Calculation of an equilibrium constant
Iodine molecules react reversibly with iodide ions to produce triiodide ions.
${\text{I}}_{2}(aq)+{\text{I}}^{\text{\u2212}}(aq)\rightleftharpoons {\text{I}}_{3}{}^{\text{\u2212}}(aq)$
If a solution with the concentrations of I
_{2} and I
^{−} both equal to 1.000
$\times $ 10
^{−3}
M before reaction gives an equilibrium concentration of I
_{2} of 6.61
$\times $ 10
^{−4}
M , what is the equilibrium constant for the reaction?
Solution
We will begin this problem by calculating the changes in concentration as the system goes to equilibrium. Then we determine the equilibrium concentrations and, finally, the equilibrium constant. First, we set up a table with the initial concentrations, the changes in concentrations, and the equilibrium concentrations using −
x as the change in concentration of I
_{2} .
Since the equilibrium concentration of I
_{2} is given, we can solve for
x . At equilibrium the concentration of I
_{2} is 6.61
$\times $ 10
^{−4}
M so that
$1.000\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}-x=6.61\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}$
$x=1.000\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}-6.61\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}$
$=3.39\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\phantom{\rule{0.2em}{0ex}}M$
Now we can fill in the table with the concentrations at equilibrium.
We now calculate the value of the equilibrium constant.
${K}_{c}={Q}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{[{\text{I}}_{3}{}^{\text{\u2212}}]}{[{\text{I}}_{2}]\left[{\text{I}}^{\text{\u2212}}\right]}$
$=\phantom{\rule{0.2em}{0ex}}\frac{3.39\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\phantom{\rule{0.2em}{0ex}}M}{(6.61\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\phantom{\rule{0.2em}{0ex}}M)(6.61\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\phantom{\rule{0.2em}{0ex}}M)}\phantom{\rule{0.2em}{0ex}}=776$
Check your learning
Ethanol and acetic acid react and form water and ethyl acetate, the solvent responsible for the odor of some nail polish removers.
${\text{C}}_{2}{\text{H}}_{5}\text{OH}+{\text{CH}}_{3}{\text{CO}}_{2}\text{H}\rightleftharpoons {\text{CH}}_{3}{\text{CO}}_{2}{\text{C}}_{2}{\text{H}}_{5}+{\text{H}}_{2}\text{O}$
When 1 mol each of C
_{2} H
_{5} OH and CH
_{3} CO
_{2} H are allowed to react in 1 L of the solvent dioxane, equilibrium is established when
$\frac{1}{3}$ mol of each of the reactants remains. Calculate the equilibrium constant for the reaction. (Note: Water is not a solvent in this reaction.)
Got questions? Get instant answers now!
If we know the equilibrium constant for a reaction and know the concentrations at equilibrium of all reactants and products except one, we can calculate the missing concentration.
Calculation of a missing equilibrium concentration
Nitrogen oxides are air pollutants produced by the reaction of nitrogen and oxygen at high temperatures. At 2000 °C, the value of the equilibrium constant for the reaction,
${\text{N}}_{2}(g)+{\text{O}}_{2}(g)\rightleftharpoons 2\text{NO}(g),$ is 4.1
$\times $ 10
^{−4} . Find the concentration of NO(
g ) in an equilibrium mixture with air at 1 atm pressure at this temperature. In air, [N
_{2} ] = 0.036 mol/L and [O
_{2} ] 0.0089 mol/L.
Solution
We are given all of the equilibrium concentrations except that of NO. Thus, we can solve for the missing equilibrium concentration by rearranging the equation for the equilibrium constant.
${K}_{c}={Q}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{{\left[\text{NO}\right]}^{2}}{\left[{\text{N}}_{2}\right]\phantom{\rule{0.2em}{0ex}}\left[{\text{O}}_{2}\right]}$
${\left[\text{NO}\right]}^{2}={K}_{c}\left[{\text{N}}_{2}\right]\phantom{\rule{0.2em}{0ex}}\left[{\text{O}}_{2}\right]$
$\left[\text{NO}\right]=\phantom{\rule{0.2em}{0ex}}\sqrt{{K}_{c}\left[{\text{N}}_{2}\right]\phantom{\rule{0.2em}{0ex}}\left[{\text{O}}_{2}\right]}$
$=\phantom{\rule{0.2em}{0ex}}\sqrt{(4.1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}})(0.036)(0.0089)}$
$=\phantom{\rule{0.2em}{0ex}}\sqrt{1.31\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}}$
$=3.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}$
Thus [NO] is 3.6
$\times $ 10
^{−4} mol/L at equilibrium under these conditions.
We can check our answer by substituting all equilibrium concentrations into the expression for the reaction quotient to see whether it is equal to the equilibrium constant.
${Q}_{c}=\phantom{\rule{0.2em}{0ex}}\frac{{\left[\text{NO}\right]}^{2}}{\left[{\text{N}}_{2}\right]\phantom{\rule{0.2em}{0ex}}\left[{\text{O}}_{2}\right]}$
$=\phantom{\rule{0.2em}{0ex}}\frac{{(3.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}})}^{2}}{(0.036)(0.0089)}$
${Q}_{c}=4.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}={K}_{c}$
The answer checks; our calculated value gives the equilibrium constant within the error associated with the significant figures in the problem.
Check your learning
The equilibrium constant for the reaction of nitrogen and hydrogen to produce ammonia at a certain temperature is 6.00
$\times $ 10
^{−2} . Calculate the equilibrium concentration of ammonia if the equilibrium concentrations of nitrogen and hydrogen are 4.26
M and 2.09
M , respectively.
Got questions? Get instant answers now!