The equilibrium constant \(K_{\mathrm{p}}\) for the reaction $$\mathrm{CCl}_{4}(g) \rightleftharpoons \mathrm{C}(s)+2 \mathrm{Cl}_{2}(g)$$ at \(700^{\circ} \mathrm{C}\) is \(0.76 .\) Determine the initial pressure of carbon tetrachloride that will produce a total equilibrium pressure of 1.20 \(\mathrm{atm}\) at \(700^{\circ} \mathrm{C} .\)

Let x be the moles of CCl₄ that reacted at equilibrium. The initial pressure of CCl₄ is P(CCl₄) and since there was no Cl₂ in the beginning, we can write the pressures at equilibrium as: P(CCl₄) - x for CCl₄ 2x for Cl₂ Note that, since Carbon in the reaction is in solid state, it doesn't affect the pressures. The total equilibrium pressure is given as 1.20 atm. So, (P(CCl₄) - x) + 2x = 1.20 Simplifying the equation, we get: P(CCl₄) + x = 1.20

We are given the equilibrium constant Kp = 0.76. The expression for Kp, using the pressures of gases at equilibrium, is: Kp = \(\dfrac{(P(Cl_2))^2}{P(CCl_4)}\) At equilibrium, the pressure of Cl₂ is 2x. Thus, we can rewrite the expression as: 0.76 = \(\dfrac{(2x)^2}{P(CCl_4) - x}\)

Now, we can rewrite our Kp equation in terms of x and P(CCl₄) only: 0.76 = \(\dfrac{4x^2}{P(CCl_4) - x}\) Using the equation from Step 1 (P(CCl₄) + x = 1.20), we can eliminate P(CCl₄) and solve for x: P(CCl₄) - x = 1.20 - 2x Now, we can substitute this expression in our Kp equation: 0.76 = \(\dfrac{4x^2}{1.20 - 2x}\) Let's isolate x in the equation: 0.76(1.20 - 2x) = 4x² After simplifying the equation, we get: 0.912 - 1.52x = 4x² Rearranging terms, we obtain the quadratic equation: 4x² + 1.52x - 0.912 = 0 We can solve this quadratic equation for x using any method (e.g., quadratic formula, factoring, etc.). Here, we will use the quadratic formula: x = \(\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where a = 4, b = 1.52, and c = -0.912. Calculating x, we get two possible values: x₁ ≈ 0.2304 and x₂ ≈ -0.991. Since x represents moles reacting and cannot be negative, we discard x₂ and use x₁: x ≈ 0.2304

Now that we have the value of x, we can find the initial pressure of CCl₄ using the equation from Step 1: P(CCl₄) + x = 1.20 P(CCl₄) + 0.2304 = 1.20 P(CCl₄) ≈ 1.20 - 0.2304 P(CCl₄) ≈ 0.9696 atm The initial pressure of carbon tetrachloride that will produce a total equilibrium pressure of 1.20 atm at 700°C is approximately 0.9696 atm.