# Question 377c6

Dec 23, 2016

["Cl"_2] = "0.0239 M"

#### Explanation:

You are dealing with the following equilibrium reaction

${\text{PCl"_ (5(g)) rightleftharpoons "PCl" _ (3(g)) + "Cl}}_{2 \left(g\right)}$

You also know that the equilibrium constant for this reaction at ${327}^{\circ} \text{C}$ is equal to

${K}_{e q} = 2.24 \cdot {10}^{- 2}$

Notice that you have ${K}_{e q} < 1$; this should tell you that at equilibrium, the reaction vessel should contain more reactant than products.

In other words, you can expect the equilibrium concentration of phosphorus pentachloride, ${\text{PCl}}_{5}$, to decrease only slightly compared to its initial value.

You will have less reactant at equilibrium than what you started with because the vessel doesn't contain chlorine gas, ${\text{Cl}}_{2}$, so right from the start, you can predict that the reaction will produce some chlorine gas.

Given the value of ${K}_{e q}$, we can expect a very low concentration of chlorine gas at equilibrium.

Since the vessel has a volume of $\text{1.00 L}$, you can treat moles and molarity interchangeably. Set up an ICE table to help you find the equilibrium concentration of the three species

${\text{ ""PCl"_ (5(g)) " "rightleftharpoons" " "PCl" _ (3(g))" " + " ""Cl}}_{2 \left(g\right)}$

color(purple)("I")color(white)(aaaacolor(black)(0.235)aaaaaaaaaacolor(black)(0.174)aaaaaaaaacolor(black)(0)
color(purple)("C")color(white)(aaacolor(black)((-x))aaaaaaaaaacolor(black)((+x))aaaaaacolor(black)((+x))
color(purple)("E")color(white)(aacolor(black)(0.235-x)aaaaaaacolor(black)(0.174+x)aaaaaaacolor(black)(x)

By definition, the equilibrium constant is equal to

${K}_{e q} = \left(\left[{\text{PCl"_3] * ["Cl"_2])/(["PCl}}_{5}\right]\right)$

In your case, you will have

$2.24 \cdot {10}^{- 2} = \frac{\left(0.174 + x\right) \cdot x}{0.235 - x}$

This is equivalent to

${x}^{2} + 0.1964 x - 0.005264 = 0$

Tis quadratic equation ahs two solutiuons, one positive and one negative. Since $x$ represents concentration, discard the negative value and go with

$x = 0.0239$

Therefore, you can say that at equilibrium, the reaction vessel will contain

color(darkgreen)(ul(color(black)(["Cl"_2] = "0.0239 M")))#

The answer is rounded to three sig figs.

As predicted, the equilibrium concentration of chlorine gas is significantly lower than that of phosphorus pentachloride and that of phosphorus trichloride.