For the autoionization of water at \(25^{\circ} \mathrm{C}\), $$ \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}^{+}(a q)+\mathrm{OH}^{-}(a q) $$ \(K_{\mathrm{w}}\) is \(1.0 \times 10^{-14} .\) What is \(\Delta G^{\circ}\) for the process?

Understanding chemical equilibrium is fundamental in explaining the autoionization of water. It's a state in a reversible chemical reaction where the rates of the forward reaction (water breaking into ions) and the backward reaction (ions recombining into water) are equal, so the concentrations of the reactants and products remain constant over time. In the case of water, the reaction can be represented as:
\[ \mathrm{H}_2\mathrm{O(l)} \rightleftharpoons \mathrm{H}^+(\mathrm{aq}) + \mathrm{OH}^-(\mathrm{aq}) \]
Here, the equilibrium constant, \( K_\mathrm{w} \), expresses the extent to which water autoionizes at a given temperature. Simplifying the process of understanding equilibrium involves recognizing it as a balance of the two opposing reactions, where no net change is observed in the system. Autoionization is a special kind of equilibrium because it involves only one species (water) undergoing self-ionization.

The concept of Gibbs free energy, \( \Delta G \), is crucial in thermodynamics and helps to predict the direction of chemical reactions. It represents the maximum amount of work a system can perform at constant temperature and pressure. A negative \( \Delta G \) signifies a spontaneous reaction, while a positive value indicates a non-spontaneous reaction that requires external energy. For the autoionization of water, this concept tells us about the feasibility of the reaction under standard conditions. The relationship between the Gibbs free energy change for a reaction and the equilibrium constant, \( K \), is given by the equation:
\[ \Delta G^\circ = -RT \ln K \]
Here, \( R \) is the universal gas constant, \( T \) is the temperature in Kelvin, and \( K \) is the equilibrium constant of the reaction. By calculating \( \Delta G^\circ \), we can assess the spontaneity of the water's autoionization at 25°C.

The equilibrium constant, denoted as \( K \), plays a vital role in quantifying the position of equilibrium in a chemical reaction. For the autoionization of water, \( K_\mathrm{w} \) specifically denotes the ion product of water, a unique form of the equilibrium constant that applies to water's self-ionization. It is calculated using the concentrations of the hydrogen ions, \( [\mathrm{H}^+] \), and hydroxide ions, \( [\mathrm{OH}^-] \):
\[ K_\mathrm{w} = [\mathrm{H}^+] \times [\mathrm{OH}^-] \]
At 25°C, \( K_\mathrm{w} \) has a value of \( 1.0 \times 10^{-14} \), signifying a very low degree of ionization, which is expected for a weakly ionizing solvent such as water. The equilibrium constant not only reflects the concentrations of ions at equilibrium but also allows us to calculate the Gibbs free energy change for the autoionization process.

At the heart of understanding the autoionization of water is thermodynamics, the study of energy and its transformations. It tells us how the energy in a system changes during chemical reactions and phase changes. For chemical systems, there are key properties, such as enthalpy (heat content), entropy (disorder), and the Gibbs free energy, which we've already discussed. These properties help scientists predict reaction spontaneity and equilibrium positions.
In the context of water's autoionization, thermodynamics reveals the balance between energy terms and allows us to use temperatures, such as 25°C, and equilibrium constants to find the Gibbs free energy change. It points out the intrinsic connection between temperature, energy, and the extent of a reaction. The autoionization example demonstrates how seemingly inactive water is, in fact, a dynamic system undergoing constant microscopic reactions that fit within the larger framework of thermodynamic principles.