# 13.2 Equilibrium constants  (Page 4/14)

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## Homogeneous equilibria

A homogeneous equilibrium is one in which all of the reactants and products are present in a single solution (by definition, a homogeneous mixture). In this chapter, we will concentrate on the two most common types of homogeneous equilibria: those occurring in liquid-phase solutions and those involving exclusively gaseous species. Reactions between solutes in liquid solutions belong to one type of homogeneous equilibria. The chemical species involved can be molecules, ions, or a mixture of both. Several examples are provided here.

${\text{C}}_{2}{\text{H}}_{2}\left(aq\right)+2{\text{Br}}_{2}\left(aq\right)⇌{\text{C}}_{2}{\text{H}}_{2}{\text{Br}}_{4}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{\left[{\text{C}}_{2}{\text{H}}_{2}{\text{Br}}_{4}\right]}{\left[{\text{C}}_{2}{\text{H}}_{2}\right]\phantom{\rule{0.2em}{0ex}}{\left[{\text{Br}}_{2}\right]}^{2}}$
${\text{I}}_{2}\left(aq\right)+{\text{I}}^{\text{−}}\left(aq\right)⇌{\text{I}}_{3}{}^{\text{−}}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{\left[{\text{I}}_{3}{}^{\text{−}}\right]}{\left[{\text{I}}_{2}\right]\left[{\text{I}}^{\text{−}}\right]}$
${\text{Hg}}_{2}{}^{2+}\left(aq\right)+{\text{NO}}_{3}{}^{\text{−}}\left(aq\right)+3{\text{H}}_{3}{\text{O}}^{\text{+}}\left(aq\right)⇌2{\text{Hg}}^{2+}\left(aq\right)+{\text{HNO}}_{2}\left(aq\right)+4{\text{H}}_{2}\text{O}\left(l\right)$
${K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{{\left[{\text{Hg}}^{\text{2+}}\right]}^{2}\left[{\text{HNO}}_{2}\right]}{\left[{\text{Hg}}_{2}{}^{2+}\right]\left[{\text{NO}}_{3}{}^{\text{−}}\right]{\left[{\text{H}}_{3}{\text{O}}^{\text{+}}\right]}^{3}}$
$\text{HF}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)⇌{\text{H}}_{3}{\text{O}}^{\text{+}}\left(aq\right)+{\text{F}}^{\text{−}}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{\left[{\text{H}}_{3}{\text{O}}^{\text{+}}\right]\left[{\text{F}}^{\text{−}}\right]}{\left[\text{HF}\right]}$
${\text{NH}}_{3}\left(aq\right)+{\text{H}}_{2}\text{O}\left(l\right)⇌{\text{NH}}_{4}{}^{\text{+}}\left(aq\right)+{\text{OH}}^{\text{−}}\left(aq\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{\left[{\text{NH}}_{4}{}^{\text{+}}\right]\left[{\text{OH}}^{\text{−}}\right]}{\left[{\text{NH}}_{3}\right]}$

In each of these examples, the equilibrium system is an aqueous solution, as denoted by the aq annotations on the solute formulas. Since H 2 O( l ) is the solvent for these solutions, its concentration does not appear as a term in the K c expression, as discussed earlier, even though it may also appear as a reactant or product in the chemical equation.

Reactions in which all reactants and products are gases represent a second class of homogeneous equilibria. We use molar concentrations in the following examples, but we will see shortly that partial pressures of the gases may be used as well.

${\text{C}}_{2}{\text{H}}_{6}\left(g\right)⇌{\text{C}}_{2}{\text{H}}_{4}\left(g\right)+{\text{H}}_{2}\left(g\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{\left[{\text{C}}_{2}{\text{H}}_{4}\right]\phantom{\rule{0.2em}{0ex}}\left[{\text{H}}_{2}\right]}{\left[{\text{C}}_{2}{\text{H}}_{6}\right]}$
$3{\text{O}}_{2}\left(g\right)⇌2{\text{O}}_{3}\left(g\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{{\left[{\text{O}}_{3}\right]}^{2}}{{\left[{\text{O}}_{2}\right]}^{3}}$
${\text{N}}_{2}\left(g\right)+3{\text{H}}_{2}\left(g\right)⇌2{\text{NH}}_{3}\left(g\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{{\left[{\text{NH}}_{3}\right]}^{2}}{\left[{\text{N}}_{2}\right]{\phantom{\rule{0.2em}{0ex}}\left[{\text{H}}_{2}\right]}^{3}}$
${\text{C}}_{3}{\text{H}}_{8}\left(g\right)+5{\text{O}}_{2}\left(g\right)\phantom{\rule{0.2em}{0ex}}⇌3{\text{CO}}_{2}\left(g\right)+4{\text{H}}_{2}\text{O}\left(g\right)\phantom{\rule{5em}{0ex}}{K}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{{\left[{\text{CO}}_{2}\right]}^{3}{\left[{\text{H}}_{2}\text{O}\right]}^{4}}{\left[{\text{C}}_{3}{\text{H}}_{8}\right]{\left[{\text{O}}_{2}\right]}^{5}}$

Note that the concentration of H 2 O( g ) has been included in the last example because water is not the solvent in this gas-phase reaction and its concentration (and activity) changes.

Whenever gases are involved in a reaction, the partial pressure of each gas can be used instead of its concentration in the equation for the reaction quotient because the partial pressure of a gas is directly proportional to its concentration at constant temperature. This relationship can be derived from the ideal gas equation, where M is the molar concentration of gas, $\frac{n}{V}.$

$PV=nRT$
$P=\left(\frac{n}{V}\right)RT$
$=MRT$

Thus, at constant temperature, the pressure of a gas is directly proportional to its concentration.

Using the partial pressures of the gases, we can write the reaction quotient for the system ${\text{C}}_{2}{\text{H}}_{6}\left(g\right)⇌{\text{C}}_{2}{\text{H}}_{4}\left(g\right)+{\text{H}}_{2}\left(g\right)$ by following the same guidelines for deriving concentration-based expressions:

${Q}_{P}=\phantom{\rule{0.2em}{0ex}}\frac{{P}_{{\text{C}}_{2}{\text{H}}_{4}}{P}_{{\text{H}}_{2}}}{{P}_{{\text{C}}_{2}{\text{H}}_{6}}}$

In this equation we use Q P to indicate a reaction quotient written with partial pressures: ${P}_{{\text{C}}_{2}{\text{H}}_{6}}$ is the partial pressure of C 2 H 6 ; ${P}_{{\text{H}}_{2}},$ the partial pressure of H 2 ; and ${P}_{{\text{C}}_{2}{\text{H}}_{6}},$ the partial pressure of C 2 H 4 . At equilibrium:

${K}_{P}={Q}_{P}=\phantom{\rule{0.2em}{0ex}}\frac{{P}_{{\text{C}}_{2}{\text{H}}_{4}}{P}_{{\text{H}}_{2}}}{{P}_{{\text{C}}_{2}{\text{H}}_{6}}}$

The subscript P in the symbol K P    designates an equilibrium constant derived using partial pressures instead of concentrations. The equilibrium constant, K P , is still a constant, but its numeric value may differ from the equilibrium constant found for the same reaction by using concentrations.

Conversion between a value for K c    , an equilibrium constant expressed in terms of concentrations, and a value for K P , an equilibrium constant expressed in terms of pressures, is straightforward (a K or Q without a subscript could be either concentration or pressure).

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