For a reversible reaction: \(\mathrm{A} \underset{K_{2}}{\stackrel{K_{1}}{K}} \mathrm{~B}\), the initial molar concentration of \(\mathrm{A}\) and \(\mathrm{B}\) are \(a \mathrm{M}\) and \(b \mathrm{M}\), respectively. If \(x \mathrm{M}\) of \(\mathrm{A}\) is reacted till the achievement of equilibrium, then \(x\) is (a) \(\frac{K_{1} a-K_{2} b}{K_{1}+K_{2}}\) (b) \(\frac{K_{1} a-K_{2} b}{K_{1}-K_{2}}\) (c) \(\frac{K_{1} a-K_{2} b}{K_{1} K_{2}}\) (d) \(\frac{K_{1} a+K_{2} b}{K_{1}+K_{2}}\)

Reversible reactions are processes that can proceed in both the forward and reverse directions. In the case of a chemical system, this means that reactants can be converted into products and, with the same facility, products can revert to reactants. A classic example includes the reaction between substance A and substance B, which can be represented as \( A \rightleftharpoons B \).

During the reaction, once some product B is formed, it can react to regenerate the reactant A. This dynamic is crucial because it eventually leads to a state where the rate of the forward reaction, turning A into B, and the reverse reaction, turning B back into A, are equal. This state is referred to as chemical equilibrium and is an essential concept in understanding chemical processes. When calculating changes in such a system, it's crucial to acknowledge the reversibility and ensure that the mathematical expressions consider both forward and reverse reactions. Additionally, when substances react, not all reactants are entirely converted into products; there is a marriage between reactants and products that is characterized through equilibrium expressions.

Chemical equilibrium is a state in a reversible reaction where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. It's important to note that reaching equilibrium does not mean that the reactants and products are present in equal concentrations, but rather that their concentrations have stabilized at a certain ratio.

Understanding the concept of equilibrium is pivotal when dealing with reactions, as it allows us to predict the concentrations of different species at any given point once the system has settled. In calculations, this is represented using an equilibrium constant, denoted as \( K \), which is a measure of the ratio of the concentration of the products to the reactants at equilibrium, each raised to the power of their stoichiometric coefficients in the balanced chemical equation.

For the given reversible reaction \( A \rightleftharpoons B \), the equilibrium constant expression would be \( K = \frac{[B]}{[A]} \), where \( [A] \) and \( [B] \) are the molar concentrations of A and B at equilibrium, respectively. The equilibrium constant provides insight into the extent of the reaction and helps determine the concentration of reactants and products at equilibrium.

Rate constants are fundamental to the kinetics of a chemical reaction, indicating how quickly a reaction proceeds. For reversible reactions, we have two rate constants: one for the forward reaction, \( K_1 \), and one for the reverse reaction, \( K_2 \). The magnitude of these rate constants can tell us about the favorability of the forward or the reverse process.

For instance, in the equilibrium expression \( K = \frac{[B]}{[A]} \) for a reversible reaction, the equilibrium constant \( K \) can also be expressed in terms of the rate constants as \( K = \frac{K_1}{K_2} \). This relationship shows that the equilibrium constant is directly proportional to the forward rate constant and inversely proportional to the reverse rate constant.

In a practical sense, if \( K_1 >>> K_2 \), it suggests that the forward reaction is much faster than the reverse, resulting in a larger concentration of products at equilibrium. Conversely, if \( K_2 >>> K_1 \), the reverse reaction predominates. Being able to calculate and understand the implications of rate constants allows chemists to manipulate conditions to favor the formation of either reactants or products as needed.