For the dissociation reaction \(2 \mathrm{H}_{2} \mathrm{S}(\mathrm{g}) \rightleftharpoons 2 \mathrm{H}_{2}(\mathrm{g})+\) \(\mathrm{S}_{2}(\mathrm{g}), K_{\mathrm{p}}=1.2 \times 10^{-2}\) at \(1065^{\circ} \mathrm{C} .\) For this same reaction at \(1000 \mathrm{K},\) (a) \(K_{\mathrm{c}}\) is less than \(K_{\mathrm{p}} ;\) (b) \(K_{\mathrm{c}}\) is greater than \(K_{\mathrm{p}} ;(\mathrm{c}) K_{\mathrm{c}}=K_{\mathrm{p}} ;\) (d) whether \(K_{\mathrm{c}}\) is less than, equal to, or greater than \(K_{\mathrm{p}}\) depends on the total gas pressure.

The equilibrium constant is a critical concept in chemistry that quantifies the ratio of concentrations of products to reactants at the point of chemical equilibrium. It is denoted as \(K\) and can take the form of \(K_c\) for concentrations or \(K_p\) for partial pressures when dealing with gases. Understanding \(K\) helps predict the extent of a reaction under given conditions. It's important to remember that the value of the equilibrium constant is dependent on temperature and will change if the temperature of the system is altered.

When a reaction has reached equilibrium, it doesn't mean the reactants and products are present in equal amounts, rather that their rates of formation are equal, leading to constant concentrations. The higher the value of \(K\), the more the equilibrium lies towards the products, indicating a 'product-favored' reaction. Conversely, a low \(K\) signals a 'reactant-favored' reaction where the formation of products is less favored.

The relationship between \(K_p\) and \(K_c\) is fundamental for dealing with gas-phase reactions. These equilibrium constants are related through the expression \(K_p = K_c(RT)^{\Delta n}\), where \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(\Delta n\) is the change in moles of gas between products and reactants. It's important to highlight that \(K_p\) pertains to partial pressures and is used when the reaction involves gases, while \(K_c\) is based on molar concentrations.

In any chemical reaction where \(\Delta n\) is not zero, \(K_p\) will not be equal to \(K_c\). For instance, if \(\Delta n\) is positive, meaning there are more gaseous products than reactants, \(K_p\) will be greater than \(K_c\) if the temperatures are above absolute zero, due to the \(RT\) term being raised to a positive power. This relationship allows us to convert between the two constants provided we know the temperature and the change in gaseous moles.

Gas-phase reactions are chemical processes that occur between substances in the gaseous state. The behavior and study of these reactions are distinctly influenced by the properties of gases, such as volume, pressure, temperature, and concentration. \(K_p\) becomes particularly useful in these cases as it deals with the equilibrium conditions involving the partial pressures of the gases involved.

For gas-phase reactions, partial pressures serve as a proxy for concentration, since according to the ideal gas law \(PV = nRT\), pressure is directly proportional to the concentration. Moreover, the unique nature of gases allows us to use the ideal gas law to relate these partial pressures to molar concentrations and thus to convert between different expressions of the equilibrium constant. The understanding and calculation of reaction directions and extents in gas-phase reactions is not just academic; it's vital for practical applications such as industrial synthesis, environmental monitoring, and even in astrophysical chemistry.

Le Chatelier's principle provides a predictive view on how a system at equilibrium reacts to external changes. It states that if an external stress, such as changes in pressure, temperature, or concentration, is applied to a system in equilibrium, the system will adjust itself to counteract the imposition of the stress and restore a new equilibrium state. This principle enables chemists to control chemical reactions to some degree and optimize yields of desired products.

Application of Le Chatelier's principle is manifold. For example, increasing the concentration of reactants will typically shift the equilibrium towards the formation of more products. Conversely, a rise in temperature for an exothermic reaction will shift the balance towards the reactants, as heat is a product of the reaction and the system aims to absorb the excess heat by forming more reactants. Understanding this principle is essential for manipulating reaction conditions to achieve optimal results in both laboratory and industrial settings.