For a reversible reaction at \(298 \mathrm{~K}\) the equilibrium constant \(K\) is \(200 .\) What is the value of \(\Delta G^{\circ}\) at \(298 \mathrm{~K}\) ? (a) \(-13.13 \mathrm{kcal}\) (b) \(-0.13 \mathrm{kcal}\) (c) \(-3.158 \mathrm{kcal}\) (d) \(-0.413 \mathrm{kcal}\)

Chemical equilibrium is a state in which a chemical reaction and its reverse are occurring at equal rates, resulting in no overall change in the amounts of substances involved. In a dynamic equilibrium, both the forward and reverse reactions continue to occur, but because they do so at the same rate, the concentrations of reactants and products remain constant over time.

Understandably, reaching equilibrium does not mean the reactants and products are in equal concentrations, but rather that their ratios do not change. It's important to note that this state can be approached from any direction of a reversible reaction, whether starting with reactants or products.

A direct implication of equilibrium is that it is influenced by various factors, including concentration, temperature, and pressure. These changes in conditions can disrupt the equilibrium state, which can be predicted by Le Chatelier's Principle.

The equilibrium constant (\(K\)) is a dimensionless value that gives us insight into the position of equilibrium for a reversible chemical reaction at a given temperature. Specifically, it is the ratio of the concentration of products to the concentration of reactants, each raised to the power of their stoichiometric coefficients as represented in the balanced chemical equation.

\begin{quote}\begin{eqnarray*}K &=& \frac{[\text{Products}]}{[\text{Reactants}]}^{\text{coefficients}}\end{eqnarray*}\end{quote}
When the value of \(K\) is much greater than 1, it indicates that at equilibrium, products are favored, and the reaction proceeds mostly to completion. Conversely, if \(K\) is much less than 1, reactants are favored and the reaction hardly proceeds. When \(K\) is close to 1, there is a significant amount of both reactants and products at equilibrium.

It's essential for students to know that the equilibrium constant is only affected by temperature; neither changing concentrations of reactants or products nor introducing a catalyst will change its value.

Thermodynamics is a branch of physics that deals with heat, work, and the forms of energy involved in chemical processes. The Gibbs free energy, denoted by \(\Delta G\), is one of its most crucial concepts when discussing spontaneity and equilibrium.

Gibbs free energy is defined as the maximum amount of work that a thermodynamic system can perform at constant temperature and pressure. It is a state function, meaning its value depends only on the current state of the system, not on how it got there. When \(\Delta G\) is negative for a process, it means the process can occur spontaneously under the given conditions.

The formula connecting Gibbs free energy to the equilibrium constant is expressed as:\begin{eqnarray*}\Delta G^{\circ} &=& -RT \ln K\end{eqnarray*}
where \(R\) is the universal gas constant, \(T\) is the absolute temperature, and \(K\) is the equilibrium constant of the reaction. This relationship helps predict how changes in temperature or the equilibrium constant will affect the Gibbs free energy and thus the spontaneity of a reaction.