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Chapter 17: Problem 40

For a problem involving the catalyzed reaction of methane and steam, the following reaction table was prepared: $$ \begin{array}{lcccc} \begin{array}{l} \text { Pressure } \\ \text { (atm) } \end{array} & \mathrm{CH}_{4}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{O}(g) & \Longrightarrow & \mathrm{CO}_{2}(g)+4 \mathrm{H}_{2}(g) \\ \hline \text { Initial } & 0.30 & 0.40 & 0 & 0 \\ \text { Change } & -x & -2 x & +x & +4 x \\ \hline \text { Equilibrium } & 0.30-x & 0.40-2 x & x & 4 x \end{array} $$ Explain the entries in the "Change" and "Equilibrium" rows.

Short Answer

Expert verified
Initial pressures decrease by proportions based on stoichiometry. Final equilibrium pressures are the initial minus the change for reactants and initial plus the change for products.

Step by step solution

01

Identify Initial Pressures

Observe that the initial pressures of \(\text{CH}_4\) and \(\text{H}_2O\) are given, whereas the initial pressures of \(\text{CO}_2\) and \(\text{H}_2\) are zero. These values set the starting conditions for the reaction.
02

Define the Change in Pressure for Reactants

When the reaction proceeds, the pressure of \(\text{CH}_4\) and \(\text{H}_2O\) will decrease. Let \(x\) be the amount by which the pressure of \(\text{CH}_4\) decreases. Since the stoichiometric coefficient for \(\text{H}_2O\) is 2, the change in pressure for \(\text{H}_2O\) will be \(-2x\).
03

Define the Change in Pressure for Products

For the products, the pressure of \(\text{CO}_2\) will increase by \(x\), and the pressure of \(\text{H}_2\) will increase by \(4x\) due to their stoichiometric coefficients.
04

Express Equilibrium Pressures

Combine the initial pressures with the changes to express the equilibrium pressures. For \(\text{CH}_4\), the equilibrium pressure is \(0.30 - x\). For \(\text{H}_2O\), it’s \(0.40 - 2x\). For \(\text{CO}_2\) and \(\text{H}_2\), the equilibrium pressures are \((x)\) and \((4x)\) respectively.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chemical Equilibrium
Chemical equilibrium is the state in which both reactants and products of a chemical reaction are present in concentrations that have no further tendency to change with time. This happens when the forward reaction rate equals the reverse reaction rate. In our example, the equilibrium positions of \(\text{CH}_4\), \(\text{H}_2O\), \(\text{CO}_2\), and \(\text{H}_2\) adjust to a point where their pressures remain constant over time. This means that the amounts of reactants depleting are exactly balanced by the amounts of products being formed.
Reaction Stoichiometry
Reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. This relationship is derived from the coefficients of the balanced chemical equation. In our example, the reaction of methane (\text{CH}_4) with steam (\text{H}_2O) follows the stoichiometric equation: \(\text{CH}_4(g) + 2 \text{H}_2O(g) → \text{CO}_2(g) + 4 \text{H}_2(g}\). This tells us that one mole of methane reacts with two moles of steam to produce one mole of carbon dioxide and four moles of hydrogen.
Gas Pressure
Gas pressure is the force exerted by gas molecules when they collide with the walls of their container. In chemical reactions involving gases, like our example, changes in gas pressure are observed as reactants are converted to products. The reaction table provided helps us understand these changes. Initially, we have 0.30 atm of \(\text{CH}_4\) and 0.40 atm of \(\text{H}_2O\). As the reaction proceeds, their pressures decrease by \-x\ and \-2x\ respectively. For the products, the pressures increase by \(+x\) for \(\text{CO}_2\) and \(+4x\) for \(\text{H}_2\). These changes are contingent on the stoichiometry of the reaction and help determine equilibrium pressures.

