Do Equilibrium Constants Have Units

Chemical property

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a country approached by a dynamic chemic arrangement after sufficient time has elapsed at which its limerick has no measurable tendency towards farther modify. For a given prepare of reaction weather condition, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values tin can exist used to make up one's mind the limerick of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

A knowledge of equilibrium constants is essential for the understanding of many chemic systems, as well as biochemical processes such as oxygen transport by hemoglobin in blood and acid–base homeostasis in the human body.

Stability constants, formation constants, binding constants, association constants and dissociation constants are all types of equilibrium constants.

Basic definitions and properties [edit]

For a system undergoing a reversible reaction described by the general chemic equation

${\displaystyle \alpha \,\mathrm {A} +\beta \,\mathrm {B} +\cdots \rightleftharpoons \rho \,\mathrm {R} +\sigma \,\mathrm {Southward} +\cdots }$

a thermodynamic equilibrium constant, denoted past ${\displaystyle K^{\ominus }}$ , is divers to exist the value of the reaction quotient Q_t when frontwards and reverse reactions occur at the same rate. At chemical equilibrium, the chemical composition of the mixture does not alter with time and the Gibbs free energy alter ${\displaystyle \Delta G}$ for the reaction is zero. If the composition of a mixture at equilibrium is changed by improver of some reagent, a new equilibrium position will exist reached, given enough fourth dimension. An equilibrium constant is related to the composition of the mixture at equilibrium past ^{[1] [2]}

${\displaystyle K^{\ominus }={\frac {\mathrm {\{R\}} ^{\rho }\mathrm {\{S\}} ^{\sigma }...}{\mathrm {\{A\}} ^{\alpha }\mathrm {\{B\}} ^{\beta }...}}={\frac {{[\mathrm {R} ]}^{\rho }{[\mathrm {S} ]}^{\sigma }...}{{[\mathrm {A} ]}^{\alpha }{[\mathrm {B} ]}^{\beta }...}}\times \Gamma ,}$

${\displaystyle \Gamma ={\frac {\gamma _{R}^{\rho }\gamma _{Southward}^{\sigma }...}{\gamma _{A}^{\alpha }\gamma _{B}^{\beta }...}},}$

where {X} denotes the thermodynamic activeness of reagent X at equilibrium, [X] the numerical value ^[3] of the respective concentration in moles per liter, and γ the corresponding activeness coefficient. If Ten is a gas, instead of [X] the numerical value of the partial pressure level ${\displaystyle P_{X}}$ in bar is used.^[3] If information technology tin can exist assumed that the quotient of activity coefficients, ${\displaystyle \Gamma }$ , is constant over a range of experimental conditions, such as pH, then an equilibrium constant can be derived every bit a quotient of concentrations.

${\displaystyle K_{c}=One thousand^{\ominus }/\Gamma ={\frac {[\mathrm {R} ]^{\rho }[\mathrm {S} ]^{\sigma }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}.}$

An equilibrium abiding is related to the standard Gibbs free free energy change of reaction ${\displaystyle \Delta G^{\ominus }}$ past

${\displaystyle \Delta G^{\ominus }=-RT\ln K^{\ominus },}$

where R is the universal gas constant, T is the absolute temperature (in kelvins), and ln is the natural logarithm. This expression implies that ${\displaystyle K^{\ominus }}$ must exist a pure number and cannot have a dimension, since logarithms can only be taken of pure numbers. ${\displaystyle K_{c}}$ must as well be a pure number. On the other hand, the reaction quotient at equilibrium

${\displaystyle {\frac {[\mathrm {R} ]^{\rho }[\mathrm {S} ]^{\sigma }...}{[\mathrm {A} ]^{\alpha }[\mathrm {B} ]^{\beta }...}}(eq)}$

does take the dimension of concentration raised to some power (meet § Dimensionality, below). Such reaction quotients are oft referred to, in the biochemical literature, as equilibrium constants.

For an equilibrium mixture of gases, an equilibrium constant can be defined in terms of partial pressure or fugacity.

An equilibrium constant is related to the forward and astern rate constants, k _f and k _r of the reactions involved in reaching equilibrium:

${\displaystyle K^{\ominus }={\frac {k_{\text{f}}}{k_{\text{r}}}}.}$

Types of equilibrium constants [edit]

Cumulative and stepwise formation constants [edit]

A cumulative or overall constant, given the symbol β, is the constant for the formation of a complex from reagents. For example, the cumulative constant for the formation of ML₂ is given by

Thousand + two L ⇌ ML₂; [ML₂] = β ₁₂[M][50]²

The stepwise constant, One thousand, for the formation of the same complex from ML and L is given by

ML + L ⇌ ML_ii; [ML₂] = K[ML][L] = Kβ ₁₁[G][L]²

It follows that

β ₁₂ = Kβ ₁₁

A cumulative constant can always be expressed equally the product of stepwise constants. There is no agreed notation for stepwise constants, though a symbol such equally K ^Fifty
_ML is sometimes plant in the literature. It is all-time always to ascertain each stability constant by reference to an equilibrium expression.

Competition method [edit]

A particular utilise of a stepwise abiding is in the determination of stability constant values exterior the normal range for a given method. For instance, EDTA complexes of many metals are outside the range for the potentiometric method. The stability constants for those complexes were determined past competition with a weaker ligand.

