Consider the equilibrium \(4 \mathrm{NO}_{2}(g)+6 \mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons 4 \mathrm{NH}_{3}(g)+7 \mathrm{O}_{2}(g)\) (a) What is the expression for the equilibrium constant \(\left(K_{c}\right)\) of the reaction? (b) How must the concentration of \(\mathrm{NH}_{3}\) change to reach equilibrium if the reaction quotient is less than the equilibrium constant? (c) If the reaction were at equilibrium, how would an increase in the volume of the reaction vessel affect the pressure of \(\mathrm{NO}_{2} ?\) (d) If the change in the pressure of \(\mathrm{NO}_{2}\) is 28 torr as a mixture of the four gases reaches equilibrium, how much will the pressure of \(\mathrm{O}_{2}\) change?

Understanding the equilibrium constant expression is fundamental when studying chemical reactions reaching a state of equilibrium. The expression is devised from the balanced chemical equation and reflects the ratio of the concentrations of the products to the reactants, raised to the power of their coefficients.
For the given reaction \(4 \mathrm{NO}_{2}(g) + 6 \mathrm{H}_{2}O(g) \rightleftharpoons 4 \mathrm{NH}_{3}(g) + 7 \mathrm{O}_{2}(g)\), the equilibrium constant expression \(K_c\) is calculated using the formula:
\[K_c = \frac{[\mathrm{NH}_{3}]^4[\mathrm{O}_{2}]^7}{[\mathrm{NO}_{2}]^4[\mathrm{H}_{2}O]^6}\]
Each concentration is measured in moles per liter (M), and the equilibrium constant itself is dimensionless. It's important to note that only gaseous and aqueous species are included in \(K_c\), while solids and liquids are omitted since their concentrations do not change.

The reaction quotient (\(Q\)) plays a crucial role in predicting the direction in which a reaction will proceed to reach equilibrium. It is calculated at any point during the reaction, using the same formula as the equilibrium constant, but with the current concentrations of the reactants and products.
\[Q = \frac{[\mathrm{NH}_{3}]^4[\mathrm{O}_{2}]^7}{[\mathrm{NO}_{2}]^4[\mathrm{H}_{2}O]^6}\]
Comparing \(Q\) to the equilibrium constant \(K_c\) indicates whether the reaction will shift towards the products (\(Q < K_c\)) or towards the reactants (\(Q > K_c\)) to achieve equilibrium. For the exercise given, if \(Q\) is less than \(K_c\), it means that the concentration of \(\mathrm{NH}_{3}\) must increase for the reaction to reach equilibrium.

Le Chatelier's principle is a foundational concept in chemistry that predicts how a change in conditions can affect the position of equilibrium. When a system at equilibrium is disturbed, it will adjust itself to counteract the disturbance and restore a new equilibrium.
For instance, altering the concentration of reactants or products will shift the equilibrium to the side that counteracts this change. A temperature change will favor the endothermic or exothermic reaction, depending on whether heat is added or removed.

Application to Volume and Pressure Changes

When the volume of a gas reaction is increased, as in our example, the pressure decreases, causing the system to shift towards the side with more gas moles to increase the pressure. Conversely, decreasing the volume will increase pressure, shifting equilibrium towards the side with fewer gas moles.

The relationship between pressure and volume for gases is described by Boyle's Law, which states that for a fixed amount of gas at constant temperature, the pressure and volume are inversely proportional. Mathematically, this is expressed as \(P_1V_1 = P_2V_2\), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the new pressure and volume after a change.
In the context of our exercise, when the volume of the reaction vessel is increased, the pressure of each gas, including \(\mathrm{NO}_{2}\), will decrease. This interplay of pressure and volume is essential to predicting how changes in the reaction vessel size will affect the reaction's direction and, ultimately, the attainment of equilibrium citing Le Chatelier’s principle.