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Chapter 13: Problem 32

At \(1100 \mathrm{~K}, K_{\mathrm{p}}=0.25\) for the reaction $$ 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{SO}_{3}(g) $$ What is the value of \(K\) at this temperature?

Short Answer

Expert verified
At 1100 K, the value of K for the given reaction is approximately \(K = 0.00002745\).

Step by step solution

01

Write the expression for the reaction quotient Qc, which is equal to K

Start by writing the expression for reaction quotient Qc in terms of concentrations for the given reaction: \( Q_c = \frac{[\mathrm{SO_3}]^2}{[\mathrm{SO_2}]^2 [\mathrm{O_2}]} \) At equilibrium, Qc = K, so \( K = \frac{[\mathrm{SO_3}]^2}{[\mathrm{SO_2}]^2 [\mathrm{O_2}]} \)
02

Write the expression for Kp and find its value

Kp can be written as: \( K_p = \frac{(\frac{P_{\mathrm{SO_3}}}{RT})^2}{(\frac{P_{\mathrm{SO_2}}}{RT})^2 (\frac{P_{\mathrm{O_2}}}{RT})} \) Given, Kp = 0.25. Now, we need to find the relationship between Kp and K.
03

Establish the relationship between Kp and K

To establish the relationship between Kp and K, we will divide both the expressions for K and Kp: \( \frac{K_p}{K} = \frac{(\frac{P_{\mathrm{SO_3}}^2}{(RT)^2})}{(\frac{P_{\mathrm{SO_2}}^2P_{\mathrm{O_2}}}{(RT)^3})} \) \( \frac{K_p}{K} = \frac{(\frac{P_{\mathrm{SO_3}}^2}{(RT)^3})}{(\frac{P_{\mathrm{SO_2}}^2P_{\mathrm{O_2}}}{(RT)^3})} \cdot \frac{RT}{RT} = \frac{P_{\mathrm{SO_2}}^2P_{\mathrm{O_2}}}{P_{\mathrm{SO_3}}^2} \cdot RT \)
04

Find the value of K

We know that Kp = 0.25, and from the reaction, the stoichiometric coefficients are equal so we will have delta n = 0, which means the ratio in the above expression will be 1. Thus, the relation becomes: \( \frac{0.25}{K} = RT\) Now, find the value of K at 1100 K. The temperature is given in Kelvin, so we will use the gas constant value R = 8.314 J/(mol·K): \( K = \frac{0.25}{(8.314 \,\mathrm{J/(mol\cdot K)})(1100 \,\mathrm{K})} \) Now, calculate the value of K: \( K = 0.00002745 \) So, at 1100 K, the value of K for the given reaction is approximately 0.00002745.

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Most popular questions from this chapter

The reaction $$ 2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g) \rightleftharpoons 2 \mathrm{NOBr}(g) $$ has \(K_{\mathrm{p}}=109\) at \(25^{\circ} \mathrm{C}\). If the equilibrium partial pressure of \(\mathrm{Br}_{2}\) is \(0.0159\) atm and the equilibrium partial pressure of \(\mathrm{NOBr}\) is \(0.0768\) atm, calculate the partial pressure of \(\mathrm{NO}\) at equilibrium.

Write the equilibrium expression \((K)\) for each of the following biologically important reactions. a. formation of glucose: $$ 6 \mathrm{H}_{2} \mathrm{O}(g)+6 \mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s)+6 \mathrm{O}_{2}(g) $$ b. fermentation of glucose to give ethanol: $$ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}(s) \rightleftharpoons 2 \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(g)+2 \mathrm{CO}_{2}(g) $$ c. lactic acid formation: $$ \mathrm{C}_{3} \mathrm{H}_{3} \mathrm{O}_{3} \mathrm{H}(a q)+\mathrm{H}_{2}(a q) \rightleftharpoons \mathrm{C}_{3} \mathrm{H}_{5} \mathrm{O}_{3} \mathrm{H}(a q) $$ Pyruvate actic ar

At high temperatures, elemental nitrogen and oxygen react with each other to form nitrogen monoxide: $$ \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{NO}(g) $$ Suppose the system is analyzed at a particular temperature, and the equilibrium concentrations are found to be \(\left[\mathrm{N}_{2}\right]=0.041 M\), \(\left[\mathrm{O}_{2}\right]=0.0078 M\), and \([\mathrm{NO}]=4.7 \times 10^{-4} M .\) Calculate the value of \(K\) for the reaction.

For the reaction $$ 2 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) $$ \(K=2.4 \times 10^{-3}\) at a given temperature. At equilibrium it is found that \(\left[\mathrm{H}_{2} \mathrm{O}(g)\right]=1.1 \times 10^{-1} M\) and \(\left[\mathrm{H}_{2}(g)\right]=1.9 \times 10^{-2} M\) What is the concentration of \(\mathrm{O}_{2}(g)\) under these conditions?

Lexan is a plastic used to make compact discs, eyeglass lenses, and bullet- proof glass. One of the compounds used to make Lexan is phosgene \(\left(\mathrm{COCl}_{2}\right)\), an extremely poisonous gas. Phosgene decomposes by the reaction $$ \mathrm{COCl}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{Cl}_{2}(g) $$ for which \(K_{\mathrm{p}}=6.8 \times 10^{-9}\) at \(100^{\circ} \mathrm{C}\). If pure phosgene at an initial pressure of \(1.0\) atm decomposes, calculate the equilibrium pressures of all species.
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