ML + Fifty′ ⇌ ML′ + Fifty ${\displaystyle [\mathrm {ML} ']=K{\frac {[\mathrm {ML} ][\mathrm {L} ']}{[\mathrm {50} ]}}=M{\frac {\beta _{\mathrm {ML} }[\mathrm {M} ][\mathrm {L} ][\mathrm {L} ']}{[\mathrm {50} ]}}=K\beta _{\mathrm {ML} }[\mathrm {M} ][\mathrm {L} '];\quad \beta _{\mathrm {ML} '}=M\beta _{\mathrm {ML} }}$

The formation constant of [Pd(CN)₄]²⁻ was adamant by the competition method.

Association and dissociation constants [edit]

In organic chemistry and biochemistry it is customary to use pK _a values for acrid dissociation equilibria.

${\displaystyle \mathrm {p} K_{\mathrm {a} }=-\log K_{\mathrm {diss} }=\log \left({\frac {1}{K_{\mathrm {diss} }}}\correct)\,}$

where log denotes a logarithm to base 10 or common logarithm, and K _diss is a stepwise acid dissociation constant. For bases, the base of operations association constant, pK _b is used. For whatever given acid or base the two constants are related by pK _a + pG _b = pK _west , so pGrand _a tin ever be used in calculations.

On the other paw, stability constants for metal complexes, and binding constants for host–guest complexes are mostly expressed as association constants. When because equilibria such as

Yard + HL ⇌ ML + H

it is customary to apply association constants for both ML and HL. Also, in generalized estimator programs dealing with equilibrium constants it is general exercise to employ cumulative constants rather than stepwise constants and to omit ionic charges from equilibrium expressions. For example, if NTA, nitrilotriacetic acid, N(CH₂CO₂H)₃ is designated equally H₃L and forms complexes ML and MHL with a metal ion M, the following expressions would apply for the dissociation constants.

${\displaystyle {\begin{array}{ll}{\ce {H3L <=> {H2L}+ H}};&{\ce {p}}K_{1}=-\log \left({\frac {[{\ce {H2L}}][{\ce {H}}]}{[{\ce {H3L}}]}}\right)\\{\ce {H2L <=> {HL}+ H}};&{\ce {p}}K_{2}=-\log \left({\frac {[{\ce {HL}}][{\ce {H}}]}{[{\ce {H2L}}]}}\right)\\{\ce {HL <=> {L}+ H}};&{\ce {p}}K_{3}=-\log \left({\frac {[{\ce {50}}][{\ce {H}}]}{[{\ce {HL}}]}}\correct)\end{array}}}$

The cumulative association constants can be expressed equally

${\displaystyle {\begin{array}{ll}{\ce {{L}+ H <=> HL}};&\log \beta _{011}=\log \left({\frac {[{\ce {HL}}]}{[{\ce {L}}][{\ce {H}}]}}\correct)={\ce {p}}K_{3}\\{\ce {{L}+ 2H <=> H2L}};&\log \beta _{012}=\log \left({\frac {[{\ce {H2L}}]}{[{\ce {L}}][{\ce {H}}]^{ii}}}\right)={\ce {p}}K_{three}+{\ce {p}}K_{2}\\{\ce {{L}+ 3H <=> H3L}};&\log \beta _{013}=\log \left({\frac {[{\ce {H3L}}]}{[{\ce {L}}][{\ce {H}}]^{iii}}}\right)={\ce {p}}K_{iii}+{\ce {p}}K_{ii}+{\ce {p}}K_{ane}\\{\ce {{M}+ Fifty <=> ML}};&\log \beta _{110}=\log \left({\frac {[{\ce {ML}}]}{[{\ce {Yard}}][{\ce {L}}]}}\correct)\\{\ce {{M}+ {Fifty}+ H <=> MLH}};&\log \beta _{111}=\log \left({\frac {[{\ce {MLH}}]}{[{\ce {Grand}}][{\ce {50}}][{\ce {H}}]}}\right)\end{array}}}$

Annotation how the subscripts define the stoichiometry of the equilibrium production.

Micro-constants [edit]

When two or more than sites in an asymmetrical molecule may be involved in an equilibrium reaction there are more than one possible equilibrium constants. For instance, the molecule 50-DOPA has two not-equivalent hydroxyl groups which may be deprotonated. Denoting L-DOPA as LH₂, the following diagram shows all the species that may exist formed (X = CH
₂ CH(NH
₂ )CO
_two H).

The concentration of the species LH is equal to the sum of the concentrations of the two micro-species with the same chemic formula, labelled Fifty¹H and L^twoH. The abiding K _two is for a reaction with these 2 micro-species as products, so that [LH] = [L¹H] + [50²H] appears in the numerator, and information technology follows that this macro-constant is equal to the sum of the ii micro-constants for the component reactions.

K ₂ = one thousand ₂₁ + k ₂₂

However, the abiding One thousand _i is for a reaction with these two micro-species equally reactants, and [LH] = [L¹H] + [L²H] in the denominator, and then that in this instance^[4]

ane/Yard ₁ =1/ grand ₁₁ + ane/yard ₁₂,

and therefore G _one =one thousand ₁₁ k ₁₂ / (k _xi + one thousand ₁₂). Thus, in this example there are four micro-constants whose values are subject to two constraints; in upshot, only the two macro-constant values, for K₁ and M₂ can exist derived from experimental data.

