Question

At \(1565 \mathrm{~K}\), the equilibrium constants for the reactions (1) \(2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftharpoons 2 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g})\) and (2) \(2 \mathrm{CO}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{CO}(\mathrm{g})+\) \(\mathrm{O}_{2}(\mathrm{~g})\) are \(1.6 \times 10^{-11}\) and \(1.3 \times 10^{-10}\), respectively. (a) What is the equilibrium constant for the reaction (3) \(\mathrm{CO}_{2}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons\) \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})+\mathrm{CO}(\mathrm{g})\) at that temperature? (b) Show that the manner in which equilibrium constants are calculated is consistent with the manner in which the \(\Delta G_{\mathrm{r}}^{\circ}\) values are calculated when combining two or more equations by determining \(\Delta G_{\mathrm{r}}^{\circ}\) for reactions (1) and (2) and using those values to calculate \(\Delta G_{\mathrm{r}}^{\circ}\) and \(K_{3}\) for reaction (3).

Short answer

The equilibrium constant for reaction (3) is \(0.8125 \times 10\). This demonstrates that equilibrium constants combine in a manner consistent with the combination of \(\Delta G_{\mathrm{r}}^{\circ}\) values, reflecting the underlying thermodynamics.

Step-by-step solution

Write the balanced equation for the new reaction

Combine the given reactions (1) and (2) to obtain reaction (3). To do this, reverse reaction (1) and add it to reaction (2) such that \( \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \) and \( \mathrm{O}_{2}(\mathrm{g}) \) cancel out:\[ \mathrm{CO}_{2}(\mathrm{~g}) + \mathrm{H}_{2}(\mathrm{~g}) \rightleftharpoons \mathrm{H}_{2}\mathrm{O}(\mathrm{g}) + \mathrm{CO}(\mathrm{g}) \]

Use the equilibrium constants for reactions (1) and (2)

Write the expressions for the equilibrium constants for reactions (1) and (2): \[K_{1} = 1.6 \times 10^{-11} \,\,K_{2} = 1.3 \times 10^{-10} \] Since reaction (1) is reversed to get reaction (3), use the reciprocal of \(K_{1}\) for the reversed reaction.

Calculate the equilibrium constant for reaction (3)

The equilibrium constant for the new reaction can be calculated by multiplying the reciprocal of \(K_{1}\) with \(K_{2}\) because constants of consecutive reactions are multiplied to obtain the overall reaction's constant:\[ K_{3} = \frac{1}{K_{1}} \times K_{2} \]

Substitute the values for \(K_{1}\) and \(K_{2}\)

Substitute the values of \(K_{1}\) and \(K_{2}\) into the equation from step 3 to find \(K_{3}\):\[ K_{3} = \frac{1}{1.6 \times 10^{-11}} \times 1.3 \times 10^{-10} \]

Calculate and simplify the equilibrium constant \(K_{3}\)

Perform the calculation to determine \(K_{3}\):\[ K_{3} = \frac{1.3 \times 10^{-10}}{1.6 \times 10^{-11}} = \frac{13}{16} \times 10^{1} = \frac{13}{16} \times 10 = 0.8125 \times 10 \]

Show consistency with \(\Delta G_{\mathrm{r}}^{\circ}\) values

The change in standard Gibbs free energy \(\Delta G_{\mathrm{r}}^{\circ}\) is related to the equilibrium constant by the equation \(\Delta G_{\mathrm{r}}^{\circ} = -RT\ln K\). Since Gibbs free energy is a state function, \(\Delta G_{\mathrm{r}}^{\circ}\) for the overall reaction can be found by adding the \(\Delta G_{\mathrm{r}}^{\circ}\) for the individual reactions. This is analogous to multiplication of equilibrium constants for the overall reaction as seen in previous steps.

Calculate the \(\Delta G_{t}{ }^{\circ}\) for reactions (1) and (2)

Using the equation \(\Delta G_{\mathrm{r}}^{\circ} = -RT\ln K\), calculate \(\Delta G_{\mathrm{r}}^{\circ}\) for reactions (1) and (2) separately. The values found would then be combined to find \(\Delta G_{\mathrm{r}}^{\circ}\) for reaction (3).

