Consider the following reaction at \(25.0^{\circ} \mathrm{C}\) : $$ 2 \mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{N}_{2} \mathrm{O}_{4}(g) $$ The values of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are \(-58.03 \mathrm{~kJ} / \mathrm{mol}\) and \(-176.6\) \(\mathrm{J} / \mathrm{K} \cdot \mathrm{mol}\), respectively. Calculate the value of \(\mathrm{K}\) at \(25.0^{\circ} \mathrm{C}\). Assuming \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\) are temperature independent, estimate the value of \(K\) at \(100.0^{\circ} \mathrm{C}\).

The equilibrium constant (K) is a vital concept in chemistry that quantifies the ratio of the concentrations of products to reactants at equilibrium for a reversible chemical reaction. It is a measure of the extent to which a reaction proceeds and is determined by the specific reaction's conditions, such as temperature. In our exercise, we are asked to calculate the value of K for the dimerization of nitrogen dioxide (NO₂) to form dinitrogen tetroxide (N₂O₄) at different temperatures.

It's important when performing calculations involving K to pay attention to the units and to use Kelvin for temperature. Starting with the standard Gibbs free energy change (∆G°), we can determine the equilibrium constant at 25°C. As the temperature changes, the value of K also shifts, reflecting the temperature's effect on the reaction's equilibrium position. This is demonstrated in the calculation of K at 100°C using the van 't Hoff equation, which shows an increase in K, indicating that the reaction is more product-favored at higher temperatures.

The standard enthalpy change (∆H°) is the energy change associated with a chemical reaction taking place under standard conditions (1 atm pressure and a specified temperature, usually 25°C). It's a key factor for understanding heat transfer during a reaction—whether it's exothermic (releasing heat) or endothermic (absorbing heat). In the given exercise, ∆H° for the reaction is negative (-58.03 kJ/mol), indicating that the formation of N₂O₄ from NO₂ is exothermic and releases heat into the surroundings.

Knowing the standard enthalpy change is crucial for predicting how a reaction's equilibrium will shift with temperature changes. As the van 't Hoff equation demonstrates, an exothermic reaction's K value will decrease with increasing temperature. Conversely, endothermic reactions will have their K value increase with temperature. This relationship is derived from the Le Chatelier's principle, which states that a system at equilibrium will adjust to counteract changes imposed upon it.

Gibbs free energy (G) is a thermodynamic quantity that represents the maximum amount of work that can be performed by a thermodynamic system at constant temperature and pressure. In the realm of chemical reactions, the change in Gibbs free energy (∆G) determines the spontaneity of a process. A negative ∆G indicates that a reaction is spontaneous and can proceed without additional energy input, whereas a positive ∆G suggests that the reaction is non-spontaneous.

In our exercise, we calculate the standard Gibbs free energy change (∆G°) at 25°C which allows us to find the equilibrium constant K. The link between ∆G° and K is given by the equation K = e^-(∆G°/RT), meaning that as ∆G° becomes less negative (or more positive), K decreases, and the reaction is less product-favored. This highlights the delicate interplay between thermodynamics and reaction equilibria—a foundational concept for students studying chemical thermodynamics.

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. It encompasses all of the core concepts mentioned earlier: equilibrium constant, standard enthalpy change, and Gibbs free energy. These principles help us understand and predict how chemical reactions occur and how they are influenced by external conditions.

The van 't Hoff equation, applied in our exercise, is an application of thermodynamic principles that describes how the equilibrium constant varies with temperature. The exercise perfectly illustrates how thermodynamics can provide insights into the direction and extent of chemical reactions, aiding the students in grasping the larger picture of how energy and matter interact in chemical processes. Emphasizing connections between these thermodynamic quantities fosters a comprehensive understanding of chemical reactions and helps students develop the ability to predict the behavior of systems under various conditions.