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Chemical equilibrium

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Chemical equilibrium

In a chemical reaction, chemical equilibrium is the state in which both reactants and products are present in concentrations which have no further tendency to change with time.[1] Usually, 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 equal. Thus, there are no net changes in the concentrations of the reactant(s) and product(s). Such a state is known as dynamic equilibrium.[2][3]

Contents

  • Historical introduction 1
  • Thermodynamics 2
    • Addition of reactants or products 2.1
    • Treatment of activity 2.2
    • Concentration quotients 2.3
    • Metastable mixtures 2.4
  • Pure substances 3
  • Multiple equilibria 4
  • Effect of temperature 5
  • Effect of electric and magnetic fields 6
  • Types of equilibrium 7
  • Composition of a mixture 8
    • Mass-balance equations 8.1
      • Polybasic acids 8.1.1
      • Solution and precipitation 8.1.2
    • Minimization of free energy 8.2
  • Symbol 9
  • See also 10
  • References 11
  • Further reading 12

Historical introduction

Burette, a common laboratory apparatus for carrying out titration, an important experimental technique in equilibrium and analytical chemistry.

The concept of chemical equilibrium was developed after Berthollet (1803) found that some chemical reactions are reversible. For any reaction mixture to exist at equilibrium, the rates of the forward and backward (reverse) reactions are equal. In the following chemical equation with arrows pointing 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:

\alpha A + \beta B \rightleftharpoons \sigma S + \tau 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:

\mbox{forward reaction rate} = k_+ {A}^\alpha{B}^\beta \,\!
\mbox{backward reaction rate} = k_{-} {S}^\sigma{T}^\tau \,\!

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:

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

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

K_c=\frac{k_+}{k_-}=\frac{\{S\}^\sigma \{T\}^\tau } {\{A\}^\alpha \{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 SN1 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 failure 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][4]

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 CH3CO2 + H3O+

a proton may hop from one molecule of acetic acid on to a water molecule and then on to 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 Chatelier's principle (1884) gives an idea of 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:

K=\frac{\{CH_3CO_2^-\}\{H_3O^+\}} {\{CH_3CO_2H\}}

If {H3O+} increases {CH3CO2H} must increase and {CH3CO2} 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 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, signalling 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.[1] 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

\Delta_rG^\ominus = -RT \ln K_{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, Kc,

K_c=\frac\right)~

  • If activity of a reagent i~ increases

Q_r = \frac{\prod (a_j)^{\nu_j}}{\prod(a_i)^{\nu_i}}~, the reaction quotient decreases.

then

Q_r < K_{eq}~ and \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

Q_r > K_{eq}~ and \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, Kc and an activity coefficient quotient, Γ.

K=\frac ^\sigma ^\tau ... } ^\alpha ^\beta ...} \times \frac }

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

K=\frac } = K_c.

A particular case is the self-ionization of water itself

H_2O + H_2O \rightleftharpoons H_3O^+ + OH^-

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

K_w = [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. Kw 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 Kw[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:[10]

2CO \rightleftharpoons CO_2 + C

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

K_c=\frac
HA^- \rightleftharpoons A^{2-} + H^+ :K_2=\frac[H^+]}

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

H_2A \rightleftharpoons A^{2-} + 2H^+ :\beta_D = \frac[H^+]^2} =K_1K_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.

A^{2-} + H^+ \rightleftharpoons HA^- :\beta_1=\frac [H^+]}
A^{2-} + 2H^+ \rightleftharpoons H_2A :\beta_2=\frac [H^+]^2}

β1 and β2 are examples of association constants. Clearly β1 = 1/K2 and β2 = 1/βD; lg β1 = pK2 and lg β2 = pK2 + pK1[11] 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

\frac {d\ln K} {dT} = \frac {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 temperature. An alternative formulation is

\frac {d\ln K} {d(1/T)} = -\frac {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

  1. In the gas phase. Rocket engines[12]
  2. The industrial synthesis such as ammonia in the Haber–Bosch process (depicted right) takes place through a succession of equilibrium steps including adsorption processes.
    Haber–Bosch process
  3. atmospheric chemistry
  4. Seawater and other natural waters: Chemical oceanography
  5. Distribution between two phases
  6. LogD-Distribution coefficient: Important for pharmaceuticals where lipophilicity is a significant property of a drug
  7. Liquid–liquid extraction, Ion exchange, Chromatography
  8. Solubility product
  9. Uptake and release of oxygen by haemoglobin in blood
  10. Acid/base equilibria: Acid dissociation constant, hydrolysis, buffer solutions, indicators, acid–base homeostasis
  11. Metal-ligand complexation: sequestering agents, chelation therapy, MRI contrast reagents, Schlenk equilibrium
  12. Adduct formation: Host–guest chemistry, supramolecular chemistry, molecular recognition, dinitrogen tetroxide
  13. In certain oscillating reactions, the approach to equilibrium is not asymptotically but in the form of a damped oscillation .[10]
  14. The related Nernst equation in electrochemistry gives the difference in electrode potential as a function of redox concentrations.
  15. 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.
  16. In these applications, terms such as stability constant, formation constant, binding constant, affinity constant, association/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 any number of 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.[13]
    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:

