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Chapter 15: Problem 21

The equilibrium constant \(K_{P}\) for the reaction $$ \mathrm{PCl}_{5}(g) \rightleftharpoons \mathrm{PCl}_{3}(g)+\mathrm{Cl}_{2}(g) $$ is 1.05 at \(250^{\circ} \mathrm{C}\). The reaction starts with a mixture of \(\mathrm{PCl}_{5}, \mathrm{PCl}_{3},\) and \(\mathrm{Cl}_{2}\) at pressures of \(0.177 \mathrm{~atm}\) 0.223 atm, and 0.111 atm, respectively, at \(250^{\circ} \mathrm{C}\). When the mixture comes to equilibrium at that temperature, which pressures will have decreased and which will have increased? Explain why.

Short Answer

Expert verified
To know which pressures increased or decreased at equilibrium, we calculate the change in pressure (x). By plugging equilibrium pressures into the equilibrium constant formula and solving for x (change in pressure), we can determine whether pressures of reactants or products have increased or decreased.

Step by step solution

01

Definition of equilibrium constant

The equilibrium constant for the given reaction at 250 degrees Celsius is denoted as \(K_{P}\) and it equals 1.05. The formula to calculate the equilibrium constant for a reaction is given by \(K_{p} = \frac{[PCl_{3}]_{eq}[Cl_{2}]_{eq}}{[PCl_{5}]_{eq}}\), where [PCl_{5}]_{eq}, [PCl_{3}]_{eq} and [Cl_{2}]_{eq} represent the equilibrium pressures of PCl_{5}, PCl_{3} and Cl_{2}, respectively.
02

Application of Change in Pressure

Since we know the initial pressures and the equilibrium constant, let's denote the change in the pressure of the products (PCl_{3} and Cl_2) as x. Therefore, at equilibrium, the pressure of PCl_{5} decreases by x to become (0.177-x) atm. The pressures of PCl_{3} and Cl_{2} increase by x each to become (0.223+x) atm and (0.111+x) atm, respectively.
03

Calculating the value of x

Now substitute the equilibrium pressures into the \(K_{P}\) formula and solve for x. \[1.05 = \frac{(0.223+x)(0.111+x)}{0.177-x}\]. By solving this equation for x, we find the value that will bring this system to equilibrium.
04

Interpretation

Once we have calculated x, we can determine which pressures have decreased and increased. If x is positive, then the pressures of PCl_{3} and Cl_{2} have increased, while the pressure of PCl_{5} has decreased. If x is negative, the opposite would be true. This would be due to Le Chatelier's Principle, which says the reaction will shift in the direction that reduces any stress (such as changes in pressure or temperature).

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Most popular questions from this chapter

Consider the statement: The equilibrium constant of a reacting mixture of solid \(\mathrm{NH}_{4} \mathrm{Cl}\) and gaseous \(\mathrm{NH}_{3}\) and \(\mathrm{HCl}\) is \(0.316 .\) List three important pieces of information that are missing from this statement.

The equilibrium constant \(\left(K_{\mathrm{c}}\right)\) for the reaction $$ 2 \mathrm{HCl}(g) \rightleftharpoons \mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) $$ is \(4.17 \times 10^{-34}\) at \(25^{\circ} \mathrm{C}\). What is the equilibrium constant for the reaction $$ \mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g) $$ at the same temperature?

When a gas was heated under atmospheric conditions, its color was found to deepen. Heating above \(150^{\circ} \mathrm{C}\) caused the color to fade, and at \(550^{\circ} \mathrm{C}\) the color was barely detectable. However, at \(550^{\circ} \mathrm{C}\), the color was partially restored by increasing the pressure of the system. Which of these best fits this description? Justify your choice. (a) A mixture of hydrogen and bromine, (b) pure bromine, (c) a mixture of nitrogen dioxide and dinitrogen tetroxide. (Hint: Bromine has a reddish color and nitrogen dioxide is a brown gas. The other gases are colorless.)

A quantity of 0.20 mole of carbon dioxide was heated at a certain temperature with an excess of graphite in a closed container until the following equilibrium was reached: $$ \mathrm{C}(s)+\mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g) $$ Under this condition, the average molar mass of the gases was found to be \(35 \mathrm{~g} / \mathrm{mol}\). (a) Calculate the mole fractions of \(\mathrm{CO}\) and \(\mathrm{CO}_{2}\). (b) What is the \(K_{P}\) for the equilibrium if the total pressure was 11 atm? (Hint: The average molar mass is the sum of the products of the mole fraction of each gas and its molar mass.)

Consider this equilibrium system: $$ \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{SO}_{2} \mathrm{Cl}_{2}(g) $$ Predict how the equilibrium position would change if (a) \(\mathrm{Cl}_{2}\) gas were added to the system, (b) \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) were removed from the system, (c) \(\mathrm{SO}_{2}\) were removed from the system. The temperature remains constant
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