Micro-constant values tin can, in principle, be adamant using a spectroscopic technique, such as infrared spectroscopy, where each micro-species gives a different signal. Methods which have been used to estimate micro-constant values include

• Chemical: blocking one of the sites, for case past methylation of a hydroxyl grouping, followed by decision of the equilibrium abiding of the related molecule, from which the micro-constant value for the "parent" molecule may be estimated.
• Mathematical: applying numerical procedures to ¹³C NMR information.^{[five] [six]}

Although the value of a micro-constant cannot be determined from experimental information, site occupancy, which is proportional to the micro-constant value, can be very important for biological activity. Therefore, various methods take been developed for estimating micro-constant values. For instance, the isomerization abiding for L-DOPA has been estimated to have a value of 0.9, so the micro-species L¹H and L²H accept almost equal concentrations at all pH values.

pH considerations (Brønsted constants) [edit]

pH is divers in terms of the activity of the hydrogen ion

pH = −log₁₀ {H⁺}

In the approximation of ideal behaviour, activity is replaced by concentration. pH is measured past means of a glass electrode, a mixed equilibrium constant, as well known equally a Brønsted abiding, may upshot.

HL ⇌ L + H; ${\displaystyle \mathrm {p} Thou=-\log \left({\frac {[\mathrm {L} ]\{\mathrm {H} \}}{[\mathrm {HL} ]}}\correct)}$

It all depends on whether the electrode is calibrated by reference to solutions of known activity or known concentration. In the latter case the equilibrium constant would be a concentration caliber. If the electrode is calibrated in terms of known hydrogen ion concentrations information technology would exist better to write p[H] rather than pH, just this suggestion is not generally adopted.

Hydrolysis constants [edit]

In aqueous solution the concentration of the hydroxide ion is related to the concentration of the hydrogen ion by

${\displaystyle {\ce {{\mathit {K}}_{West}=[H][OH]}}}$

${\displaystyle {\ce {[OH]={\mathit {K}}_{West}[H]^{-1}}}}$

The first stride in metal ion hydrolysis^[7] tin can be expressed in two different ways

${\displaystyle {\begin{cases}{\ce {M(Water) <=> {1000(OH)}+ H}};&[{\ce {1000(OH)}}]=\beta ^{*}[{\ce {M}}][{\ce {H}}]^{-1}\\{\ce {{M}+ OH <=> Chiliad(OH)}};&[{\ce {Yard(OH)}}]=Chiliad[{\ce {1000}}][{\ce {OH}}]=KK_{{\ce {Due west}}}[{\ce {Yard}}][{\ce {H}}]^{-1}\terminate{cases}}}$

It follows that β ^* = KK _Westward . Hydrolysis constants are usually reported in the β ^* form and therefore often have values much less than 1. For case, if log Yard = four and log One thousand_W = −14, log β ^* = four + (−14) = −10 so that β^* = x^−ten. In full general when the hydrolysis product contains n hydroxide groups log β ^* = log K + n log Grand _Westward

Conditional constants [edit]

Conditional constants, also known as credible constants, are concentration quotients which are not truthful equilibrium constants only can exist derived from them.^[8] A very common example is where pH is fixed at a particular value. For example, in the case of fe(Three) interacting with EDTA, a conditional constant could be divers by

${\displaystyle K_{\mathrm {cond} }={\frac {[{\mbox{Total Fe bound to EDTA}}]}{[{\mbox{Full Iron not leap to EDTA}}]\times [{\mbox{Total EDTA not leap to Fe}}]}}}$

This conditional constant will vary with pH. Information technology has a maximum at a certain pH. That is the pH where the ligand sequesters the metal most effectively.

In biochemistry equilibrium constants are ofttimes measured at a pH fixed by means of a buffer solution. Such constants are, by definition, conditional and different values may be obtained when using different buffers.

Gas-stage equilibria [edit]

For equilibria in a gas phase, fugacity, f, is used in place of activeness. Withal, fugacity has the dimension of pressure, so it must be divided by a standard pressure level, usually one bar, in order to produce a dimensionless quantity, f / p ^o . An equilibrium abiding is expressed in terms of the dimensionless quantity. For example, for the equilibrium 2NO₂ ⇌ N₂O₄,

${\displaystyle {\frac {f_{\mathrm {N_{2}O_{4}} }}{p^{\ominus }}}=Chiliad\left({\frac {f_{\mathrm {NO_{2}} }}{p^{\ominus }}}\right)^{two}}$

Fugacity is related to fractional force per unit area, ${\displaystyle p_{X}}$ , by a dimensionless fugacity coefficient ϕ: ${\displaystyle f_{X}=\phi _{X}p_{X}}$ . Thus, for the example,

${\displaystyle Thou={\frac {\phi _{\mathrm {N_{two}O_{iv}} }p_{\mathrm {N_{two}O_{four}} }/{p^{\ominus }}}{\left(\phi _{\mathrm {NO_{2}} }p_{\mathrm {NO_{2}} }/{p^{\ominus }}\right)^{two}}}}$

Usually the standard pressure is omitted from such expressions. Expressions for equilibrium constants in the gas stage then resemble the expression for solution equilibria with fugacity coefficient in place of action coefficient and partial pressure in place of concentration.

