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The equilibrium constant, represented as \( K_P \), is an essential concept in chemical equilibrium. It quantifies the ratio of the concentrations or partial pressures of products and reactants at equilibrium for a given reaction. For reactions involving gases, partial pressures are used instead of concentrations. This constant is unique for every reaction at a specific temperature and provides insight into the position of equilibrium.

For the decomposition of \( \text{NO}_2 \) into \( \text{NO} \) and \( \text{O}_2 \), the equilibrium constant expresses the relationship between the partial pressures of the gases at 1000 K. The formula for \( K_P \) is:

• \( K_P = \frac{(P_{\text{NO}})^2 \cdot (P_{\text{O}_2})}{(P_{\text{NO}_2})^2} \)

This expression helps us explore the equilibrium conditions by tying together the pressures of the chemical species involved. With this equation, knowing one or two partial pressures allows us to compute others, making \( K_P \) a crucial tool in determining equilibrium states.

Partial pressure is a concept in chemistry that helps us understand the contribution of individual gases in a mixture to the total pressure. When different gases are present in a container, each gas exerts pressure independently as if the other gases were absent. This is what we refer to as partial pressure.

In an equilibrium state, knowing the partial pressure of one gas, such as \( \text{O}_2 \) in our decomposition reaction, can be instrumental in finding the partial pressures of other gases. The given partial pressure of \( \text{O}_2 \) at equilibrium is 0.25 atm. By using the reaction's stoichiometry and the equilibrium constant equation, we can determine the pressures of \( \text{NO} \) and \( \text{NO}_2 \). Understanding partial pressures is vital when dealing with gas reactions at equilibrium.

Reaction stoichiometry involves the quantitative relationships between the amounts of reactants and products in a chemical reaction. It is derived from the balanced chemical equation and indicates how moles of reactants relate to moles of products.

For the decomposition reaction \( 2 \text{NO}_2 \rightarrow 2 \text{NO} + \text{O}_2 \), the stoichiometry tells us that 2 moles of \( \text{NO}_2 \) form 2 moles of \( \text{NO} \) and 1 mole of \( \text{O}_2 \).

This mole relationship is directly used to relate changes in the pressures of these gases. If the partial pressure of \( \text{O}_2 \) increases by a certain amount, the partial pressure of \( \text{NO} \) increases twice that amount, while \( \text{NO}_2 \) decreases by the same amount. These relationships are critical for formulating equations to calculate unknown pressures at equilibrium.

Decomposition reactions are chemical reactions where a single compound breaks down into two or more simpler substances. These reactions often require an input of energy such as heat, which is the case in our decomposition of \( \text{NO}_2 \) at 1000 K.

In the given reaction, \( 2 \text{NO}_2 \) decomposes into \( 2 \text{NO} \) and \( \text{O}_2 \). This type of reaction not only changes the substances involved but also alters the system's pressure and composition as products are formed from a single reactant. Understanding decomposition reactions helps predict the behavior of reactants and products under various conditions, like temperature changes in this context.