Equilibrium constant

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In chemistry, the equilibrium constant is a quantity characterizing a chemical equilibrium in a chemical reaction. It is a useful tool in determining the concentration of various reactants and products in a system where chemical equilibrium occurs.

1 Definition
2 Derivations
- 2.1 Simple equilibrium
3 Applications
4 Thermodynamics
5 See also
6 External links

[edit] Definition

A typical equilibrium situation is as below:


$aA + bB \rightleftharpoons cC + dD$

where A and B are reactant chemical species, C and D are product species, and a, b, c, and d are the stoichiometric coefficients of the respective reactants and products.

Equilibrium occurs when the forward reaction rate equals the backward reaction rate.

The equilibrium constant is defined as

$K \ \stackrel{\mathrm{def}}{=}\ \frac{\left[C\right]^c \left[D\right]^d} {\left[A\right]^a \left[B\right]^b}$

The law of chemical equilibrium says that this constant is independent of initial concentrations or most reaction conditions except temperature. [J] represents the chemical activity of the species J at equilibrium under reaction conditions. Sometimes the activity can be replaced by molar concentration or partial pressure without significant error, yielding $K c$ or $K p$ respectively. K is dimensionless, while $K c$ and $K p$ are not necessarily so.

[edit] Derivations

For single-step reactions, where the law of mass action is valid, we can write

$k_{f} \left[A\right]^a \left[B\right]^b = k_{b} \left[C\right]^c \left[D\right]^d$

where $k f$ and $k b$ are the forward and backward reaction rate constants, respectively. Cross-dividing yields

$K \ \stackrel{\mathrm{def}}{=}\ \frac{k_{f}}{k_{b}} = \frac{\left[C\right]^c \left[D\right]^d} {\left[A\right]^a \left[B\right]^b}$

Since the rate constants are constant by definition, the law of chemical equilibrium is valid. But it should be kept in mind that this derivation is valid only for single-step elementary reactions (which are very rare) and not for general reactions where the law of mass action does not apply. Using this treatment for general reactions is an example of getting the right answer using the wrong method. The law of chemical equilibrium however holds for all reactions. The general derivation requires advanced treatment using thermodynamics.

[edit] Simple equilibrium

Concentration of A (A0 = 0.25 mole/l) and B versus time reaching equilibrium kf = 2 min-1 and kr = 1 min-1

Concentration of A (A₀ = 0.25 mole/l) and B versus time reaching equilibrium k_f = 2 min^-1 and k_r = 1 min^-1

In a simple equilibrium between two species:

$A \rightleftharpoons B$

the constant K at equilibrium is expressed as:

$K \ \stackrel{\mathrm{def}}{=}\ \frac{k_{f}}{k_{b}} = \frac{\left[B\right]_e} {\left[A\right]_e}$

When the concentration of A at equilibrium is that of the concentration at time 0 minus the conversion in moles

$\ [A]_e = [A]_0 - x$

with x equal to the concentration of B at equilibrium

$\ [B]_e = x$

then it follows that

$\ [B]_e = x = \frac{k_{-f}}{k_f+k_r}[A]_0$

and

$\ [A]_e = [A]_0 - x = \frac{k_{-r}}{k_f+k_r}[A]_0$

The reaction rate becomes:

$\ \frac{dx}{dt} = \frac{k_f[A]_0}{x_e} (x_e - x)$

which results in:

$\ ln(\frac{[A]_0 - [A]_e}{[A_t]-[A]_e}) = (k_f + k_r)t$

A plot of the negative natural logarithm of the concentration of A in time minus the concentration at equilibrium versus time t gives a straight line with slope k_f + k_r. By measurement of A_e and B_e the values of K and the two reaction rate constants will be known.

When the equilibrium constant is close to unity and the reaction rates very fast for instance in conformational analysis of molecules, other methods are required for the determination of rate constants for instance by complete lineshape analysis in NMR spectroscopy.

[edit] Applications

Equilibrium constants can be defined for many physical/chemical processes. Examples include the acidity constant (the equilibrium constant for the dissociation of protons from acids) and the solubility constant (the equilibrium constant for precipitating out of solution).

There are certain implications of the equilibrium constant. If the value is very large, over 1, the reaction is said to lie to the right (of the arrow) indicating a greater concentration of products; values less than 1 lie to the left higher formation rates of reactants, and values of one indicate equal concentrations. Knowledge of the equilibrium constant helps to determine, in an industrial setting for example, how to best produce a desirable material.

For example, in the Haber process for the formation of ammonia, the value of K is around 30 at pressures and temperatures standard for the process.

In an equilibrium between two conformers with energy difference 0, the equilibrium constant is 1 and both conformers are present in a 1:1 ratio. When the energy difference increases to 1 kcal/mol, the equilibrium constant at 25 °C becomes around 5 and the concentration of the more stable conformer gets 85%.

[edit] Thermodynamics

Relationship equilibrium constant vs Gibbs energy at three different temperatures.

It can be shown that the equilibrium constant is related to the standard Gibbs energy change of reaction as:

$K = e^{-\frac{\Delta G^\circ}{RT}}$ ,

where ΔG° is the standard Gibbs energy change of reaction, R is the gas constant, and T the absolute temperature.

This relationship is also written as:

$\ \Delta G^\circ = -RT \ln K$

A direct consequence of this important relation is the Van't Hoff equation, which relates the change in temperature to the change in the equilibrium constant given the enthalpy change. 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 in secondary reactions the final product ratio is determined according to the Curtin-Hammett principle.