${\displaystyle Thousand={\frac {\phi _{\mathrm {N_{2}O_{four}} }p_{\mathrm {N_{2}O_{4}} }}{\left(\phi _{\mathrm {NO_{2}} }p_{\mathrm {NO_{ii}} }\right)^{2}}}}$

Thermodynamic basis for equilibrium constant expressions [edit]

Thermodynamic equilibrium is characterized past the gratuitous energy for the whole (closed) system being a minimum. For systems at constant temperature and pressure the Gibbs complimentary energy is minimum.^[nine] The gradient of the reaction free energy with respect to the extent of reaction, ξ, is zero when the free energy is at its minimum value.

${\displaystyle \left({\frac {\partial One thousand}{\partial \11 }}\right)_{T,P}=0}$

The energy change, dChiliad _r, can be expressed every bit a weighted sum of change in amount times the chemical potential, the partial molar free energy of the species. The chemical potential, μ_i , of the ith species in a chemical reaction is the partial derivative of the gratuitous energy with respect to the number of moles of that species, North _i

${\displaystyle \mu _{i}=\left({\frac {\partial Thou}{\partial N_{i}}}\right)_{T,P}}$

A general chemical equilibrium tin can be written as

${\displaystyle \sum _{j}n_{j}\mathrm {Reactant} _{j}\rightleftharpoons \sum _{k}m_{g}\mathrm {Product} _{k}}$

where northward_j are the stoichiometric coefficients of the reactants in the equilibrium equation, and one thousand_j are the coefficients of the products. At equilibrium

${\displaystyle \sum _{chiliad}m_{k}\mu _{k}=\sum _{j}n_{j}\mu _{j}}$

The chemical potential, μ_i , of the ith species can be calculated in terms of its action, a_i .

${\displaystyle \mu _{i}=\mu _{i}^{\ominus }+RT\ln a_{i}}$

μ ^o
_i is the standard chemical potential of the species, R is the gas abiding and T is the temperature. Setting the sum for the reactants j to exist equal to the sum for the products, k, and then that δG _r(Eq) = 0

${\displaystyle \sum _{j}n_{j}(\mu _{j}^{\ominus }+RT\ln a_{j})=\sum _{k}m_{thou}(\mu _{grand}^{\ominus }+RT\ln a_{k})}$

Rearranging the terms,

${\displaystyle \sum _{k}m_{k}\mu _{thou}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }=-RT\left(\sum _{k}\ln {a_{k}}^{m_{g}}-\sum _{j}\ln {a_{j}}^{n_{j}}\correct)}$

${\displaystyle \Delta G^{\ominus }=-RT\ln K.}$

This relates the standard Gibbs energy modify, ΔG ^o to an equilibrium abiding, M, the reaction quotient of activity values at equilibrium.

${\displaystyle \Delta 1000^{\ominus }=\sum _{k}m_{g}\mu _{k}^{\ominus }-\sum _{j}n_{j}\mu _{j}^{\ominus }}$

${\displaystyle \ln K=\sum _{1000}\ln {a_{k}}^{m_{k}}-\sum _{j}\ln {a_{j}}^{n_{j}};M={\frac {\prod _{k}{a_{g}}^{m_{k}}}{\prod _{j}{a_{j}}^{n_{j}}}}\equiv {\frac {{\{\mathrm {R} \}}^{\rho }{\{\mathrm {S} \}}^{\sigma }...}{{\{\mathrm {A} \}}^{\alpha }{\{\mathrm {B} \}}^{\beta }...}}}$

Equivalence of thermodynamic and kinetic expressions for equilibrium constants [edit]

At equilibrium the rate of the frontward reaction is equal to the backward reaction rate. A uncomplicated reaction, such equally ester hydrolysis

${\displaystyle {\ce {AB + H2O <=> AH + B(OH)}}}$

has reaction rates given by expressions

${\displaystyle {\text{forward charge per unit}}=k_{f}{\ce {[AB][H2O]}}}$

${\displaystyle {\text{astern rate}}=k_{b}{\ce {[AH][B(OH)]}}}$

According to Guldberg and Waage, equilibrium is attained when the frontward and backward reaction rates are equal to each other. In these circumstances, an equilibrium constant is defined to be equal to the ratio of the forwards and backward reaction rate constants

${\displaystyle Grand={\frac {k_{f}}{k_{b}}}={\frac {{\ce {[AH][B(OH)]}}}{{\ce {[AB][H2O]}}}}}$ .

The concentration of h2o may be taken to be abiding, resulting in the simpler expression

${\displaystyle M^{c}={\frac {{\ce {[AH][B(OH)]}}}{{\ce {[AB]}}}}}$ .

This particular concentration quotient, ${\displaystyle K^{c}}$ , has the dimension of concentration, simply the thermodynamic equilibrium constant, K, is always dimensionless.

Unknown activity coefficient values [edit]

Variation of logK _c of acetic acid with ionic strength

It is very rare for action coefficient values to have been adamant experimentally for a system at equilibrium. There are iii options for dealing with the state of affairs where activity coefficient values are not known from experimental measurements.

