Question

Using \(\Delta G_{f}^{\circ}\) for \(\mathrm{NO}\) and \(\mathrm{NO}_{2},\) from Appendix \(\mathrm{C},\) calculate the equilibrium constant for the oxidation of \(\mathrm{NO}\) to \(\mathrm{NO}_{2}\) at \(298.0 \mathrm{~K}\) as described in equation \(22.38 .\)

Short answer

The equilibrium constant (K) for the oxidation of NO to NO2 at 298.0 K is approximately 3.47 × 10⁻³. This was calculated using the standard Gibbs free energy of formation values and the equation ΔG° = -RTln(K).

Step-by-step solution

Write the balanced chemical equation

We must first write the balanced chemical equation for the oxidation of NO to NO2: \[ 2NO (g) \rightarrow NO_{2} (g) \]

Find the standard Gibbs free energy change of the reaction

To find the standard Gibbs free energy change (ΔG°) for the reaction, we will use the standard Gibbs free energy of formation (ΔGf°) values for the products and reactants from Appendix C: ΔG° = (ΔGf°[NO2]) - 2(ΔGf°[NO]) Now, plug in the values given: ΔG° = (-33.18kJ/mol) - 2(-86.57kJ/mol)

Calculate the standard Gibbs free energy change

Perform the calculation: ΔG° = -33.18kJ/mol + 173.14kJ/mol ΔG° = 139.96 kJ/mol

Use the Gibbs free energy change to calculate the equilibrium constant (K) at 298.0 K

Using the equation ΔG° = -RTln(K), we can find the equilibrium constant K. First, convert the temperature to Kelvin and rearrange the formula to solve for K: \[ K = e^{-\frac{ΔG°}{RT}} \] Where: R = Gas constant = 8.314 J/(mol·K) T = Temperature = 298.0 K ΔG° = 139960 J/mol (convert kJ/mol to J/mol) Plug in the values: \[ K = e^{\frac{-139960J/mol}{8.314J/(mol\cdot K) \cdot 298.0K}} \]

Calculate the equilibrium constant (K)

Perform the calculation: \[ K = e^{-5.65} \] \[ K = 3.47 \times 10^{-3} \] The equilibrium constant (K) for the oxidation of NO to NO2 at 298.0 K is approximately 3.47 × 10⁻³.

Key concepts

Gibbs Free Energy

Gibbs free energy, represented as \( \Delta G \), is a crucial concept in chemical thermodynamics that helps us understand the spontaneity of a reaction. It's like the energy currency for reactions, indicating whether a process can occur without external input.
The formula \( \Delta G = \Delta H - T\Delta S \) relates it to enthalpy (\( \Delta H \)) and entropy (\( \Delta S \)). A negative \( \Delta G \) means a reaction is spontaneous, while a positive \( \Delta G \) implies it needs energy to proceed.

• In the problem, the reaction involves the oxidation of \( \text{NO} \) to \( \text{NO}_2 \).
• We use the standard Gibbs free energy change (\( \Delta G^\circ \)) to calculate equilibrium properties.

In essence, \( \Delta G \) provides a snapshot of the energy changes in a reaction, determining its direction and feasibility.

Oxidation Reaction

An oxidation reaction is a chemical process where a substance loses electrons. Typically, it involves the increase of an oxidation state. Understanding oxidation reactions is vital in chemistry, especially when dealing with reactions involving gases like nitrogen oxides.
In this particular exercise, we're looking at the oxidation of \( \text{NO} \) to \( \text{NO}_2 \):

• The balanced chemical equation is \( 2 \text{NO} \rightarrow \text{NO}_2 \).
• Electrons are transferred, with \( \text{NO} \) being oxidized in the process.

Observing oxidation reactions like this helps in analyzing atmospheric chemistry and industrial processes where nitrogen oxides are involved.

Chemical Thermodynamics

Chemical thermodynamics is the study of energy transformations in chemical reactions. It provides insight into the heat and work involved in a system and includes principles like enthalpy, entropy, and Gibbs free energy.
These principles help us understand equilibrium, spontaneity, and energy dynamics of reactions. In this context, we use thermodynamics to calculate the equilibrium constant (\( K \)) for a reaction by employing the equation \( \Delta G^\circ = -RT \ln(K) \).

• \( R \) is the universal gas constant.
• \( T \) is the temperature in Kelvin.
• The equilibrium constant indicates the ratio of products to reactants at equilibrium.

By applying these principles, we can determine whether a chemical reaction can progress and to what extent it will reach equilibrium. This understanding is crucial for fields such as chemistry, engineering, and environmental science.