Developing an Equilibrium Constant Expression
In 1864, the Norwegian chemists Cato Guldberg (1836-1902) and Peter Waage (1833-1900) carefully measured the compositions of many reaction systems at equilibrium. They discovered that for any reversible reaction of the general form
[aA+bB rightleftharpoons cC+dD label{Eq6}]
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where A and B are reactants, C and D are products, and a, b, c, and d are the stoichiometric coefficients in the balanced chemical equation for the reaction, the ratio of the product of the equilibrium concentrations of the products (raised to their coefficients in the balanced chemical equation) to the product of the equilibrium concentrations of the reactants (raised to their coefficients in the balanced chemical equation) is always a constant under a given set of conditions. This relationship is known as the law of mass action and can be stated as follows:
[K=dfrac{[C]^c[D]^d}{[A]^a[B]^b} label{Eq7}]
where (K) is the equilibrium constant for the reaction. Equation (ref{Eq6}) is called the equilibrium equation, and the right side of Equation (ref{Eq7}) is called the equilibrium constant expression. The relationship shown in Equation (ref{Eq7}) is true for any pair of opposing reactions regardless of the mechanism of the reaction or the number of steps in the mechanism.
The equilibrium constant can vary over a wide range of values. The values of (K) shown in Table (PageIndex{2}), for example, vary by 60 orders of magnitude. Because products are in the numerator of the equilibrium constant expression and reactants are in the denominator, values of K greater than (10^3) indicate a strong tendency for reactants to form products. In this case, chemists say that equilibrium lies to the right as written, favoring the formation of products. An example is the reaction between (H_2) and (Cl_2) to produce (HCl), which has an equilibrium constant of (1.6 times 10^{33}) at 300 K. Because (H_2) is a good reductant and (Cl_2) is a good oxidant, the reaction proceeds essentially to completion. In contrast, values of (K) less than (10^{-3}) indicate that the ratio of products to reactants at equilibrium is very small. That is, reactants do not tend to form products readily, and the equilibrium lies to the left as written, favoring the formation of reactants.
Table (PageIndex{2}): Equilibrium Constants for Selected Reactions* Reaction Temperature (K) Equilibrium Constant (K) *Equilibrium constants vary with temperature. The K values shown are for systems at the indicated temperatures. (S_{(s)}+O_{2(g)} rightleftharpoons SO_{2(g)}) 300 (4.4 times 10^{53}) (2H_{2(g)}+O_{2(g)} rightleftharpoons 2H2O_{(g)}) 500 (2.4 times 10^{47}) (H_{2(g)}+Cl_{2(g)} rightleftharpoons 2 HCl_{(g)}) 300 (1.6 times 10^{33}) (H_{2(g)}+Br_{2(g)} rightleftharpoons 2HBr_{(g)}) 300 (4.1 times 10^{18}) (2NO_{(g)}+O_{2(g)} rightleftharpoons 2NO_{2(g)}) 300 (4.2 times 10^{13}) (3H_{2(g)}+N_{2(g)} rightleftharpoons 2NH_{3(g)}) 300 (2.7 times 10^{8}) (H_{2(g)}+D_{2(g)} rightleftharpoons 2HD_{(g)}) 100 (1.92) (H_{2(g)}+I_{2(g)} rightleftharpoons 2HI_{(g)}) 300 (2.9 times 10^{−1}) (I_{2(g)} rightleftharpoons 2I_{(g)}) 800 (4.6 times^{ 10−7}) (Br_{2(g)} rightleftharpoons 2Br_{(g)}) 1000 (4.0 times 10^{−7}) (Cl_{2(g)} rightleftharpoons 2Cl_{(g)}) 1000 (1.8 times 10^{−9}) (F_{2(g)} rightleftharpoons 2F_{(g)}) 500 (7.4 times 10^{−13})
You will also notice in Table (PageIndex{2}) that equilibrium constants have no units, even though Equation (ref{Eq7}) suggests that the units of concentration might not always cancel because the exponents may vary. In fact, equilibrium constants are calculated using “effective concentrations,” or activities, of reactants and products, which are the ratios of the measured concentrations to a standard state of 1 M. As shown in Equation (ref{Eq8}), the units of concentration cancel, which makes (K) unitless as well:
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[ dfrac{[A]_{measured}}{[A]_{standard; state}}=dfrac{cancel{M}}{cancel{M}} = dfrac{cancel{frac{mol}{L}}}{cancel{frac{mol}{L}}} label{Eq8}]
In fact, equilibrium constants are calculated using “effective concentrations,” or activities, of reactants and products, which are the ratios of the measured concentrations to a standard state of 1 M.
Many reactions have equilibrium constants between 1000 and 0.001 ((10^3 ge K ge 10^{−3})), neither very large nor very small. At equilibrium, these systems tend to contain significant amounts of both products and reactants, indicating that there is not a strong tendency to form either products from reactants or reactants from products. An example of this type of system is the reaction of gaseous hydrogen and deuterium, a component of high-stability fiber-optic light sources used in ocean studies, to form (HD):
[H_{2(g)}+D_{2(g)} rightleftharpoons 2HD_{(g)} label{Eq9}]
The equilibrium constant expression for this reaction is
[K= dfrac{[HD]^2}{[H_2][D_2]}]
with (K) varying between 1.9 and 4 over a wide temperature range (100-1000 K). Thus an equilibrium mixture of (H_2), (D_2), and (HD) contains significant concentrations of both product and reactants.
Figure (PageIndex{3}) summarizes the relationship between the magnitude of K and the relative concentrations of reactants and products at equilibrium for a general reaction, written as reactants (rightleftharpoons) products. Because there is a direct relationship between the kinetics of a reaction and the equilibrium concentrations of products and reactants (Equations (ref{Eq8}) and (ref{Eq7})), when (k_f >> k_r), (K) is a large number, and the concentration of products at equilibrium predominate. This corresponds to an essentially irreversible reaction. Conversely, when (k_f << k_r), (K) is a very small number, and the reaction produces almost no products as written. Systems for which (k_f ≈ k_r) have significant concentrations of both reactants and products at equilibrium.
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Writing an equation in different but chemically equivalent forms also causes both the equilibrium constant expression and the magnitude of the equilibrium constant to be different. For example, we could write the equation for the reaction
[2NO_2 rightleftharpoons N_2O_4]
as
[NO_2 rightleftharpoons frac{1}{2}N_2O_4]
with the equilibrium constant K″ is as follows:
[ K′′=dfrac{[N_2O_4]^{1/2}}{[NO_2]} label{Eq14}]
The values for K′ (Equation (ref{Eq13})) and K″ are related as follows:
[ K′′=(K’)^{1/2}=sqrt{K’} label{Eq15}]
In general, if all the coefficients in a balanced chemical equation were subsequently multiplied by (n), then the new equilibrium constant is the original equilibrium constant raised to the (n^{th}) power.
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