At a particular temperature, \(K=4.0 \times 10^{-7}\) for the reaction $$ \mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g) $$ In an experiment, \(1.0\) mole of \(\mathrm{N}_{2} \mathrm{O}_{4}\) is placed in a \(10.0-\mathrm{L}\) vessel. Calculate the concentrations of \(\mathrm{N}_{2} \mathrm{O}_{4}\) and \(\mathrm{NO}_{2}\) when this reaction reaches equilibrium.

Understanding chemical equilibrium can sometimes be a complex task, but with an ICE table (Initial, Change, Equilibrium), it becomes a lot more manageable. The ICE table is a systematic way to organize the changes in concentration of reactants and products during a reaction as it approaches equilibrium.

Let's walk through an example. Suppose we start with 1.0 mole of N₂O₄ in a 10.0 L vessel. Initially, we have no NO₂, so its concentration is 0. We represent the initial concentration of N₂O₄ as 0.1 M and NO₂ as 0 M. As the reaction proceeds towards equilibrium, N₂O₄ decomposes into NO₂, and we define this change in concentration as 'x' and '2x', respectively, in line with the stoichiometry of the reaction.

Therefore, at equilibrium, the concentration of N₂O₄ will be '0.1 - x' M and for NO₂, it will be '2x' M. Using an ICE table efficiently categorizes these stages, providing clear visualization of the process, which greatly simplifies the computation of equilibrium concentrations.

A cornerstone of chemical equilibrium is the equilibrium constant, denoted as K. It's a number that expresses the ratio of the concentration of the products to the reactants at equilibrium, each raised to the power of their stoichiometric coefficients.

For our N₂O₄ and NO₂ reaction, the equilibrium constant, K, would be formulated as follows:
\[ K = \frac{[\mathrm{NO}_{2}]^{2}}{[\mathrm{N}_{2} \mathrm{O}_{4}]} \]
In the real world of chemistry, a large K (much greater than 1) suggests that at equilibrium, mostly products are present, indicating the forward reaction is favored. Conversely, a small K (much less than 1) means that reactants predominate, and the reverse reaction is favored. For our reaction, K is given to be 4.0 x 10⁻⁷, indicating the equilibrium heavily favors the reactants, N₂O₄.

Before a system reaches equilibrium, we can predict the direction in which the reaction will proceed using the reaction quotient (Q). Like the equilibrium constant, it represents a ratio of product and reactant concentrations at any point in time before the system reaches equilibrium, taking into account their stoichiometric coefficients.

The reaction quotient is calculated using the formula:
\[ Q = \frac{[\mathrm{NO}_{2}]^{2}}{[\mathrm{N}_{2} \mathrm{O}_{4}]} \]
just as with K. However, Q is evaluated with current concentrations, not those at equilibrium. When comparing Q to K, if \( Q > K \), the reaction will proceed in the reverse direction to reach equilibrium. If \( Q < K \), the reaction will move forward, and if \( Q = K \), the system is already at equilibrium. Using the reaction quotient helps in predicting the behavior of a reaction under different initial conditions and is a powerful tool in the study of dynamic chemical processes.