A mixture of \(0.0560 \mathrm{~mol} \mathrm{O}_{2}\) and \(0.0200 \mathrm{~mol}\) \(\mathrm{N}_{2} \mathrm{O}\) is placed in a \(1.00-\mathrm{L}\) reaction vessel at \(25^{\circ} \mathrm{C}\). When the reaction \(2 \mathrm{~N}_{2} \mathrm{O}(\mathrm{g})+3 \mathrm{O}_{2}(\mathrm{~g})=4 \mathrm{NO}_{2}(\mathrm{~g})\) is at equilibrium, \(0.0200\) mol \(\mathrm{NO}_{2}\) is present. (a) What are the equilibrium concentrations? (b) What is the value of \(K_{c}\) ?

Understanding equilibrium concentrations is crucial when studying chemical reactions in a closed system. At equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, meaning that the concentrations of reactants and products remain constant over time.

To calculate equilibrium concentrations, we need to know the initial concentrations and the changes in concentrations due to the reaction proceeding. For example, in a reaction where 0.0200 moles of product is formed in a 1.00 L vessel, if the stoichiometry of the reaction is known, we can calculate the change in amounts for reactants and subsequently find their equilibrium concentrations by dividing the remaining moles by the volume.

It's essential to keep track of the stoichiometry, as the amounts of reactants consumed and products formed will be in ratios defined by the balanced chemical equation. After computing the amount of each reactant and product at equilibrium, these values can be converted into molar concentrations which are used in calculating the equilibrium constant, discussed in the upcoming section.

The equilibrium constant, denoted as Kc, is a number representing the ratio of product concentrations to reactant concentrations at equilibrium, with each raised to the power of their coefficients in the balanced equation. For gas-phase reactions, Kc is especially important as it reflects the ratio of partial pressures of the products and reactants.

In the provided exercise, the Kc expression based on the balanced chemical equation is \[K_c = \frac{[NO_2]^4}{[N_2O]^2[O_2]^3}\] To calculate Kc, you simply plug in the equilibrium concentrations obtained previously into this expression. Each concentration is raised to the power of the respective coefficient from the balanced equation. This calculation tells us the tendency of a reaction to proceed toward products or reactants under given conditions. A higher Kc value indicates that, at equilibrium, the reaction favors the products. Conversely, a lower Kc suggests a preference towards the reactants.

Reaction stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. It relies on the balanced chemical equation, which provides the mole ratio of reactants to products. Correctly understanding and applying stoichiometry is vital for predicting the amounts of reactants needed and products formed.

For instance, the balanced equation in our exercise indicates that 2 moles of \(N_2O\) react with 3 moles of \(O_2\) to produce 4 moles of \(NO_2\). This information is essential to determine the changes in amounts of reactants and products when the system reaches equilibrium. It also informs us on how to relate the decrease in the concentration of reactants to the increase in the concentration of products, which we need for the calculation of equilibrium concentrations and the equilibrium constant.

Gas phase reactions, such as the one featured in the exercise, involve substances that are in the gaseous state. The behavior of gases in a reaction can be influenced by factors such as temperature, pressure, and volume, which makes understanding the ideal gas law and partial pressures important.

For gas-phase reactions at equilibrium, it is common to express the equilibrium constant in terms of partial pressures, known as \(K_p\). However, if concentrations are given, as in our case, we use \(K_c\). It is essential to remember that the volume of the gas will affect the concentration and hence the equilibrium position. The unique properties of gases require careful consideration when calculating reaction yields and equilibrium states, as they can vary significantly with changes in the system's conditions.