Problem 108 - Chemical Principles: The Quest for Insight

Question

The temperature dependence of the equilibrium constant of the reaction \(\mathrm{N}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})\), which makes an important contribution to atmospheric nitrogen oxides, can be expressed as \(\ln K=2.5-\) \((21700 \mathrm{~K}) \cdot T^{-1}\). What is the standard enthalpy of the forward reaction?

Step-by-Step Solution

Answer

\(\Delta H^{\circ} = 21700 \, \mathrm{K} \times 8.314 \, \mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1} = 180482 \, \mathrm{J}\,\mathrm{mol}^{-1} = 180.482 \, \mathrm{kJ}\,\mathrm{mol}^{-1}\)

Step 1: Identify the Van't Hoff Equation

The given equation \(\ln K = 2.5 - (21700 \, \mathrm{K}) \cdot T^{-1}\)is a form of the Van't Hoff equation which relates the equilibrium constant to temperature: \(\ln K = - \frac{\Delta H^{\circ}}{R} \cdot T^{-1} + \frac{\Delta S^{\circ}}{R}\)Where \(\Delta H^{\circ}\)is the standard enthalpy change, \(\Delta S^{\circ}\)is the standard entropy change, \(R\)is the gas constant, and \(T\)is the temperature.

Step 2: Relate to Standard Enthalpy Change

Comparing the given equation to the standard form of Van't Hoff equation, it is noticeable that the term \(- \frac{\Delta H^{\circ}}{R} \) corresponds to the given term \(-21700 \, \mathrm{K}\). We are interested in finding \(\Delta H^{\circ}\).

Step 3: Calculate the Standard Enthalpy Change

To find the standard enthalpy change \(\Delta H^{\circ}\),we can equate the coefficients of \(T^{-1}\): \(- \frac{\Delta H^{\circ}}{R} = -21700 \, \mathrm{K}\)We can solve for \(\Delta H^{\circ}\) by multiplying both sides with \(-R\), knowing that \(R\) (the gas constant) is \(8.314 \, \mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1}\):\(\Delta H^{\circ} = 21700 \, \mathrm{K} \times 8.314 \, \mathrm{J}\,\mathrm{mol}^{-1}\mathrm{K}^{-1}\)

Key Concepts

Chemical Equilibrium

The state of chemical equilibrium is reached when the rates of the forward and reverse reactions in a chemical system become equal. Consequently, the concentrations of reactants and products remain constant over time. This does not mean that the reactions have ceased, but rather that they are proceeding at an equal pace in both directions, resulting in no overall concentration change.

Understanding chemical equilibrium is crucial for predicting the composition of a reaction mixture under specific conditions. It's especially important in cases where the reaction plays a significant role in biological or environmental systems, such as the formation of nitrogen oxides in the atmosphere from nitrogen and oxygen gases.

Standard Enthalpy Change

Standard enthalpy change, denoted as \(\Delta H^{\circ}\), refers to the heat absorbed or released when a reaction occurs at standard conditions (usually 1 bar of pressure and 298.15 K). This thermodynamic quantity is vital in determining whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).

In the given exercise, the standard enthalpy change for the formation of nitrogen monoxide (NO) from nitrogen (N2) and oxygen (O2) can be deduced from the relationship presented in the Van't Hoff equation. By isolating and calculating \(\Delta H^{\circ}\), we can quantify the heat energy change for this reaction which can be helpful to understand the potential impacts on the surrounding environment when considering energy balance and reaction spontaneity.

Equilibrium Constant

The equilibrium constant, denoted as \(K\), is a dimensionless value that expresses the ratio of product concentrations to reactant concentrations at equilibrium, each raised to the power of their respective coefficients in the balanced equation. It's a reflection of the position of equilibrium and is temperature-dependent.

In the exercise example, the Van't Hoff equation illustrates that \(K\) can be influenced by temperature changes, which affects the balance between reactants and products. A larger \(K\) indicates a reaction that favors product formation at equilibrium, while a smaller \(K\) indicates a reaction that favors the reactants.

Standard Entropy Change

Standard entropy change, represented by \(\Delta S^{\circ}\), is the change in entropy, a measure of disorder or randomness, during a process where all reagents and products are in their standard states. It is also a temperature-dependent variable and can be a decisive factor for the spontaneity of a reaction when combined with the enthalpy change according to the Gibbs free energy equation.

In studying the Van't Hoff equation, which was presented in the textbook's exercise, the term \(\frac{\Delta S^{\circ}}{R}\) encapsulates the entropy contribution to the equilibrium state. Entropy changes can suggest how the molecular randomness varies from reactants to products, which in turn affects the magnitude and direction of a reaction under given conditions.