1. Apply calculated action coefficients, together with concentrations of reactants. For equilibria in solution estimates of the activity coefficients of charged species can be obtained using Debye–Hückel theory, an extended version, or Sit down theory. For uncharged species, the activity coefficient γ ₀ mostly follows a "salting-out" model: log₁₀ γ ₀ = bI where I stands for ionic forcefulness.^[10]
2. Assume that the activeness coefficients are all equal to i. This is acceptable when all concentrations are very low.
3. For equilibria in solution employ a medium of high ionic strength. In outcome this redefines the standard state every bit referring to the medium. Activity coefficients in the standard country are, by definition, equal to 1. The value of an equilibrium constant adamant in this fashion is dependent on the ionic strength. When published constants refer to an ionic forcefulness other than the ane required for a particular application, they may exist adjusted by means of specific ion theory (SIT) and other theories.^[11]

Dimensionality [edit]

An equilibrium abiding is related to the standard Gibbs costless energy modify, ${\displaystyle \Delta G^{\ominus }}$ , for the reaction by the expression

${\displaystyle \Delta Thou^{\ominus }=-RT\ln Thousand}$

Therefore, K, must be a number from which a logarithm can exist derived. In the example of a simple equilibrium

${\displaystyle {\ce {A + B <=> AB}}}$

the thermodynamic equilibrium constant is defined in terms of the activities, {AB}, {A} and {B}, of the species in equilibrium with each other.

${\displaystyle One thousand={\frac {\{AB\}}{\{A\}\{B\}}}}$

Now, each activity term can be expressed every bit a product of a concentration ${\displaystyle [X]}$ and a corresponding activity coefficient, ${\displaystyle \gamma (10)}$ . Therefore,

${\displaystyle K={\frac {[AB]}{[A][B]}}\times {\frac {\gamma (AB)}{\gamma (A)\gamma (B)}}={\frac {[AB]}{[A][B]}}\times \Gamma }$

When ${\displaystyle \Gamma }$ , the caliber of activity coefficients, is set equal to 1, nosotros get

${\displaystyle Grand={\frac {[AB]}{[A][B]}}}$ .

K and so appears to accept the dimension of 1/concentration. This is what unremarkably happens in practice when an equilibrium constant is calculated every bit a quotient of concentration values. This can exist avoided by dividing each concentration by its standard-state value (usually mol/L or bar), which is standard practice in chemistry.^[iii]

The assumption underlying this practise is that the quotient of activities is constant nether the weather condition in which the equilibrium abiding value is determined. These conditions are usually accomplished by keeping the reaction temperature constant and by using a medium of relatively loftier ionic strength as the solvent. It is non unusual, particularly in texts relating to biochemical equilibria, to see an equilibrium constant value quoted with a dimension. The justification for this practice is that the concentration calibration used may exist either mol dm⁻³ or mmol dm⁻³, and so that the concentration unit of measurement has to be stated in society to avoid there being any ambivalence.

Note. When the concentration values are measured on the mole fraction scale all concentrations and activity coefficients are dimensionless quantities.

In general equilibria between ii reagents tin exist expressed as

${\displaystyle {\ce {{{\mathit {p}}A}+{\mathit {q}}B<=>A_{\mathit {p}}B_{\mathit {q}}}}}$

in which instance the equilibrium constant is defined, in terms of numerical concentration values, every bit

${\displaystyle Chiliad={\frac {[{\ce {A}}_{p}{\ce {B}}_{q}]}{[{\ce {A}}]^{p}[{\ce {B}}]^{q}}}}$

The credible dimension of this K value is concentration^1−p−q; this may be written as One thousand^{(ane−p−q)} or mM^(1−p−q), where the symbol Chiliad signifies a molar concentration (1M = 1 mol dm⁻ⁱⁱⁱ ). The apparent dimension of a dissociation abiding is the reciprocal of the apparent dimension of the respective association abiding, and vice versa.

When discussing the thermodynamics of chemical equilibria information technology is necessary to accept dimensionality into business relationship. There are two possible approaches.

1. Set the dimension of Γ to exist the reciprocal of the dimension of the concentration quotient. This is nearly universal exercise in the field of stability constant determinations. The "equilibrium constant" ${\displaystyle {\frac {K}{\Gamma }}}$ , is dimensionless. It will be a function of the ionic forcefulness of the medium used for the determination. Setting the numerical value of Γ to exist one is equivalent to re-defining the standard states.
2. Replace each concentration term ${\displaystyle [X]}$ by the dimensionless caliber ${\displaystyle {\frac {[Ten]}{[X^{0}]}}}$ , where ${\displaystyle [X^{0}]}$ is the concentration of reagent Ten in its standard state (commonly 1 mol/L or ane bar).^[3] By definition the numerical value of ${\displaystyle \gamma (Ten^{0})}$ is 1, so Γ also has a numerical value of ane.

In both approaches the numerical value of the stability constant is unchanged. The first is more useful for practical purposes; in fact, the unit of measurement of the concentration quotient is often attached to a published stability constant value in the biochemical literature. The second approach is consistent with the standard exposition of Debye–Hückel theory, where ${\displaystyle \gamma (AB)}$ , etc. are taken to exist pure numbers.

