Problem 45 - Chemistry and Chemical Reactivity

Question

The standard free energy change, \(\Delta_{\mathrm{r}} G^{\circ},\) for the formation of \(\mathrm{NO}(\mathrm{g})\) from its elements is \(+86.58 \mathrm{kJ} / \mathrm{mol}-\mathrm{rxn}\) at \(25^{\circ} \mathrm{C} .\) Calculate \(K_{\mathrm{p}}\) at this temperature for the equilibrium $$1 / 2 \mathrm{N}_{2}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{NO}(\mathrm{g})$$ Comment on the sign of \(\Delta_{\mathrm{r}} G^{\circ}\) and the magnitude of \(K_{\mathrm{p}}.\)

Step-by-Step Solution

Answer

Kp at 298.15 K is approximately 7.1 × 10⁻¹⁶. The positive ΔrG° indicates non-spontaneity, and the small Kp suggests reactants predominate.

Step 1: Understand the Relationship Between ΔrG° and Kp

The relationship between the standard free energy change of a reaction, ΔrG°, and the equilibrium constant, K, can be determined using the equation \( \Delta_{\mathrm{r}} G^{\circ} = -RT \ln K \), where R is the universal gas constant (8.314 J/mol·K) and T is the temperature in Kelvin.

Step 2: Convert Temperature to Kelvin

Since the temperature is given in Celsius, convert it to Kelvin using the formula \( T(K) = T(°C) + 273.15 \). Thus, \( 25^{\circ} \mathrm{C} \) becomes \( 298.15 \text{ K} \).

Step 3: Solve for Kp Using ΔrG°

Substitute the given ΔrG° and the converted temperature into the equation: \( 86580 \text{ J/mol} = -(8.314 \text{ J/mol·K})(298.15 \text{ K}) \ln K_p \). Rearrange to find \( \ln K_p \) and solve for \( K_p \).

Step 4: Calculate ln Kp

Rearrange the equation: \( \ln K_p = \frac{-86580}{8.314 \times 298.15} \approx -34.92 \).

Step 5: Calculate Kp

Take the exponential of both sides to solve for \( K_p \): \( K_p = e^{-34.92} \approx 7.1 \times 10^{-16} \).

Step 6: Interpret the Sign of ΔrG° and Magnitude of Kp

Given that \( \Delta_{\mathrm{r}} G^{\circ} \) is positive, this indicates that the formation of NO is not spontaneous under standard conditions. Additionally, the very small magnitude of \( K_p \) (7.1 × 10⁻¹⁶) suggests that at equilibrium, the concentrations of reactants dominate over products, indicating a far shift to the left.

Key Concepts

Free Energy Change

Free energy change, denoted as \( \Delta_{\text{r}} G^{\circ} \), is an important concept in thermodynamics and chemistry that describes the amount of energy available to do work during a chemical reaction. In the context of a reaction, if \( \Delta_{\text{r}} G^{\circ} \) is negative, the reaction can proceed spontaneously under standard conditions.However, if it is positive, like in the formation of NO from N₂ and O₂ with a value of 86.58 kJ/mol, the reaction is non-spontaneous.
The free energy change is a central measure used to determine both the spontaneity and feasibility of reactions:

• A negative \( \Delta_{\text{r}} G^{\circ} \) means the process releases energy, spontaneously moving forward.
• A positive \( \Delta_{\text{r}} G^{\circ} \) indicates energy must be added for the reaction to occur.

The relationship between \( \Delta_{\text{r}} G^{\circ} \) and the equilibrium constant\( K \) is given by the formula \( \Delta_{\text{r}} G^{\circ} = -RT \ln K \), where \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin.

Equilibrium Equation

Equilibrium equations express the state of a chemical reaction where the rates of the forward and backward reactions are equal. This means the concentrations of reactants and products remain constant over time. Equilibrium can be represented by a constant known as the equilibrium constant \( K \).
The equilibrium constant is derived from the ratio of the concentrations of products to reactants, each raised to the power of their stoichiometric coefficients. For the given reaction:\[ \frac{1}{2} \text{N}_2(\text{g}) + \frac{1}{2} \text{O}_2(\text{g}) \rightleftharpoons \text{NO}(\text{g}) \]the constant \( K_p \) can be determined using the equation:\( \Delta_{\text{r}} G^{\circ} = -RT \ln K_p \).
In this exercise, \( \Delta_{\text{r}} G^{\circ} \) was calculated as +86.58 kJ/mol, indicating a non-spontaneous reaction, and leading to an exceptionally low \( K_p \) value of approximately \( 7.1 \times 10^{-16} \). This signifies that at equilibrium, the reactants are much more prevalent than the products, indicating the equilibrium strongly favors the reactants.

Reaction Thermodynamics

Thermodynamics is the study of energy transformations in reaction processes. It provides insights into whether chemical reactions will occur spontaneously and to what extent. Using the principles of reaction thermodynamics, one can determine the spontaneity and equilibrium position of a reaction.
The key thermodynamic parameters are:

• Enthalpy (\( \Delta H \)): The total energy of a system. It is not directly addressed in this exercise, but it plays a crucial role in conjunction with entropy to determine spontaneity.
• Entropy (\( \Delta S \)): The degree of randomness or disorder in a system. It influences the free energy change, and hence the equilibrium of a reaction.
• Free Energy (\( \Delta G \)): Combines enthalpy and entropy changes to assess whether a process can be spontaneous.

Through the equation-related energy changes and equilibrium constant, reacting systems can be evaluated for feasibility and direction.