Consider the reaction and the associated equilibrium constant: $$ a A(g)+l B(g) \Longrightarrow c C(g) \quad K_{c}-5.0 $$ And the equilibrium concentrations of \(A, B,\) and \(C\) for each value of \(a, b,\) and \(c\). Assume that the initial concentrations of \(A\) and \(B\) are each \(1.0 \mathrm{M}\) and that no product is present at the begin. ning of the reaction. a. \(a-1 ; b-1 ; c-2\) b. \(a-1 ; b-1 ; c-1\) c. \(a-2 ; b-1 ; c=1\) (set up equation for \(x ;\) don't solve)

The equilibrium constant, represented as \( K_c \), is a crucial concept in chemical equilibrium that quantifies the ratio of the concentrations of the products to the reactants at equilibrium. It is a static value for a particular reaction at a given temperature and does not change unless the temperature changes. As shown in our textbook exercise, for the general reaction \( aA(g) + bB(g) \longleftrightarrow cC(g) \), the equilibrium constant is expressed as \( K_c = \frac{[C]^c}{[A]^a[B]^b} \).

The value of the equilibrium constant provides insights into the position of the equilibrium. A large \( K_c \) indicates a reaction with a higher concentration of products at equilibrium, suggesting that the reaction favors the forward process. Conversely, a smaller \( K_c \) signals that the reactants are more favored, and thus the reaction tends toward the reverse process. In the exercise given, the equilibrium constant of 5.0 suggests that the products are relatively favored when equilibrium is achieved.

While the equilibrium constant is a measure of the ratio of concentrations at equilibrium, the reaction quotient, \( Q \), applies to any point in time during the reaction. It has the same form as the equilibrium constant but uses the current concentrations of reactants and products rather than their concentrations at equilibrium. The reaction quotient is calculated using \( Q = \frac{[C]^c}{[A]^a[B]^b} \) for the given reaction at any stage before reaching equilibrium.

The value of \( Q \) determines the direction in which the reaction will proceed to reach equilibrium. If \( Q < K_c \), the forward reaction is favored, and the system will shift towards the products to reach equilibrium. If \( Q > K_c \), the opposite is true, and the system will shift towards the reactants. If \( Q = K_c \), the system is already at equilibrium, and no net change will occur.

Finding the equilibrium concentration involves calculating the concentrations of all reactants and products once the system has reached a state of equilibrium. At the beginning of the given reaction, reactants \( A \) and \( B \) are present at initial concentrations, but as the reaction progresses, their concentrations decrease while the concentration of product \( C \) increases until equilibrium is established.

In the step-by-step solution, changes in concentration were denoted using \( x \) to gauge how the shifting levels of reactants and products would affect the equilibrium constant calculation. By setting up balanced equations, the exercise demonstrated the manipulation of the initial and equilibrium concentrations to solve for \( x \). This variable \( x \) is subsequently used to determine the final equilibrium concentrations of all species involved in the reaction. Therefore, understanding how to manipulate these variables is essential for mastering equilibrium problems in chemistry.