Question

In a study of the reaction $$3 \mathrm{Fe}(s)+4 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{Fe}_{3} \mathrm{O}_{4}(s)+4 \mathrm{H}_{2}(g)$$,at \(1200 \mathrm{K}\) it was observed that when the equilibrium partial pressure of water vapor is 15.0 torr, the total pressure at equilibrium is 36.3 torr. Calculate the value of \(K_{\mathrm{p}}\) for this reaction at \(1200 \mathrm{K}\). (Hint: Apply Dalton's law of partial pressures.)

Short answer

The value of the equilibrium constant, \(K_{\mathrm{p}}\), for the given reaction at 1200 K is approximately 1.92.

Step-by-step solution

Write down the balanced chemical equation and the equilibrium constant expression

The balanced chemical equation is given by: \[3 \mathrm{Fe}(s) + 4 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{Fe}_{3} \mathrm{O}_{4}(s) + 4 \mathrm{H}_{2}(g)\] Now, we will write down the equilibrium constant expression for Kp. Since the reaction has both gaseous and solid species, only the gaseous moles will be considered in the Kp expression. In this case, it becomes: \[K_{\mathrm{p}} = \frac{(P_{\mathrm{H}_2})^4}{(P_{\mathrm{H}_2\mathrm{O}})^4}\] where \(P_{\mathrm{H}_2}\) and \(P_{\mathrm{H}_2\mathrm{O}}\) are the equilibrium partial pressures of hydrogen and water vapor, respectively.

Use Dalton's law of partial pressures to relate the given data

We are given that the equilibrium partial pressure of water vapor is 15.0 torr, and the total pressure at equilibrium is 36.3 torr. Let \(P_{\mathrm{H}_2}\) be the equilibrium partial pressure of hydrogen gas. Using Dalton's law of partial pressures, we can write the following equation for the total pressure: \[P_{\mathrm{total}} = P_{\mathrm{H}_2\mathrm{O}} + P_{\mathrm{H}_2}\] Now, we can substitute the given values into the equation: \[36.3 \, \mathrm{torr} = 15.0 \, \mathrm{torr} + P_{\mathrm{H}_2}\]

Calculate the equilibrium partial pressure of hydrogen gas

Solve the equation from step 2 for \(P_{\mathrm{H}_2}\): \[P_{\mathrm{H}_2} = 36.3 \, \mathrm{torr} - 15.0 \, \mathrm{torr} = 21.3 \, \mathrm{torr}\] So, the equilibrium partial pressure of hydrogen gas is 21.3 torr.

Substitute the pressures in the Kp expression and solve for Kp

Now, we can substitute the obtained equilibrium partial pressures of hydrogen gas and water vapor in the Kp expression: \[K_{\mathrm{p}} = \frac{(21.3)^4}{(15.0)^4}\] Calculate Kp: \[K_{\mathrm{p}} \approx 1.92\] Hence, the value of the equilibrium constant, Kp, for the given reaction at 1200 K is approximately 1.92.

Key concepts

Dalton's Law of Partial Pressures

Understanding the behavior of gases in chemical reactions is critical, and Dalton's law of partial pressures is key in this context. Dalton's law states that in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. A partial pressure is the pressure that gas would exert if it alone occupied the entire volume. This is crucial when dealing with chemical reactions in the gas phase, particularly at equilibrium, as it enables us to calculate the pressure contributed by each gas independently.

For instance, if we have a container with hydrogen and oxygen gases not reacting with one another, the total pressure in the container is the sum of the pressures each gas would exert if it was alone in the container. This principle becomes indispensable when trying to calculate equilibrium constants in reactions involving gases, as seen in the study of the iron and water vapor reaction.

Chemical Equilibrium

The state of chemical equilibrium is reached when the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of the reactants and products over time. It's a dynamic process, meaning the reactions continue to occur, but the concentrations remain constant.

The equilibrium constant, denoted as K or Kp when referring to partial pressures, quantitatively expresses the extent of the reaction at equilibrium. For the given reaction involving iron and water vapor, interpretation of the equilibrium constant provides insight into the favorability of the reaction towards the production of iron oxide and hydrogen gas at a given temperature. It’s important to note that each reaction has its unique equilibrium constant value at a constant temperature.

Partial Pressure

Partial pressure, an important term in gas laws and equilibrium, refers to the pressure that a single gas in a mixture of gases would exert if it were the only gas present in the entire volume of the mixture. The partial pressure depends on the temperature, the volume of the container, and the number of moles of the gas according to the ideal gas law. Understanding this concept is particularly important when we want to predict the behavior of a specific gas in a reaction or investigate the properties of a gas mixture.

In chemical reactions, especially those occurring in the gaseous state, partial pressure directly influences the behavior of the reactants and products. For example, in the reaction between solid iron and water vapor to form iron oxide and hydrogen gas, knowing the partial pressure of the water vapor is critical to calculating the equilibrium constant for the reaction. If we're dealing with mixtures of gases, as often the case in real-world applications, Dalton's law aids in calculating these partial pressures.