Question

A mixture containing 3.9 moles of \(\mathrm{NO}\) and 0.88 mole of \(\mathrm{CO}_{2}\) was allowed to react in a flask at a certain temperature according to the equation $$ \mathrm{NO}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{NO}_{2}(g)+\mathrm{CO}(g) $$ At equilibrium, 0.11 mole of \(\mathrm{CO}_{2}\) was present. Calculate the equilibrium constant \(K_{\mathrm{c}}\) of this reaction.

Short answer

The equilibrium constant \(K_c\) for the reaction is approximately 17.2.

Step-by-step solution

Reactant and Product Mole Calculation

We first need to determine the change in moles for each reactant and product. Initially, we have 3.9 moles of \(\mathrm{NO}\) and 0.88 mole of \(\mathrm{CO_2}\). At equilibrium, we have 0.11 moles of \(\mathrm{CO_2}\). From the reaction equation, we know for every mole of \(\mathrm{NO}\) used, one mole of \(\mathrm{CO_2}\) is also used. So, the change in moles for \(\mathrm{NO}\) is the same as the change in moles for \(\mathrm{CO_2}\).

Change in Moles Calculation

The change in moles for \(\mathrm{CO_2}\) is the initial moles minus the moles at equilibrium (0.88 - 0.11 = 0.77 moles). So, the change in moles for \(\mathrm{NO}\) is also 0.77 moles. Consequently, the moles of \(\mathrm{NO}\) at equilibrium become 3.9 - 0.77 = 3.13 moles. Because for each mole of reactant consumed, one mole of product is created (according to the equation), the moles of \(\mathrm{NO_2}\) and \(\mathrm{CO}\) at equilibrium would be 0.77 moles.

Finding Equilibrium Constant \(K_c\)

The equilibrium constant \(K_c\) for the reaction is given by the ratio of the concentrations (in moles) of the products to the reactants each raised to the power of their stoichiometric coefficients. Algebraically, this becomes \[K_c = \frac{[\mathrm{NO_2}][\mathrm{CO}]}{[\mathrm{NO}][\mathrm{CO_2}]}\]. Substituting the calculated equilibrium moles of the reactants and products into the equation, we find \(K_c = \frac{(0.77)(0.77)}{(3.13)(0.11)}\).

Calculating the Numerical Value of \(K_c\)

After performing the mentioned arithmetic operation, the value of \(K_c\) will be obtained.

Key concepts

Chemical Equilibrium

In chemical reactions, reaching a point where the forward and reverse reactions occur at the same rate results in a state known as chemical equilibrium. This implies that the concentrations of the reactants and products remain constant over time, although they are not necessarily equal. In our given reaction, NO(g) + CO₂(g) ⇌ NO₂(g) + CO(g), we achieve chemical equilibrium when the rate at which NO and CO₂ are converted into NO₂ and CO matches the reverse process.
This balance is crucial for calculating the equilibrium constant, represented as K_c. The constant gives insight into the position of the equilibrium—specifically, whether the products or reactants are favored. A larger value of K_c indicates a product-favored equilibrium, while a smaller value points to one that favors reactants. Understanding this concept is essential for interpreting reactions within closed systems, such as those involving gases in a flask like in this case.

Reaction Moles

The concept of moles is fundamental in chemistry, allowing us to quantify the amount of a substance involved in a reaction. Initially in our problem, we start with 3.9 moles of NO and 0.88 moles of CO₂. When the mixture is allowed to reach equilibrium, changes in the moles of these substances help determine the amounts of products and reactants still present.
The key idea is to track how the moles change from the initial state to equilibrium. We observe that 0.77 moles of CO₂ are used up, indicating the same change (0.77 moles) in NO because they react on a one-to-one basis as per the reaction equation. Similarly, this change corresponds to a production of 0.77 moles each of NO₂ and CO, as products form in equal amounts. Understanding reaction moles aids in calculating equilibrium concentrations, a necessary step for determining the equilibrium constant, K_c.

Concentration

Concentration, usually expressed in moles per liter (M), is a critical parameter in studying chemical equilibria. It describes how crowded the molecules are in a given volume, directly influencing the reaction's progression tendencies. In the problem at hand, while specific volumes aren't provided, we still calculate changes in moles that would eventually be converted to concentrations through volume if needed.
The importance of concentration stems from its role in computing the equilibrium constant, K_c. Typically, K_c is determined using concentrations of products over reactants, each raised to their respective stoichiometric coefficients, as evident in the formula:

• K_c = [NO₂][CO] / [NO][CO₂]

In equilibrium chemistry, knowing how concentrations change as reactions proceed helps in predicting the direction of a reaction, designing experiments, and customizing industrial processes for desired outcomes.