Water as both reactant and solvent [edit]

For reactions in aqueous solution, such as an acid dissociation reaction

AH + H₂O ⇌ A⁻ + H₃O⁺

the concentration of water may be taken as being constant and the germination of the hydronium ion is implicit.

AH ⇌ A⁻ + H⁺

Water concentration is omitted from expressions defining equilibrium constants, except when solutions are very full-bodied.

${\displaystyle K={\frac {[A][H]}{[AH]}}}$ (K defined every bit a dissociation constant)

Similar considerations apply to metal ion hydrolysis reactions.

Enthalpy and entropy: temperature dependence [edit]

If both the equilibrium constant, ${\displaystyle M}$ and the standard enthalpy modify, ${\displaystyle \Delta H^{\ominus }}$ , for a reaction have been determined experimentally, the standard entropy change for the reaction is easily derived. Since ${\displaystyle \Delta G=\Delta H-T\Delta S}$ and ${\displaystyle \Delta G=-RT\ln One thousand}$

${\displaystyle \Delta S^{\ominus }={\frac {\Delta H^{\ominus }+RT\ln K}{T}}}$

To a first approximation the standard enthalpy change is independent of temperature. Using this approximation, definite integration of the van 't Hoff equation

${\displaystyle \Delta H^{\ominus }=-R{\frac {d\ln K}{d(1/T)}}\ }$

gives^[12]

${\displaystyle \ln K_{2}=\ln K_{one}-{\frac {\Delta H^{\ominus }}{R}}\left({\frac {1}{T_{2}}}-{\frac {1}{T_{1}}}\right)}$

This equation can be used to summate the value of log K at a temperature, T₂, knowing the value at temperature T₁.

The van 't Hoff equation as well shows that, for an exothermic reaction ( ${\displaystyle \Delta H<0}$ ), when temperature increases 1000 decreases and when temperature decreases K increases, in accordance with Le Chatelier's principle. The opposite applies when the reaction is endothermic.

When K has been adamant at more than than two temperatures, a straight line plumbing equipment procedure may be applied to a plot of ${\displaystyle \ln K}$ against ${\displaystyle ane/T}$ to obtain a value for ${\displaystyle \Delta H^{\ominus }}$ . Error propagation theory can be used to testify that, with this procedure, the fault on the calculated ${\displaystyle \Delta H^{\ominus }}$ value is much greater than the error on individual log 1000 values. Consequently, K needs to be determined to high precision when using this method. For example, with a silver ion-selective electrode each log Chiliad value was adamant with a precision of ca. 0.001 and the method was applied successfully.^[13]

Standard thermodynamic arguments tin exist used to evidence that, more more often than not, enthalpy will change with temperature.

${\displaystyle \left({\frac {\partial H}{\fractional T}}\right)_{p}=C_{p}}$

where C _p is the estrus chapters at constant pressure level.

A more circuitous formulation [edit]

The calculation of K at a particular temperature from a known K at another given temperature can be approached equally follows if standard thermodynamic properties are available. The outcome of temperature on equilibrium constant is equivalent to the effect of temperature on Gibbs free energy because:

${\displaystyle \ln K={{-\Delta _{\mathrm {r} }One thousand^{\ominus }} \over {RT}}}$

where Δ_r G ^o is the reaction standard Gibbs energy, which is the sum of the standard Gibbs energies of the reaction products minus the sum of standard Gibbs energies of reactants.

Here, the term "standard" denotes the ideal behaviour (i.east., an infinite dilution) and a hypothetical standard concentration (typically 1 mol/kg). Information technology does not imply whatsoever particular temperature or pressure because, although contrary to IUPAC recommendation, it is more user-friendly when describing aqueous systems over wide temperature and pressure ranges.^[14]

The standard Gibbs energy (for each species or for the entire reaction) tin be represented (from the basic definitions) as:

${\displaystyle G_{T_{2}}^{\ominus }=G_{T_{ane}}^{\ominus }-S_{T_{one}}^{\ominus }(T_{ii}-T_{i})-T_{two}\int _{T_{i}}^{T_{2}}{{C_{p}^{\ominus }} \over {T}}\,dT+\int _{T_{ane}}^{T_{2}}C_{p}^{\ominus }\,dT}$

In the higher up equation, the consequence of temperature on Gibbs energy (and thus on the equilibrium constant) is ascribed entirely to rut capacity. To evaluate the integrals in this equation, the grade of the dependence of heat capacity on temperature needs to exist known.

If the standard tooth oestrus capacity C ^o
_p can be approximated past some analytic function of temperature (e.g. the Shomate equation), and so the integrals involved in calculating other parameters may be solved to yield analytic expressions for them. For example, using approximations of the following forms:^[fifteen]

• For pure substances (solids, gas, liquid):

  $${\displaystyle C_{p}^{\ominus }\approx A+BT+CT^{-2}}$$

  

• For ionic species at T < 200 °C:

  $${\displaystyle C_{p}^{\ominus }\approx (four.186a+b{\breve {South}}_{T_{1}}^{\ominus }){{(T_{2}-T_{ane})} \over {\ln \left({\frac {T_{2}}{T_{1}}}\correct)}}}$$

  

then the integrals can be evaluated and the following terminal form is obtained:

${\displaystyle G_{T_{2}}^{\ominus }\approx G_{T_{1}}^{\ominus }+(C_{p}^{\ominus }-S_{T_{1}}^{\ominus })(T_{2}-T_{1})-T_{ii}\ln \left({\frac {T_{ii}}{T_{1}}}\right)C_{p}^{\ominus }}$

The constants A, B, C, a, b and the absolute entropy, S̆ ^o
_298 K , required for evaluation of C ^o
_p (T), as well as the values of G _298 Grand and South _298 K for many species are tabulated in the literature.

