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Chapter 14: Problem 115

The forward and reverse rate constants for the reaction \(\mathrm{A}(g)+\mathrm{B}(g) \rightleftharpoons \mathrm{C}(g)\) are \(3.6 \times 10^{-3} / M \cdot \mathrm{s}\) and \(8.7 \times 10^{-4} \mathrm{~s}^{-1},\) respectively, at \(323 \mathrm{~K}\). Calculate the equilibrium pressures of all the species starting at \(P_{\mathrm{A}}=1.6 \mathrm{~atm}\) and \(P_{\mathrm{B}}=0.44 \mathrm{~atm}\).

Short Answer

Expert verified
Compute Kp and use it along with the initial pressures mentioned to set up a quadratic equation for ∆P. Solve this equation, and replace ∆P in the expressions from Step 2 to obtain the equilibrium pressures for gases A, B, and C.

Step by step solution

01

Determine the equilibrium constant

The equilibrium constant (Kp) can be determined using the equation \(Kp = k_{forward} / k_{backward}\), where \(k_{forward} = 3.6 \times 10^{-3} M^{-1} s^{-1}\) and \(k_{backward} = 8.7 \times 10^{-4} s^{-1}\). Substituting these values into the equation gives \(Kp = (3.6 \times 10^{-3}) / (8.7 \times 10^{-4})\). Evaluate this expression to find Kp.
02

Formulate the expressions for pressures at equilibrium

Next, express the pressures at equilibrium in terms of the reaction's stoichiometry and a common variable, which could be the change in pressure (∆P). For gases A and B, the pressure decreases from the initial pressure by ∆P as they react to form gas C. Hence, their pressures at equilibrium can be expressed as \(P_A = 1.6 - ∆P\) and \(P_B = 0.44 - ∆P\), respectively. For gas C, the pressure increases by ∆P as it is formed, so its pressure at equilibrium can be expressed as \(P_C = ∆P\).
03

Set up and solve the equilibrium condition

The equilibrium condition is that the ratio of the pressures of the products to the pressures of the reactants equals to the equilibrium constant Kp, which can be expressed as \(Kp = P_C / (P_A \times P_B)\). Substituting the expressions for pressures at equilibrium from Step 2 into this equation gives \(Kp = ∆P / ((1.6 - ∆P) \times (0.44 - ∆P))\). Substituting Kp from Step 1 into this equation gives a quadratic equation for ∆P. Solve this quadratic equation to find the value of ∆P, and then substitute ∆P into expressions from Step 2 to find the equilibrium pressures of A, B, and C.

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Most popular questions from this chapter

A student placed a few ice cubes in a drinking glass with water. A few minutes later she noticed that some of the ice cubes were fused together. Explain what happened.

A sealed glass bulb contains a mixture of \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) gases. Describe what happens to the following properties of the gases when the bulb is heated from \(20^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}:\) (a) color, (b) pressure, (c) average molar mass, (d) degree of dissociation (from \(\mathrm{N}_{2} \mathrm{O}_{4}\) to \(\mathrm{NO}_{2}\) ), (e) density. Assume that volume remains constant. (Hint: \(\mathrm{NO}_{2}\) is a brown gas; \(\mathrm{N}_{2} \mathrm{O}_{4}\) is colorless.)

A quantity of 1.0 mole of \(\mathrm{N}_{2} \mathrm{O}_{4}\) was introduced into an evacuated vessel and allowed to attain equilibrium at a certain temperature $$\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)$$ The average molar mass of the reacting mixture was \(70.6 \mathrm{~g} / \mathrm{mol} .\) (a) Calculate the mole fractions of the gases. (b) Calculate \(K_{P}\) for the reaction if the total pressure was 1.2 atm. (c) What would be the mole fractions if the pressure were increased to 4.0 atm by reducing the volume at the same temperature?

Write equilibrium constant expressions for \(K_{\mathrm{c}},\) and for \(K_{P}\), if applicable, for the following processes: (a) \(2 \mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)+\mathrm{O}_{2}(g)\) (b) \(3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}_{3}(g)\) (c) \(\mathrm{CO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{COCl}_{2}(g)\) (d) \(\mathrm{H}_{2} \mathrm{O}(g)+\mathrm{C}(s) \rightleftharpoons \mathrm{CO}(g)+\mathrm{H}_{2}(g)\) (e) \(\mathrm{HCOOH}(a q) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{HCOO}^{-}(a q)\) (f) \(2 \mathrm{HgO}(s) \rightleftharpoons 2 \mathrm{Hg}(l)+\mathrm{O}_{2}(g)\)

Write the equilibrium constant expressions for \(K_{\mathrm{c}}\) and \(K_{P}\), if applicable, for the following reactions: (a) \(2 \mathrm{NO}_{2}(g)+7 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g)+4 \mathrm{H}_{2} \mathrm{O}(l)\) (b) \(2 \mathrm{ZnS}(s)+3 \mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{ZnO}(s)+2 \mathrm{SO}_{2}(g)\) (c) \(\mathrm{C}(s)+\mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)\) (d) \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}(a q) \rightleftharpoons\) $$\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}^{-}(a q)+\mathrm{H}^{+}(a q)$$
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