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Chapter 13: Problem 49

An initial mixture of nitrogen gas and hydrogen gas is reacted in a rigid container at a certain temperature by the reaction $$ 3 \mathrm{H}_{2}(g)+\mathrm{N}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) $$ At equilibrium, the concentrations are \(\left[\mathrm{H}_{2}\right]=5.0 \mathrm{M},\left[\mathrm{N}_{2}\right]=\) \(8.0 M\), and \(\left[\mathrm{NH}_{3}\right]=4.0 M .\) What were the concentrations of nitrogen gas and hydrogen gas that were reacted initially?

Short Answer

Expert verified
The initial concentrations of nitrogen gas and hydrogen gas were 9 M and 6 M, respectively.

Step by step solution

01

The balanced chemical equation for this reaction is: \( 3 H_2(g) + N_2(g) \rightleftharpoons 2 NH_3(g) \) #Step 2: Write the expression for the equilibrium constant (K) #

The expression for the equilibrium constant (K) can be written as: \( K = \frac{[NH_3]^2}{([H_2]^3 [N_2])} \) #Step 3: Calculate K using the given equilibrium concentrations#
02

We are given the equilibrium concentrations: [ \( H_2 \) ] = 5.0 M [ \( N_2 \) ] = 8.0 M [ \( NH_3 \) ] = 4.0 M Then, calculate K: \( K = \frac{4.0^2}{(5.0^3 \times 8)} \) \( K = \frac{16}{(125 \times 8)} \) \( K = \frac{16}{1000} = 0.016 \) #Step 4: Set up an ICE table (Initial, Change, Equilibrium)#

Let x be the change in the concentration of reactants and products from their initial concentrations to at the equilibrium. [ \( H_2 \) ] [ \( N_2 \) ] [ \( NH_3 \) ] Initial: 5+x 8+x 0-2x Change: -3x -x +2x Equilibrium: 5 8 4 #Step 5: Substitute the concentrations back into the K expression#
03

Substitute the equilibrium concentrations back into the K expression and solve for x: \( K = 0.016 = \frac{(4)^2}{((5+x)^3 \times (8+x))} \) #Step 6: Solve for x and find initial concentrations#

As this equation can be hard to solve algebraically, we will approximate the solution (assuming x is small compared to 5 and 8): \( 0.016 = \frac{16}{(5^3 \times 8)} \) Solve for x: \( x = \frac{16}{(125 \times 8 \times 0.016 )} \) \( x \approx 1 \) Initial concentrations: [ \( H_2 \) ] = 5 + x = 5 + 1 = 6 M [ \( N_2 \) ] = 8 + x = 8 + 1 = 9 M Hence, the initial concentrations of nitrogen gas and hydrogen gas were 9 M and 6 M, respectively.

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Most popular questions from this chapter

What will happen to the number of moles of \(\mathrm{SO}_{3}\) in equilibrium with \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\) in the reaction $$ 2 \mathrm{SO}_{3}(g) \rightleftharpoons 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) $$ in each of the following cases? a. Oxygen gas is added. b. The pressure is increased by decreasing the volume of the reaction container. c. In a rigid reaction container, the pressure is increased by adding argon gas. d. The temperature is decreased (the reaction is endothermic). e. Gaseous sulfur dioxide is removed.

At \(35^{\circ} \mathrm{C}, K=1.6 \times 10^{-5}\) for the reaction $$ 2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) $$ If \(2.0\) moles of \(\mathrm{NO}\) and \(1.0\) mole of \(\mathrm{Cl}_{2}\) are placed into a \(1.0-\mathrm{L}\) flask, calculate the equilibrium concentrations of all species.

At \(25^{\circ} \mathrm{C}, K_{\mathrm{p}}=5.3 \times 10^{5}\) for the reaction $$ \mathrm{N}_{2}(g)+3 \mathrm{H}_{2}(g) \rightleftharpoons 2 \mathrm{NH}_{3}(g) $$ When a certain partial pressure of \(\mathrm{NH}_{3}(g)\) is put into an otherwise empty rigid vessel at \(25^{\circ} \mathrm{C}\), equilibrium is reached when \(50.0 \%\) of the original ammonia has decomposed. What was the original partial pressure of ammonia before any decomposition occurred?

A \(1.604-g\) sample of methane \(\left(\mathrm{CH}_{4}\right)\) gas and \(6.400 \mathrm{~g}\) oxygen gas are sealed into a 2.50-L vessel at \(411^{\circ} \mathrm{C}\) and are allowed to reach equilibrium. Methane can react with oxygen to form gaseous carbon dioxide and water vapor, or methane can react with oxygen to form gaseous carbon monoxide and water vapor. At equilibrium, the pressure of oxygen is \(0.326 \mathrm{~atm}\), and the pressure of water vapor is \(4.45\) atm. Calculate the pressures of carbon monoxide and carbon dioxide present at equilibrium.

A sample of gaseous nitrosyl bromide (NOBr) was placed in a container fitted with a frictionless, massless piston, where it decomposed at \(25^{\circ} \mathrm{C}\) according to the following equation: $$ 2 \mathrm{NOBr}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Br}_{2}(g) $$ The initial density of the system was recorded as \(4.495 \mathrm{~g} / \mathrm{L}\). After equilibrium was reached, the density was noted to be \(4.086 \mathrm{~g} / \mathrm{L}\) a. Determine the value of the equilibrium constant \(K\) for the reaction. b. If \(\operatorname{Ar}(g)\) is added to the system at equilibrium at constant temperature, what will happen to the equilibrium position? What happens to the value of \(K\) ? Explain each answer.
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