Question

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?

Short answer

Overall equation: \( \mathrm{C}_3\mathrm{H}_8(g) + 10\mathrm{H}_2\mathrm{O}(g) \rightleftharpoons 3 \mathrm{CO}_2(g) + 10 \mathrm{H}_2(g) \).\[ K_p = 5.68\times10^{15} \text{ at } 1200 \text{ K}\].Final pressure: 10 atm. Unreacted \(C_3H_8\): 0%.

Step-by-step solution

Title - Write the overall equation

Combine the given equations to find the overall reaction. The first reaction is \(\mathrm{C}_{3} \mathrm{H}_{8}(g)+3 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons 3 \mathrm{CO}(g)+7 \mathrm{H}_{2}(g) \). The second reaction is \( \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{CO}_{2}(g)+\mathrm{H}_{2}(g) \). Adding these two reactions: \[\mathrm{C}_{3} \mathrm{H}_{8}(g) + 3\mathrm{CO}(g) + 10 \mathrm{H}_{2}O(g) \rightarrow 3\mathrm{CO}_{2}(g) + 10 \mathrm{H}_{2}(g)\]

Title - Calculate the equilibrium constant for the overall reaction

Multiply the equilibrium constants for the individual reactions to find the equilibrium constant for the overall reaction.\[ K_p = K_{p1} \times K_{p2}\]Where \(K_{p1} = 8.175 \times 10^{15}\) and \(K_{p2} = 0.6944\). Thus,\[ K_p = 8.175 \times 10^{15} \times 0.6944 = 5.68 \times 10^{15} \].

Title - Find the final pressure in the container

Use the ideal gas law and stoichiometry to determine the final pressure. Starting with 1.00 volume of \(\mathrm{C}_{3} \mathrm{H}_{8}\) and 4.00 volumes of \(\mathrm{H}_{2}O\), the total initial moles of gases is \(1+ 4 = 5 \text{ volumes}\). Since each mole of \(\mathrm{C}_{3}\mathrm{H}_{8}\) produces 10 moles of gas, the total final moles is \[1\times(10)=10 \text{ moles total}\]. Therefore, final pressure, using P_final = P_initial (V_final/V_initial)\[P_{final} = 5.0 \text{ atm} \times (10 /5)\ = 10 \text{ atm}\]

Title - Calculate the percentage of \( \mathrm{C}_3\mathrm{H}_8 \) unreacted

Given that the reaction is assumed to be complete, the percentage of \( \mathrm{C}_3\mathrm{H}_8 \) that remains unreacted is 0%.

Key concepts

chemical equilibrium

Chemical equilibrium refers to the state where the rate of the forward reaction equals the rate of the reverse reaction. In other words, the concentrations of reactants and products remain constant over time. When propane reacts with steam to produce carbon dioxide and hydrogen, as outlined in the given reactions, the system eventually reaches a point where the reactants and products are formed at the same rate. Due to this balance, no net change in the concentrations of the substances occurs. This state can be described using the equilibrium constant, which provides a mathematical way to quantify the position of the equilibrium.

ideal gas law

The ideal gas law is a crucial concept used in calculating properties of gases. It is represented by the equation: \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles of gas, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin. In the given problem, we use this law to find the final pressure after the reaction. By assuming the gases behave ideally, we can accurately use this law to predict how the pressure changes as the reaction takes place. This helps us understand how the proportion of reactants and products affects the overall properties of the gas mixture.

reaction stoichiometry

Reaction stoichiometry deals with the quantitative relationships between the reactants and products in a chemical reaction. In the given exercise, we start with 1 volume of propane (\(\mathrm{C}_{3}\mathrm{H}_{8}\)) and 4 volumes of steam (\(\mathrm{H}_{2}\mathrm{O}\)), which react to form carbon dioxide (\(\mathrm{CO}_{2}\)) and hydrogen (\(\mathrm{H}_{2}\)). By applying stoichiometric calculations, we can find the total initial and final volumes, and thereby determine the final pressure of the container. This ensures that the law of conservation of mass is followed, meaning the amount of each element is the same before and after the reaction.

equilibrium constant

The equilibrium constant, \(K_p\), is a value that expresses the ratio of the concentrations of products to reactants at equilibrium for reactions involving gases. For the overall process in the exercise, \(K_p\) is calculated by multiplying the equilibrium constants of the individual reactions. Specifically, \(K_{p1} = 8.175 \times 10^{15}\) and \(K_{p2} = 0.6944\). Therefore, the overall \(K_p\) for the process is \(5.68 \times 10^{15}\). This high value indicates that, at equilibrium, the formation of products is highly favored. Understanding and calculating the equilibrium constant assists in predicting the extent of reactions and the concentrations of different species involved.