Pressure dependence [edit]

The pressure dependence of the equilibrium constant is unremarkably weak in the range of pressures normally encountered in industry, and therefore, it is usually neglected in practice. This is true for condensed reactant/products (i.e., when reactants and products are solids or liquid) equally well as gaseous ones.

For a gaseous-reaction example, i may consider the well-studied reaction of hydrogen with nitrogen to produce ammonia:

Due north₂ + iii H₂ ⇌ ii NH₃

If the pressure is increased by the addition of an inert gas, so neither the limerick at equilibrium nor the equilibrium constant are appreciably affected (considering the fractional pressures remain constant, bold an ideal-gas behaviour of all gases involved). Withal, the composition at equilibrium will depend appreciably on pressure when:

• the force per unit area is changed by compression or expansion of the gaseous reacting organisation, and
• the reaction results in the change of the number of moles of gas in the arrangement.

In the example reaction above, the number of moles changes from iv to 2, and an increase of force per unit area past system compression will result in appreciably more ammonia in the equilibrium mixture. In the full general example of a gaseous reaction:

α A + β B ⇌ σ South + τ T

the change of mixture composition with pressure can be quantified using:

${\displaystyle K_{p}={\frac {{p_{\mathrm {S} }}^{\sigma }{p_{\mathrm {T} }}^{\tau }}{{p_{\mathrm {A} }}^{\alpha }{p_{\mathrm {B} }}^{\beta }}}={\frac {{X_{\mathrm {S} }}^{\sigma }{X_{\mathrm {T} }}^{\tau }}{{X_{\mathrm {A} }}^{\alpha }{X_{\mathrm {B} }}^{\beta }}}P^{\sigma +\tau -\alpha -\beta }=K_{X}P^{\sigma +\tau -\alpha -\beta }}$

where p denote the partial pressures and X the mole fractions of the components, P is the total organization pressure, Thousand_p is the equilibrium abiding expressed in terms of partial pressures and K₁₀ is the equilibrium constant expressed in terms of mole fractions.

The above alter in composition is in accordance with Le Chatelier'southward principle and does not involve any change of the equilibrium constant with the total arrangement pressure. Indeed, for platonic-gas reactions K_p is independent of pressure.^[16]

Pressure dependence of the water ionization abiding at 25 °C. In full general, ionization in aqueous solutions tends to increase with increasing pressure.

In a condensed phase, the force per unit area dependence of the equilibrium constant is associated with the reaction volume.^[17] For reaction:

α A + β B ⇌ σ S + τ T

the reaction volume is:

${\displaystyle \Delta {\bar {Five}}=\sigma {\bar {V}}_{\mathrm {S} }+\tau {\bar {Five}}_{\mathrm {T} }-\alpha {\bar {V}}_{\mathrm {A} }-\beta {\bar {V}}_{\mathrm {B} }}$

where V̄ denotes a partial molar volume of a reactant or a product.

For the above reaction, one can expect the change of the reaction equilibrium abiding (based either on mole-fraction or molal-concentration scale) with pressure at constant temperature to be:

${\displaystyle \left({\frac {\partial \ln K_{Ten}}{\partial P}}\right)_{T}={\frac {-\Delta {\bar {Five}}}{RT}}.}$

The matter is complicated equally partial tooth volume is itself dependent on force per unit area.

Effect of isotopic substitution [edit]

Isotopic substitution can atomic number 82 to changes in the values of equilibrium constants, particularly if hydrogen is replaced by deuterium (or tritium).^[eighteen] This equilibrium isotope consequence is analogous to the kinetic isotope consequence on rate constants, and is primarily due to the alter in zero-bespeak vibrational energy of H–X bonds due to the alter in mass upon isotopic substitution.^[eighteen] The naught-indicate energy is inversely proportional to the square root of the mass of the vibrating hydrogen cantlet, and will therefore be smaller for a D–X bond that for an H–X bond.

An case is a hydrogen cantlet abstraction reaction R' + H–R ⇌ R'–H + R with equilibrium constant Thousand_H, where R' and R are organic radicals such that R' forms a stronger bond to hydrogen than does R. The decrease in zilch-bespeak free energy due to deuterium substitution volition then be more important for R'–H than for R–H, and R'–D will be stabilized more than R–D, so that the equilibrium constant K_D for R' + D–R ⇌ R'–D + R is greater than K_H. This is summarized in the rule the heavier atom favors the stronger bail.^[18]

Similar effects occur in solution for acid dissociation constants (K_a) which describe the transfer of H⁺ or D⁺ from a weak aqueous acid to a solvent molecule: HA + H₂O = H₃O⁺ + A⁻ or DA + D₂O ⇌ D₃O⁺ + A⁻. The deuterated acid is studied in heavy h2o, since if information technology were dissolved in ordinary water the deuterium would rapidly exchange with hydrogen in the solvent.^[eighteen]

The product species H₃O⁺ (or D_threeO⁺) is a stronger acrid than the solute acrid, then that it dissociates more easily, and its H–O (or D–O) bond is weaker than the H–A (or D–A) bond of the solute acrid. The decrease in zero-point energy due to isotopic commutation is therefore less important in D_threeO⁺ than in DA so that K_D < K_H, and the deuterated acid in D₂O is weaker than the non-deuterated acid in H_iiO. In many cases the difference of logarithmic constants pK_D – pK_H is about 0.6,^[18] so that the pD corresponding to fifty% dissociation of the deuterated acid is most 0.6 units college than the pH for l% dissociation of the non-deuterated acid.

