91Ó°ÊÓ

The equilibrium constant, denoted as \( K_c \), is central to understanding chemical equilibrium. It provides insight into the extent to which a chemical reaction will proceed. At a given temperature, the equilibrium constant is calculated using the concentrations of the products and reactants when the reaction is at equilibrium.
For the reaction involving phosgene (\(\text{COCl}_2\)), which dissociates into carbon monoxide (\(\text{CO}\)) and chlorine gas (\(\text{Cl}_2\)), the equilibrium expression is defined by the formula:

• \( K_c = \frac{[\text{CO}][\text{Cl}_2]}{[\text{COCl}_2]} \)

This equation states that the equilibrium constant is the ratio of the product of the concentrations of the reaction products to the concentration of the reactant. The magnitude of \( K_c \) reveals whether the reactants or products are favored at equilibrium:

• If \( K_c \) is large, products are favored.
• If \( K_c \) is small, reactants are favored.

In the given exercise, \( K_c = 3.63 \times 10^{-3} \), which suggests that dissociation into \(\text{CO}\) and \(\text{Cl}_2\) is not extensive under these conditions.
A low equilibrium constant indicates that only a small proportion of phosgene dissociates at \(720^\circ \text{C}\). Keeping track of these values helps understand the chemical behavior under specific conditions.

Dissociation in chemistry refers to the process where molecules split into smaller particles, such as atoms or ions. For phosgene (\(\text{COCl}_2\)), dissociation occurs when it breaks down into carbon monoxide (\(\text{CO}\)) and chlorine gas (\(\text{Cl}_2\)). This process can be represented by the equation:

• \( \text{COCl}_2(g) \rightleftharpoons \text{CO}(g) + \text{Cl}_2(g) \)

Understanding dissociation is crucial because it influences the concentrations of different species in the reaction. These concentrations, in turn, affect the equilibrium constant. It's essential to set up an initial concentration of the reactant—in this case, \(\text{COCl}_2\)—and determine how much of it dissociates.
During the dissociation process, changes occur in the reactant and product concentrations. Initially, you start with a certain amount of phosgene and, as it dissociates, the concentration of \(\text{COCl}_2\) decreases while the concentrations of \(\text{CO}\) and \(\text{Cl}_2\) increase by the same amount. In this exercise, we denote this change by \( x \), where:

• \( \text{Concentration of COCl}_2 = 0.317\,\text{M} - x \)
• \( \text{Concentration of CO} = x \)
• \( \text{Concentration of Cl}_2 = x \)

Calculating the value of \( x \) allows us to find the percentage of phosgene that dissociates, giving a comprehensive view of how the equilibrium state is achieved. Understanding dissociation is fundamental in predicting how much of the starting material remains and how much is converted into products.

Quadratic equations often arise in chemistry when dealing with equilibrium expressions involving dissociation. Here, deriving a quadratic equation is a key step in determining how much of the reactant dissociates in a reaction. The formula \( ax^2 + bx + c = 0 \) is used to describe such equations.
In the exercise, after establishing the equilibrium concentrations, we substitute them into the equilibrium expression for \( K_c \), resulting in the equation:

• \( x^2 = (3.63 \times 10^{-3})(0.317 - x) \)

Rearranging this gives a standard quadratic equation:

• \( x^2 + (3.63 \times 10^{-3})x - 1.15071 \times 10^{-3} = 0 \)

To solve this equation, we apply the quadratic formula:\

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

where \( a = 1 \), \( b = 3.63 \times 10^{-3} \), and \( c = -1.15071 \times 10^{-3} \). Using this formula, we find two potential solutions for \( x \).
However, only the positive solution is physically meaningful in the context of dissociation since negative concentrations do not make sense. Solving gives us \( x \approx 0.0324 \, \text{M} \), which is used to find the percentage of initial phosgene that dissociates.
The quadratic equation thus serves as a mathematical tool to bridge the balance of the chemical equilibrium with the actual concentrations of reactants and products in a reaction at equilibrium.