    T_A = [A] + [HA] +[H_2A] \,
    T_H = [H] + [HA] + 2[H_2A] - [OH] \,

    With TA 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] = Kw[H]−1

    T_A = [A] + \beta_1[A][H] + \beta_2[A][H]^2 \,
    T_H = [H] + \beta_1[A][H] + 2\beta_2[A][H]^2 - K_w[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

    T_A=[A]+\sum_i{p_i \beta_i[A]^{p_i}[B]^{q_i}}
    T_B=[B]+\sum_i{q_i \beta_i[A]^{p_i}[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[14] shows the species concentrations for a 5×10−6M 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 5μM solution of Al3+ are aluminium hydroxides Al(OH)2+, Al(OH)2+ and Al13(OH)327+, 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 Chatelier'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 free energy

    At equilibrium, G is at a minimum:

    dG= \sum_{j=1}^m \mu_j\,dN_j = 0

    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:

    \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 bi0 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.

    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, also known as undetermined multipliers (though other methods may be used).

    Define:

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

    \frac{\partial \mathcal{G}}{\partial N_j}=0     and     \frac{\partial \mathcal{G}}{\partial \lambda_i}=0

    (For proof see Lagrange multipliers)

    This is a set of (m+k) equations in (m+k) unknowns (the N_j and the \lambda_i) and may, therefore, be solved for the equilibrium concentrations N_j as long as the chemical potentials are known as functions of the concentrations at the given temperature and pressure. (See Thermodynamic databases for pure substances).

    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.[12]

    Symbol

    In Unicode, a suitable symbol is registered as .[15] It can be typed in Microsoft Windows as Alt + + 2, 1, C, C on the numeric keypad, and in most Linux distributions with Ctrl + Shift + u, 2, 1, C, C, Enter.

    See also

    References

    1. ^ a b c Peter Atkins and Julio de Paula, Atkins' Physical Chemistry, 8th edition (W.H. Freeman 2006, ISBN 0-7167-8759-8) p.200-202
    2. ^ a b Atkins, Peter W and Jones, Loretta Chemical Principles: The Quest for Insight 2nd Ed. ISBN 0-7167-9903-0
    3. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "chemical equilibrium".
    4. ^ Chemistry: Matter and Its Changes James E. Brady, Fred Senese 4th Ed. ISBN 0-471-21517-1
    5. ^
    6. ^
    7. ^ C.W. Davies, Ion Association, Butterworths, 1962
    8. ^ a b I. Grenthe and H. Wanner, Guidelines for the extrapolation to zero ionic strength
    9. ^ F.J.C. Rossotti and H. Rossotti, The Determination of Stability Constants, McGraw-Hill, 1961
    10. ^ a b c Concise Encyclopedia Chemistry 1994 ISBN 0-89925-457-8
    11. ^ M.T. Beck, Chemistry of Complex Equilibria, Van Nostrand, 1970. 2nd. Edition by M.T. Beck and I Nagypál, Akadémiai Kaidó, Budapest, 1990.
    12. ^ a b NASA Reference publication 1311 (1994), Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications
    13. ^ This approach is described in detail in W. R. Smith and R. W. Missen, Chemical Reaction Equilibrium Analysis: Theory and Algorithms, Krieger Publishing, Malabar, Fla, 1991 (a reprint, with corrections, of the same title by John Wiley & Sons, 1982). A comprehensive treatment of the theory of chemical reaction equilibria and its computation. Details at http://www.mathtrek.com/
    14. ^ The diagram was created with the program HySS
    15. ^

    Further reading

    • F. Van Zeggeren and S.H. Storey, The Computation of Chemical Equilibria, Cambridge University Press, 1970. Mainly concerned with gas-phase equilibria.
    • D. J. Leggett (editor), Computational Methods for the Determination of Formation Constants, Plenum Press, 1985.
    • A.E. Martell and R.J. Motekaitis, The Determination and Use of Stability Constants, Wiley-VCH, 1992.
    • P. Gans, Stability Constants: Determination and Uses, an interactive CD, Protonic Software (Leeds), 2004
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