At \(25^{\circ} \mathrm{C}\), gaseous \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) decomposes to \(\mathrm{SO}_{2}\left(\mathrm{~g}\right.\) ) and \(\mathrm{Cl}_{2}(\mathrm{~g})\) to the extent that \(12.5 \%\) of the original \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) (by moles) has decomposed to reach equilibrium. The total pressure (at equilibrium) is \(0.900\) atm. Calculate the value of \(K_{\mathrm{p}}\) for this system.

The equilibrium constant, denoted as K or K_eq, is a fundamental aspect of chemical equilibrium. It numerically quantifies the ratio of the concentrations of products to reactants for a reversible chemical reaction at a specific temperature.
The calculation relies on the balanced chemical equation. For a general reaction:
\[aA + bB \rightleftharpoons cC + dD\], the equilibrium constant expression in terms of concentration (K_c) is
\[K_{c} = \frac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}}\],
where [A], [B], [C], and [D] represent the molar concentrations of the reactants and products at equilibrium, and a, b, c, and d are their respective stoichiometric coefficients.
For reactions involving gases, the equilibrium constant expression in terms of partial pressure (K_p) is often used, and it is given by
\[K_{p} = \frac{P(C)^{c}P(D)^{d}}{P(A)^{a}P(B)^{b}}\].
This exercise illustrates the calculation of K_p for a decomposition reaction. Understanding K can indicate the extent of the reaction and predict the direction of the reaction: a high K suggests a predominance of products, while a low K indicates a higher concentration of reactants.

In the context of chemical equilibrium of gases, partial pressure plays a significant role. It refers to the pressure that a gas would exert if it occupied the entire volume of a mixture on its own. Dalton's Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of its individual gases.
For a gas mixture at equilibrium:
\[P_{total} = P_{1} + P_{2} + ... + P_{n}\].
Where P₁, P₂, ..., P_n are the partial pressures of the n gasses in the mixture.
In our exercise, we needed the partial pressures of the reactant and products to find the equilibrium constant K_p. These pressures are proportional to the mole fractions of the respective components in the gas mixture. Knowing how to calculate partial pressures is essential for solving various gas-related equilibrium problems and understanding the behavior of gases in different conditions.

An ICE table—standing for Initial, Change, and Equilibrium—is a systematic method to organize the quantitative data for substances in a chemical reaction. It's an incredibly useful tool for solving equilibrium problems.
You start by filling out initial concentrations or pressures, then indicate the change that occurs as the system moves towards equilibrium, and finally the equilibrium concentrations or pressures.

• Initial: The starting concentrations or pressures of reactants and products; often products are zero for a reaction starting from reactants only.
• Change: How much the concentrations or pressures change from initial to equilibrium, typically represented by a variable like x, often based on the stoichiometry of the reaction.
• Equilibrium: The concentrations or pressures at equilibrium, found by combining initial amounts and the change.

In this exercise, the ICE table helped us deduce the changes in concentration of reactants and products which then allowed us to calculate the partial pressures needed to determine the equilibrium constant, K_p. Including an ICE table in solutions can greatly facilitate students' understanding of the progress of the reaction and how the initial composition of a system alters to achieve equilibrium.