For the reaction: \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons\) \(2 \mathrm{NOCl}(\mathrm{g}), \mathrm{NO}\) and \(\mathrm{Cl}_{2}\) are initially taken in mole ratio of \(2: 1 .\) The total pressure at equilibrium is found to be \(1 \mathrm{~atm}\). If the moles of \(\mathrm{NOCl}\) are one-fourth of that of \(\mathrm{Cl}_{2}\) at equilibrium, \(K_{\mathrm{p}}\) for the reaction is (a) \(\frac{13}{36}\) (b) \(\frac{13}{256}\) (c) \(\frac{13}{512}\) (d) \(\frac{13}{128}\)

Chemical equilibrium is a state in which a chemical reaction and its reverse reaction occur at the same rate, resulting in no overall change in the concentrations of the reactants and products. This balance can be disturbed by changing the conditions, such as temperature, pressure, or concentration, leading to a shift in the equilibrium position according to Le Châtelier's Principle.

When a reaction reaches equilibrium, it does not mean the reactants and products have stopped reacting. Instead, they are being formed and consumed at equal rates, leading to what we describe as a dynamic equilibrium. Understanding this helps explain observations where, for example, a change in pressure will cause the system to adjust until a new equilibrium is established.

In a mixture of gases, each gas exerts a pressure independently of the other gases present. This pressure is known as the partial pressure of the gas. It is proportional to the mole fraction of the gas in the mixture, which is a consequence of Dalton's Law of Partial Pressures.

For instance, in an equilibrium mixture, the total pressure is the sum of the individual partial pressures of the gases involved in the reaction. The partial pressure of a component can be used to calculate its concentration at equilibrium when employing the ideal gas law, which relates pressure, volume, temperature, and the number of moles of a gas.

The equilibrium constant expression, denoted as Kp when using partial pressures, quantitatively describes the proportions of reactants and products in equilibrium. For the reaction \(aA(g) + bB(g) \rightleftharpoons cC(g) + dD(g)\), the expression is \(K_p = \frac{[P(C)]^c [P(D)]^d}{[P(A)]^a [P(B)]^b}\), where P indicates partial pressures of the gases.

When calculating the equilibrium constant, one must remember that it is a dimensionless quantity, and the powers in the expression correspond to the coefficients in the balanced chemical equation. The value of Kp is constant for a given reaction at a fixed temperature and provides insight into the extent of the reaction at equilibrium.

The ideal gas law combines several gas laws into one equation that relates the pressure (P), volume (V), temperature (T), and number of moles (n) of an ideal gas: \(PV = nRT\). The constant R is known as the ideal gas constant.

This law assumes that gases consist of particles with negligible volume and no intermolecular forces. In real-life applications, the ideal gas law provides close approximations for many gases under non-extreme conditions. When applying this law to solve equilibrium problems involving gases, one can relate changes in conditions to changes in the system's behavior, assisting in the prediction and understanding of the system's response at equilibrium.