For similar reasons the self-ionization of heavy water is less than that of ordinary water at the same temperature.

See also [edit]

• Determination of equilibrium constants
• Stability constants of complexes
• Equilibrium fractionation

References [edit]

1. ^ IUPAC Aureate Book.
2. ^ Rossotti, F. J. C.; Rossotti, H. (1961). The Decision of Stability Constants. McGraw-Hill. p. 5.
3. ^ ^{a b c d} Atkins, P.; Jones, 50.; Laverman, Fifty. (2016).Chemic Principles, 7th edition, pp. 399 & 461. Freeman. ISBN 978-1-4641-8395-ix
4. ^ Splittgerber, A. Chiliad.; Chinander, L.L. (one Feb 1988). "The spectrum of a dissociation intermediate of cysteine: a biophysical chemistry experiment". Journal of Chemical Educational activity. 65 (2): 167. doi:10.1021/ed065p167.
5. ^ Hague, David N.; Moreton, Anthony D. (1994). "Protonation sequence of linear aliphatic polyamines by 13C NMR spectroscopy". J. Chem. Soc., Perkin Trans. two (two): 265–lxx. doi:10.1039/P29940000265.
6. ^ Borkovec, Michal; Koper, Ger J. Yard. (2000). "A Cluster Expansion Method for the Consummate Resolution of Microscopic Ionization Equilibria from NMR Titrations". Anal. Chem. 72 (fourteen): 3272–9. doi:10.1021/ac991494p. PMID 10939399.
7. ^ Baes, C. F.; Mesmer, R. E. (1976). "Affiliate eighteen. Survey of Hydrolysis Behaviour". The Hydrolysis of Cations. Wiley. pp. 397–430.
8. ^ Schwarzenbach, G.; Flaschka, H. (1969). Complexometric titrations. Methuen. ^{[ page needed ]}
9. ^ Denbigh, K. (1981). "Chapter 4". The principles of chemic equilibrium (4th ed.). Cambridge: Cambridge University Press. ISBN978-0-521-28150-8.
10. ^ Butler, J. Northward. (1998). Ionic Equilibrium. John Wiley and Sons.
11. ^ "Project: Ionic Force Corrections for Stability Constants". International Union of Pure and Applied Chemistry. Archived from the original on 29 October 2008. Retrieved 2008-11-23 .
12. ^ Atkins, Peter; de Paula, Julio (2006). Physical Chemistry . Oxford. p. 214. ISBN978-0198700722.
13. ^ Barnes, D.Southward.; Ford, G.J; Pettit, L.D.; Sherringham, C. (1971). "Ligands containing elements of group VIB. Part Five. Thermodynamics of silver complex formation of some saturated and unsaturated (alkyl-thio)acerb and (alkylseleno)acetic acids". J. Chem. Soc. A: 2883–2887. doi:x.1039/J19710002883.
14. ^ Majer, 5.; Sedelbauer, J.; Woods (2004). "Calculations of standard thermodynamic properties of aqueous electrolytes and nonelectrolytes". In Palmer, D. A.; Fernandez-Prini, R.; Harvey, A. (eds.). Aqueous Systems at Elevated Temperatures and Pressures: Concrete Chemistry of Water, Steam and Hydrothermal Solutions. Elsevier. ^{[ page needed ]}
15. ^ Roberge, P. R. (November 2011). "Appendix F". Handbook of Corrosion Technology. McGraw-Hill. p. 1037ff.
16. ^ Atkins, P. W. (1978). Physical Chemistry (6th ed.). Oxford University Printing. p. 210.
17. ^ Van Eldik, R.; Asano, T.; Le Noble, Westward. J. (1989). "Activation and reaction volumes in solution. two". Chem. Rev. 89 (3): 549–688. doi:10.1021/cr00093a005.
18. ^ ^{a b c d e} Laidler Thou.J. Chemical Kinetics (3rd ed., Harper & Row 1987), p.428–433 ISBN 0-06-043862-two

Data sources [edit]

• IUPAC SC-Database Archived 2017-06-19 at the Wayback Machine A comprehensive database of published information on equilibrium constants of metal complexes and ligands
• NIST Standard Reference Database 46 Archived 2010-07-05 at the Wayback Car: Critically selected stability constants of metallic complexes
• Inorganic and organic acids and bases pK _a data in water and DMSO
• NASA Glenn Thermodynamic Database webpage with links to (self-consistent) temperature-dependent specific rut, enthalpy, and entropy for elements and molecules

Do Equilibrium Constants Have Units,

Source: https://en.wikipedia.org/wiki/Equilibrium_constant

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