, if applicable, pH of the aqueous phase. Partition coefficients are very important in dielectric constant the species with no electrical charge will be the most likely one to pass from the aqueous phase to the organic phase. Even at pH 7-7.2, the range of biological pH values, the aqueous phase may support an equilibrium between more than one protonated form. Log p is determined from the analytical concentration of the substance in the aqueous phase, that is, the sum of the concentration of the different species in equilibrium.

Solvent extraction is used extensively in separation and purification processes. In its simplest form a reaction is performed in an organic solvent and unwanted by-products are removed by extraction into water at a particular pH.

A metal ion may be extracted from an aqueous phase into an organic phase in which the salt is not soluble, by adding a ion pair. The additional ligand is not always required. For example, uranyl nitrate, UO2(NO3)2, is soluble in diethyl ether because the solvent itself acts as a ligand. This property was used in the past for separating uranium from other metals whose salts are not soluble in ether. Currently extraction into kerosene is preferred, using a ligand such as tri-n-butyl phosphate, TBP. In the PUREX process, which is commonly used in nuclear reprocessing, uranium(VI) is extracted from strong nitric acid as the electrically neutral complex [UO2(TBP)2(NO3)2]. The strong nitric acid provides a high concentration of nitrate ions which pushes the equilibrium in favour of the weak nitrato complex. Uranium is recovered by back-extraction (stripping) into weak nitric acid. Plutonium(IV) forms a similar complex, [PuO2(TBP)2(NO3)2] and the plutonium in this complex can be reduced to separate it from uranium.

Another important application of solvent extraction is in the separation of the lanthanoids. This process also uses TBP and the complexes are extracted into kerosene. Separation is achieved because the stability constant for the formation of the TBP complex increases as the size of the lanthanoid ion decreases.

An instance of ion-pair extraction is in the use of a ligand to enable oxidation by crown ether is added to an aqueous solution of KMnO4, it forms a hydrophobic complex with the potassium cation which allows the uncharged ion-pair, }} is sometimes used, as in Nernst equation

• ^ The distinction between a partition coefficient and a distribution coefficient is of historical significance only.
• ^ Feeding babies formula made up with sodium rich water can lead to hypernatremia.
• ## References

1. ^ Denbeigh, K (1981). The principles of chemical equilibrium (4th. ed.). Cambridge, U.K.: Cambridge University Press.
2. ^ de Nevers, N. (2002). Physical and Chemical Equilibrium for Chemical Engineers.
3. ^ Denbigh, Chapter 4
4. ^ Denbigh, Chapter 5
5. ^ Atkins, p 203
6. ^ Atkins, p 149
7. ^ Schultz, M.J. (1999). "Why Equilibrium? Understanding the Role of Entropy of Mixing". J. Chem. Educ. 76 (10): 1391.
8. ^ a b Rossotti, F.J.C; Rossotti, H. (1961). The Determination of Stability Constants. McGraw-Hill. Chapter 2, Activity and concentration quotients
9. ^ Atkins, p 208
10. ^ Blandamer, M.J. (1992). Chemical equilibria in solution : dependence of rate and equilibrium constants on temperature and pressure. New York: Ellis Horwood/PTR Prentice Hall.
11. ^ Atkins, p 111
12. ^ Damköhler, G; Edse, R. (1943). "Composition of dissociating combustion gases and the calculation of simultaneous equilibria". Z. Elektrochem. 49: 178–802.
13. ^ Van Zeggeren, F.; Storey, S. H. (1970). The computation of chemical equilibria. London: Cambridge University Press.
14. ^ Smith, W.R.; Missen, R.W. (1991). Chemical reaction equilibrium analysis : theory and algorithms. Malabar, Fla.: Krieger.
15. ^ *Hartley, F.R.; Burgess, C.; Alcock., R. M. (1980). Solution equilibria. New York : Halsted Press: Ellis Horwood.
16. ^ a b Leggett, D.J. (Editor) (1985). Computational methods for the determination of formation constants. New York: Plenum Press.
17. ^ Martell, A.E.; Motekaitis. (1992). Determination and use of stability constants. R.J. (2nd ed.). New York: VCH Publishers.
18. ^ Bell, R.P. (1973). The Proton in Chemistry (2nd ed.). London: Chapman & Hall. Includes discussion of many organic Brønsted acids.
19. ^ Shriver, D.F; Atkins, P.W. (1999). Inorganic Chemistry (3rd ed.). Oxford: Oxford University Press. Chapter 5: Acids and Bases
20. ^ Housecroft, C.E.; Sharpe, A.G. (2008). Inorganic Chemistry (3rd ed.). Prentice Hall. Chapter 6: Acids, Bases and Ions in Aqueous Solution
21. ^ Headrick, J.M.; Diken, E.G.; Walters, R. S.; Hammer, N. I.; Christie, .A.; Cui, J.; Myshakin, E.M.; Duncan, M.A.; Johnson, M.A.; Jordan, K.D. (2005). "Spectral Signatures of Hydrated Proton Vibrations in Water Clusters". Science 308 (5729): 1765–69.
22. ^ Smiechowski, M.; Stangret, J. (2006). "Proton hydration in aqueous solution: Fourier transform infrared studies of HDO spectra". J. Chem. Phys. 125 (20): 204508–204522.
23. ^ Burgess, J. (1978). Metal Ions in Solution. Ellis Horwood. Section 9.1 "Acidity of Solvated Cations" lists many pKa values.
24. ^ Lehn, J.-M. (1995). Supramolecular Chemistry. Wiley-VCH.
25. ^ Steed, J.W.; Atwood, L.J. (2000). Supramolecular chemistry. Wiley.
26. ^ Cattrall, R.W. (1997). Chemical sensors. Oxford University Press.
27. ^ "Drug discovery today". Retrieved 23 March 2010.
28. ^ Beck, M.T.; Nagypál, I. (1990). Chemistry of Complex Equilibria. Horwood. Section 2.2, Types of complex equilibrium constants
29. ^ Hartley, F.R.; Burgess, C.; Alcock., R. M. (1980). Solution equilibria. New York : Halsted Press: Ellis Horwood.
30. ^ Atkins, Chapter 7, section "Equilibrium electrochemistry"
31. ^ Mendham, pp 59-64
32. ^ Mendham, section 2.33, p63 for details
33. ^ Hefter, G.T.; Tomkins, R.P.T. (editors), (2003). The Experimental Determination of Solubilities. Wiley.
34. ^ Mendham, pp 37-45
35. ^ Skoog, D.A; West, D.M.; Holler, J.F.; Crouch, S.R. (2004). Fundamentals of Analytical Chemistry (8th ed.). Thomson Brooks/Cole. Section 30E, Chromatographic separations
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