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Chapter 14: Problem 117

Consider the reaction between \(\mathrm{NO}_{2}\) and \(\mathrm{N}_{2} \mathrm{O}_{4}\) in a closed container: $$\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)$$ Initially, 1 mole of \(\mathrm{N}_{2} \mathrm{O}_{4}\) is present. At equilibrium, \(\alpha\) mole of \(\mathrm{N}_{2} \mathrm{O}_{4}\) has dissociated to form \(\mathrm{NO}_{2}\). (a) Derive an expression for \(K_{P}\) in terms of \(\alpha\) and \(P\), the total pressure. (b) How does the expression in (a) help you predict the shift in equilibrium due to an increase in \(P ?\) Does your prediction agree with Le Châtelier's principle?

Short Answer

Expert verified
The expression for \(K_P\) in terms of \(\alpha\) and \(P\) is \(K_P = 4\alpha^2P/(1 - \alpha)\). An increase in pressure would drive the equilibrium to the right in favor of dissociation, consistent with Le Châtelier's principle.

Step by step solution

01

Understand the given equilibrium reaction

The chemical equilibrium reaction given is: \(\mathrm{N}_{2}\mathrm{O}_{4}(g) \rightleftharpoons 2\mathrm{NO}_{2}(g)\). It is mentioned that one mole of \(\mathrm{N}_{2}\mathrm{O}_{4}\) is present initially and at equilibrium, \(\alpha\) mole of \(\mathrm{N}_{2}\mathrm{O}_{4}\) has dissociated to form \(\mathrm{NO}_{2}\). Since for every one mole that dissociates we form 2 moles of NO2, the concentration of NO2 at equilibrium will be \(2\alpha\). The concentration of \(\mathrm{N}_{2}\mathrm{O}_{4}\) left at equilibrium will be \(1-\alpha\).
02

Derive the expression for \(K_P\)

We know that at equilibrium the partial pressure of a gaseous substance is directly proportional to its molar concentration. Therefore, \(P_{NO2} = 2\alpha P\) and \(P_{N2O4} = (1 - \alpha)P\). Now \(K_P\) for the reaction is given by: \(K_P = [P_{NO2}]^2/[P_{N2O4}]\), plugging in the expressions obtained for \(P_{NO2}\) and \(P_{N2O4}\), \(K_P = ((2\alpha P)^2)/((1 - \alpha)P)\) = \(4\alpha^2P/(1 - \alpha)\).
03

Predict the effect of increases in pressure.

Looking at our expression for \(K_P\), \(K_P = 4\alpha^2P/(1 - \alpha)\), an increase in \(P\) would increase \(K_P\). According to Le Chatelier’s principle, the system will shift to counteract the change, or in this case, it will shift to the right to increase the number of gaseous moieties and offshore the increased pressure. Our prediction agrees with Le Chatelier's principle.

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Most popular questions from this chapter

When dissolved in water, glucose (corn sugar) and fructose (fruit sugar) exist in equilibrium as follows: fructose \(\rightleftharpoons\) glucose A chemist prepared a \(0.244 M\) fructose solution at \(25^{\circ} \mathrm{C}\). At equilibrium, it was found that its concentration had decreased to \(0.113 M .\) (a) Calculate the equilibrium constant for the reaction. (b) At equilibrium, what percentage of fructose was converted to glucose?

What is the rule for writing the equilibrium constant for the overall reaction involving two or more reactions?

A 2.50 -mole quantity of \(\mathrm{NOCl}\) was initially in a \(1.50-\mathrm{L}\) reaction chamber at \(400^{\circ} \mathrm{C}\). After equilibrium was established, it was found that 28.0 percent of the NOCl had dissociated: $$2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g)$$ Calculate the equilibrium constant \(K_{\mathrm{c}}\) for the reaction.

For the reaction $$\mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g)$$ at \(700^{\circ} \mathrm{C}, K_{\mathrm{c}}=0.534 .\) Calculate the number of moles of \(\mathrm{H}_{2}\) that are present at equilibrium if a mixture of 0.300 mole of \(\mathrm{CO}\) and \(0.300 \mathrm{~mole}\) of \(\mathrm{H}_{2} \mathrm{O}\) is heated to \(700^{\circ} \mathrm{C}\) in a 10.0 -L container.

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) \(2 \mathrm{HgO}(s) \rightleftharpoons 2 \mathrm{Hg}(l)+\mathrm{O}_{2}(g)\)
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