Combine the \(\Delta G_{\mathrm{r}}^{\circ}\) values

Since reaction (1) is reversed to obtain reaction (3), negate the \(\Delta G_{\mathrm{r}}^{\circ}\) value for reaction (1) before adding it to that of reaction (2). This combined \(\Delta G_{\mathrm{r}}^{\circ}\) should match the \(\Delta G_{\mathrm{r}}^{\circ}\) calculated using the \(K_{3}\) obtained in previous steps, proving consistency in the calculations.

Key concepts

Understanding Chemical Equilibrium

Chemical equilibrium is a fundamental principle in chemistry where the rate of the forward reaction equals the rate of the backward reaction, leading to a state where the concentrations of reactants and products remain constant over time. It is an essential concept in chemical thermodynamics, governing how reactants convert to products and vice versa.

In a typical equilibrium scenario, such as the combustion of hydrogen and oxygen to form water, both the formation of water from hydrogen and oxygen and the decomposition of water back into hydrogen and oxygen occur simultaneously. When these two rates are equal, the system reaches equilibrium, and the concentration of each species remains unchanged. Understanding this principle is crucial when predicting how a system will respond to changes in conditions, a concept known as Le Chatelier's Principle.

To properly grasp equilibrium constants calculation, students should keep in mind that the constants are a ratio of the concentration of products to reactants, raised to the power of their stoichiometric coefficients. Symbols such as Kc and Kp represent these constants in molar concentrations and partial pressures, respectively. It is important to recognize that equilibrium constants are temperature-dependent, as seen in the exercise where the constants are given at 1565 K.

Gibbs Free Energy and Its Role in Chemical Processes

Gibbs Free Energy, symbolized as \( G \), is a thermodynamic potential that provides invaluable information about the spontaneity of chemical reactions at constant temperature and pressure. Named after Josiah Willard Gibbs, the concept introduces the notion of 'free' energy available to do work during a chemical process.

The equation \( \Delta G_{\mathrm{r}}^{\circ} = -RT\ln K \) is imperative as it links the Gibbs Free Energy change under standard conditions to the equilibrium constant (K). A negative \( \Delta G_{\mathrm{r}}^{\circ} \) signifies a spontaneous reaction, while a positive value indicates non-spontaneity. In the exercise solution steps, we see the application of this equation in action when calculating the Gibbs Free Energy changes for reactions (1) and (2) to ultimately derive \( \Delta G_{\mathrm{r}}^{\circ} \) for reaction (3).

Students should be aware that being a state function, Gibbs Free Energy change does not depend on the path of the transformation but only on initial and final states. This principle allows us to add or subtract individual \( \Delta G_{\mathrm{r}}^{\circ} \) values from multiple steps to get the total \( \Delta G_{\mathrm{r}}^{\circ} \) for the overall reaction. Consistency in calculating \( \Delta G_{\mathrm{r}}^{\circ} \) and K, as demonstrated in the exercise, is vital for predicting the direction and extent of chemical reactions.

Thermodynamics: The Backdrop for Understanding Chemical Reactions

Thermodynamics, the branch of physical science concerned with heat and its relation to other forms of energy and work, is the backdrop against which we understand chemical reactions, including those in equilibrium. Its laws serve as guidelines for the conservation of energy and the direction in which reactions occur.

The first and second laws of thermodynamics are pivotal for grasping concepts like enthalpy, entropy, and the Gibbs Free Energy. They explain that energy cannot be created or destroyed, and that there is a natural tendency for systems to move towards increased disorder or entropy.

The second law, in particular, is closely related to the concept of Gibbs Free Energy, stating that for a process to occur spontaneously, there must be an increase in the entropy of the universe. Consequently, this laws leads to the criterion for spontaneous processes in terms of Gibbs Free Energy, nicely tying it to the chemical equilibrium constant. Through the exercise on equilibrium constants calculation, students witness the practical application of thermodynamic principles. They learn not only about stability and reactivity but also about the quantitative aspects of reaction equilibria intimately connected to these thermodynamic concepts.