Determine \(K_{c}\) from the value of \(K\) for cach of the following equilibria. (a) \(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g})=2 \mathrm{SO}_{3}(\mathrm{~g}), K=3.4\) at \(1000 \mathrm{~K}\) (b) \(\mathrm{NH}_{4} \mathrm{HS}(\mathrm{s})=\mathrm{NH}_{3}(\mathrm{~g})+\mathrm{H}_{2} \mathrm{~S}(\mathrm{~g}), K=9.4 \times 10^{-2}\) at \(24^{\circ} \mathrm{C}\)

Determining the equilibrium constant, commonly represented as K_c, is a fundamental concept in chemical equilibrium that permits chemists to understand the extent of a reaction under a set of conditions. K_c is used when dealing with concentrations of reactants and products in the gaseous state or dissolved in solution.

To determine K_c, one must look at the balanced chemical equation and use the equilibrium concentrations of the reactants and products. For a general reaction such as aA(g) + bB(g) ⇌ cC(g) + dD(g), the equilibrium expression in terms of concentration is written as
\[ K_{c} = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]
where the square brackets indicate concentration in molarity. The coefficients a, b, c, and d from the balanced chemical equation become the exponents in the expression.

In the exercise, for scenario (a), K_c is given directly as the reactants and products are all in gas phase, while in scenario (b), despite having a solid reactant, the K value represents K_c since it is written in terms of gases' partial pressures and the equilibrium involves no change in moles of gas.

An equilibrium expression is a mathematical relationship derived from the law of mass action, which states that at a constant temperature, a chemical system reaches a state where the ratio of reactant and product concentrations raised to the power of their stoichiometric coefficients achieves constancy.

When setting up an equilibrium expression, it's crucial to follow specific rules: only gases and dissolved substances are included, and each concentration is raised to the power of its coefficient in the balanced equation. For instance, if water (H₂O) and solids, like NH₄HS as seen in the given exercise, do not change their concentration significantly during the reaction, they are omitted from the expression as their activities are considered to be constant.

The process of writing equilibrium expressions aids in predicting the direction of the reaction and helps in calculating K_c, thereby contributing to a deeper understanding of the equilibrium state of a system.

The phases of reactants and products are crucial when considering equilibrium. In chemical equilibria, the state of matter—whether gas, liquid, or solid—plays a significant role in formulating the equilibrium expression.

Solid and Liquid Phases

For solids and liquids, their concentration remains relatively constant and independent of the amount present. This is why solids and liquids do not appear in equilibrium expressions in terms of concentrations. In the second scenario of the exercise, NH₄HS(s) is a solid and thus does not factor into the K_c value.

Gas Phase

For the gas phase, the concentration changes with the number of moles and the volume of the container. This changeability is why gases appear in equilibrium expressions and their partial pressures or concentrations are important when calculating equilibrium constants.

Understanding the phase consideration is imperative to accurately determine the equilibrium constant and to derive meaningful conclusions about the state of a chemical system in equilibrium.