Calculate the equilibrium constant \(K_{\mathrm{p}}\) for the reaction given below if \(\Delta G^{\circ}=-10.632 \mathrm{~kJ}\) at \(300 \mathrm{~K}\) $$ \mathrm{CO}_{2(\mathrm{~g})}+\mathrm{H}_{2(\mathrm{~g})} \rightleftharpoons \mathrm{CO}_{(\mathrm{g})}+\mathrm{H}_{2} \mathrm{O}_{(\mathrm{g})} $$

Gibbs free energy, represented by the symbol \( \Delta G \), is a thermodynamic quantity that provides valuable insight into the spontaneity of a chemical reaction at constant pressure and temperature. It incorporates entropy (\textbf{disorder}) and enthalpy (\textbf{energy content}) into one value, defining the energy available to do work. A negative \( \Delta G \) implies that a reaction can proceed spontaneously, which means it can occur without any additional energy input. In the context of a chemical reaction involving gases, the relationship between Gibbs free energy and the equilibrium constant is crucial for predicting the extent of the reaction.

The equation \( \Delta G^\degree = -RT \ln(K_p) \) connects the standard Gibbs free energy change \( \Delta G^\degree \) for a reaction to the equilibrium constant \(K_p\) when the system is at equilibrium. Here, \( R \) is the universal gas constant and \( T \) is the temperature in Kelvin. This formula allows chemists to predict how changes in conditions like temperature can affect the direction and extent of a chemical reaction.

The equilibrium constant, denoted as \(K_p\), is a dimensionless number that quantifies the position of equilibrium for a gaseous reaction. It is calculated using the partial pressures of the gases involved in the reaction. When a reaction reaches equilibrium, the rates of the forward and reverse reactions are equal, and the concentrations (or partial pressures) of reactants and products remain constant over time. The value of \(K_p\) provides a ratio of the products to reactants at equilibrium, raised to the power of their respective stoichiometric coefficients in the balanced chemical equation.

\(K_p\) is critical for chemists as it allows them to understand the proportions of reactants and products that will be present when the reaction is at equilibrium. A larger \(K_p\) value generally indicates that at equilibrium, products predominate over reactants, while a smaller value suggests that reactants are more abundant. Calculating \(K_p\) from \( \Delta G^\degree \) is essential for processes where controlling the yield of a product is necessary.

Chemical reaction equilibrium is a state in a reversible reaction where the rate of the forward reaction equals the rate of the backward reaction, resulting in no net change in concentration of reactants and products over time. This dynamic equilibrium is essential in understanding how reactions can be manipulated and controlled in a laboratory or industrial setting.

While reactions may appear to have 'stopped' at equilibrium, in reality, they are constantly occurring, but the rates are equal, so the overall concentrations remain steady. Determining the point of equilibrium is critical for maximizing yields in chemical manufacturing, as well as for predicting the behavior of chemical systems under different conditions. The position of equilibrium is represented by the equilibrium constant \(K_p\) for reactions involving gases, or \(K_c\) for reactions in solution, and this position can shift in response to changes in temperature, pressure, or concentration according to Le Chatelier's principle.

Thermodynamics is the branch of physical science that deals with the relationships between heat and other forms of energy such as work. It governs the principles underlying chemical reactions, phase changes, and the physical behavior of matter. In chemistry, thermodynamics helps us understand the energy changes that accompany reactions and phase changes, allowing us to predict the spontaneity of a process.

The three main laws of thermodynamics set the stage for energy conservation, the directionality of energy flow, and the absolute scale of entropy. One of thermodynamics' most important applications in chemistry is establishing the connection between Gibbs free energy (\textbf{\( \Delta G \)}), equilibrium constants (\textbf{\(K_p\)} or \textbf{\(K_c\)}), and the fundamental properties of a system. By understanding these concepts, chemists can tailor reactions to optimize conditions for desired outcomes, such as the highest yield of a product or the most efficient use of energy.