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Most popular questions from this chapter

At a particular temperature, \(K_{c}=1.6 \times 10^{-2}\) for $$ 2 \mathrm{H}_{2} \mathrm{~S}(g) \rightleftharpoons 2 \mathrm{H}_{2}(g)+\mathrm{S}_{2}(g) $$ Calculate \(K_{c}\) for each of the following reactions: (a) \(\frac{1}{2} \mathrm{~S}_{2}(g)+\mathrm{H}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{~S}(g)\) (b) \(5 \mathrm{H}_{2} \mathrm{~S}(g) \rightleftharpoons 5 \mathrm{H}_{2}(g)+\frac{5}{2} \mathrm{~S}_{2}(g)\)

A sealed 2.0 -L container initially contains 0.12 mol each of \(\mathrm{H}_{2} \mathrm{O}(g), \mathrm{Cl}_{2} \mathrm{O}(g),\) and \(\mathrm{HClO}(g) ;\) the reaction mixture is allowed to come to equilibrium according to the reaction $$ \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{Cl}_{2} \mathrm{O}(g) \rightleftharpoons 2 \mathrm{HClO}(g) \quad K_{\mathrm{c}}=0.090 $$ Calculate the equilibrium concentrations of all three compounds.

Using \(\mathrm{CH}_{4}\) and steam as a source of \(\mathrm{H}_{2}\) for \(\mathrm{NH}_{3}\) synthesis requires high temperatures. Rather than burning \(\mathrm{CH}_{4}\) separately to heat the mixture, it is more efficient to inject some \(\mathrm{O}_{2}\) into the reaction mixture. All of the \(\mathrm{H}_{2}\) is thus released for the synthesis, and the heat of reaction for the combustion of \(\mathrm{CH}_{4}\) helps maintain the required temperature. Imagine the reaction occurring in two steps: $$ \begin{array}{r} 2 \mathrm{CH}_{4}(g)+\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g)+4 \mathrm{H}_{2}(g) \\ K_{\mathrm{p}}=9.34 \times 10^{28} \mathrm{at} 1000 . \mathrm{K} \\ \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) \quad K_{\mathrm{p}}=1.374 \text { at } 1000 . \mathrm{K} \end{array} $$ (a) Write the overall equation for the reaction of methane, steam, and oxygen to form carbon dioxide and hydrogen. (b) What is \(K_{\mathrm{p}}\) for the overall reaction? (c) What is \(K_{c}\) for the overall reaction? (d) A mixture of \(2.0 \mathrm{~mol}\) of \(\mathrm{CH}_{4}, 1.0 \mathrm{~mol}\) of \(\mathrm{O}_{2},\) and \(2.0 \mathrm{~mol}\) of steam with a total pressure of \(30 .\) atm reacts at \(1000 . \mathrm{K}\) at constant volume. Assuming that the reaction is complete and the ideal gas law is a valid approximation, what is the final pressure?

The methane used to obtain \(\mathrm{H}_{2}\) for \(\mathrm{NH}_{3}\) manufacture is impure and usually contains other hydrocarbons, such as propane, \(\mathrm{C}_{3} \mathrm{H}_{8}\). Imagine the reaction of propane occurring in two steps: \(\mathrm{C}_{3} \mathrm{H}_{8}(g)+3 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons 3 \mathrm{CO}(g)+7 \mathrm{H}_{2}(g)\) $$ \begin{array}{r} K_{\mathrm{p}}=8.175 \times 10^{15} \text { at } 1200 . \mathrm{K} \\ \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) \\ K_{\mathrm{p}}=0.6944 \text { at } 1200 . \mathrm{K} \end{array} $$ (a) Write the overall equation for the reaction of propane and steam to produce carbon dioxide and hydrogen. (b) Calculate \(K_{p}\) for the overall process at \(1200 .\) K. (c) When 1.00 volume of \(\mathrm{C}_{3} \mathrm{H}_{8}\) and 4.00 volumes of \(\mathrm{H}_{2} \mathrm{O},\) each at \(1200 . \mathrm{K}\) and \(5.0 \mathrm{~atm},\) are mixed in a container, what is the final pressure? Assume the total volume remains constant, that the reaction is essentially complete, and that the gases behave ideally. (d) What percentage of the \(\mathrm{C}_{3} \mathrm{H}_{8}\) remains unreacted?

Balance each of the following examples of heterogeneous equilibria and write each reaction quotient, \(Q_{c}\) (a) \(\mathrm{NaHCO}_{3}(s) \rightleftharpoons \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{CO}_{2}(g)+\mathrm{H}_{2} \mathrm{O}(g)\) (b) \(\operatorname{SnO}_{2}(s)+\mathrm{H}_{2}(g) \rightleftharpoons \operatorname{Sn}(s)+\mathrm{H}_{2} \mathrm{O}(g)\) (c) \(\mathrm{H}_{2} \mathrm{SO}_{4}(l)+\mathrm{SO}_{3}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{~S}_{2} \mathrm{O}_{7}(l)\)
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