# Given K_c = 0.36 at 2000^@ "C" for "N"_2"O"_4(g) rightleftharpoons "NO"_2(g), if the initial concentration of "NO"_2 is "1 M", what are the equilibrium concentrations of "N"_2"O"_4(g) and "NO"_2(g)?

Aug 23, 2017

${\text{N"_2"O}}_{4} : 0.33$ $M$

${\text{NO}}_{2} : 0.34$ $M$

#### Explanation:

We're asked to find the equilibrium concentrations of ${\text{N"_2"O}}_{4}$ and ${\text{NO}}_{2}$, given the initial ${\text{NO}}_{2}$ concentration.

The equilibrium constant expression is given by

K_c = (["NO"_2]^2)/(["N"_2"O"_4]) = ul(0.36)" " $\left(2000 \textcolor{w h i t e}{l} \text{^"o""C}\right)$

We'll do our I.C.E. chart in the form of bullet points, for fun. Then, our initial concentrations are

INITIAL

• ${\text{N"_2"O}}_{4} :$ $0$

• ${\text{NO}}_{2} :$ $1$ $M$

According to the coefficients of the reaction, the amount by which ${\text{NO}}_{2}$ decreases is two times as much as the amount by which ${\text{N"_2"O}}_{4}$ increases:

CHANGE

• ${\text{N"_2"O}}_{4} :$ $+ x$

• ${\text{NO}}_{2} :$ $- 2 x$

And so the final concentrations are

FINAL

• ${\text{N"_2"O}}_{4} :$ $x$

• ${\text{NO}}_{2} :$ $1$ $M$ $- 2 x$

Plugging these into the equilibrium constant expression gives us

${K}_{c} = \frac{{\left(1 - 2 x\right)}^{2}}{x} = \underline{0.36} \text{ }$ (2000color(white)(l)""^"o""C", excluding units)

Now we solve for $x$:

$\frac{4 {x}^{2} - 4 x + 1}{x} = 0.36$

$4 {x}^{2} - 4 x + 1 = 0.36 x$

$4 {x}^{2} - 4.36 x + 1 = 0$

$x = \frac{4.36 \pm \sqrt{{\left(4.36\right)}^{2} - 4 \left(4\right) \left(1\right)}}{8} = 0.328 \textcolor{w h i t e}{l} \text{or} \textcolor{w h i t e}{l} 0.762$
If we plug the larger solution in for $x$ in the final ${\text{NO}}_{2}$ concentration $\left(1 - 2 x\right)$, we would obtain a worthless negative value. Therefore, we use the smaller solution to calculate the final concentrations:
color(red)("final N"_2"O"_4) = x = color(red)(ulbar(|stackrel(" ")(" "0.33color(white)(l)M" ")|)
color(blue)("final NO"_2) = 1-2x = 1-2(0.328) = color(blue)(ulbar(|stackrel(" ")(" "0.34color(white)(l)M" ")|)
each rounded to $2$ significant figures.