Show that as long as the extent of the reaction, α, for the disassociation reaction X2 ⇌2 X is s

step 1 All right, guys, in order to answer this question, the actual form of the reaction can be writte

Show that as long as the extent of the reaction, α, for the...

Show that as long as the extent of the reaction, $\alpha,$ for the disassociation reaction $\mathrm{X}_{2} \rightleftharpoons 2 \mathrm{X}$ is smaller than one, $\alpha$ is given by $\alpha=\sqrt{\frac{K_{P}}{4+K_{P}}}$

Key Concepts

Mass and Pressure Balances

Mass and pressure balances are conservation principles that ensure the total amount of substance or pressure remains constant throughout a closed system during a reaction. These balances enable the formulation of relationships between the initial conditions and the changes undergone by the system as the reaction progresses to equilibrium, which are essential when deriving analytical expressions for quantities such as the extent of reaction.

Reaction Stoichiometry

Reaction stoichiometry involves the quantitative relationships between reactants and products as given by the balanced chemical equation. It provides the necessary coefficients that are used to relate the changes in amounts or pressures of the species involved in the reaction, thus forming the basis for deriving expressions for the equilibrium constant in relation to the extent of reaction.

Extent of Reaction (?)

The extent of reaction, denoted by ?, quantifies the progress of a reaction from initial reactants toward products, typically expressed as a fraction of the total amount of reactant that has been converted. This parameter is crucial for linking the changes in concentrations or pressures of individual species to the overall reaction progress as defined by stoichiometry.

Equilibrium Constant (Kp)

The equilibrium constant, particularly when expressed in terms of partial pressures (Kp), is a dimensionless value that indicates the ratio of the product of the pressures of the products to the product of the pressures of the reactants, each raised to the power of their stoichiometric coefficients. It is a fundamental parameter for predicting the position and extent of the equilibrium in a gaseous reaction.

Chemical Equilibrium

Chemical equilibrium refers to the state in a reversible reaction where the rates of the forward and reverse reactions are equal, meaning that the concentrations (or partial pressures) of the reactants and products remain constant over time. This concept is essential in analyzing reactions, as it allows us to apply the equilibrium constant, which quantitatively expresses the balance between products and reactants at a given temperature.

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