Problem: At 700 K, CCl4 decomposes to carbon and chlorine. The Kp for the decomposition is 0.76. Find the starting pressure of CCl4 at this temperature that produces a total pressure of 1.6 atm at equilibrium.        

Reaction:         CCl₄(g) ⇌ C(s) + 2 Cl₂(g)             K_p = 0.76

Step 1: Construct an ICE chart for the equilibrium reaction.

Let y = initial pressure of CCl₄

Step 2: Calculate the change (x) in the reaction using the equilibrium constant.

(y – x) + 2x = total pressure

y – x + 2x = 1.6

y + x = 1.6

y = 1.6 – x

$$\begin{gathered} \boxed{{\mathbf{K}}_{\mathbf{p}}\mathbf{=}\frac{\mathbf{products}}{\mathbf{reactants}}} \\ {\mathbf{K}}_{\mathbf{p}}\mathbf{=}\frac{{\left(P_{{\mathrm{Cl}}_2}\right)}^{\mathbf{2}}}{{\mathbf{P}}_{{\mathbf{CCl}}_{\mathbf{4}}}} \\ \mathbf{0}\mathbf{.}\mathbf{76}\mathbf{=}\frac{{\left(2x\right)}^{\mathbf{2}}}{\mathbf{y}\mathbf{-}\mathbf{x}} \\ \mathbf{0}\mathbf{.}\mathbf{76}\mathbf{=}\frac{{\left(2x\right)}^2}{(1.6-x)\mathbf{-}\mathbf{x}} \\ \mathbf{0}\mathbf{.}\mathbf{76}(1.6-2x)\mathbf{=}\frac{4x^2}{\cancel{1.6-2x}}\left(\cancel{1.6-2x}\right) \\ \mathbf{1}\mathbf{.}\mathbf{216}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{52}\mathbf{x}\mathbf{=}\mathbf{4}{\mathbf{x}}^{\mathbf{2}} \\ \boxed{\mathbf{0}\mathbf{=}\mathbf{4}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{52}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{216}} \end{gathered}$$

Now, we will use the quadratic formula to solve for x. The quadratic formula is:

$\boxed{\mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}}$

Solving for x:

$$\begin{gathered} \mathbf{x}\mathbf{=}\frac{\mathbf{-}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{52}\mathbf{)}\mathbf{\pm }\sqrt{\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{52}{\mathbf{)}}^{\mathbf{2}}\mathbf{-}\mathbf{4}\mathbf{(}\mathbf{4}\mathbf{\left)}\mathbf{\right(}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{216}\mathbf{)}}}{\mathbf{2}\mathbf{\left(}\mathbf{4}\mathbf{\right)}} \\ \mathbf{x}\mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{52}\mathbf{\pm }\sqrt{\mathbf{2}\mathbf{.}\mathbf{3104}\mathbf{+}\mathbf{19}\mathbf{.}\mathbf{456}}}{\mathbf{8}} \\ \mathbf{x}\mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{52}\mathbf{\pm }\sqrt{\mathbf{21}\mathbf{.}\mathbf{7664}}}{\mathbf{8}} \\ \mathbf{x}\mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{52}\mathbf{\pm }\mathbf{4}\mathbf{.}\mathbf{66}}{\mathbf{8}} \end{gathered}$$

The calculation then splits into two ways: