91Ó°ÊÓ

The dissociation constant, often represented as \(K_c\), is a crucial value in chemical equilibrium. It quantifies the extent of dissociation of a compound in a chemical reaction. In an equilibrium reaction, like the dissociation of phosgene \(\mathrm{COCl}_{2}\) into carbon monoxide \(\mathrm{CO}\) and chlorine gas \(\mathrm{Cl}_{2}\), the dissociation constant helps determine how much of the original compound actually breaks apart. For the reaction given:\[\mathrm{COCl}_{2} \rightleftharpoons \mathrm{CO} + \mathrm{Cl}_{2}\]The dissociation constant \(K_c\) is calculated based on the concentrations of the products and reactants at equilibrium:
\[K_c = \frac{[\text{CO}][\text{Cl}_2]}{[\text{COCl}_2]}\]A small \(K_c\) value, such as \(8.05 \times 10^{-4}\) in our case, indicates that the equilibrium heavily favors the reactants, and little \(\mathrm{COCl}_{2}\) dissociates. The dissociation constant is essential because it informs us about the direction and state of the equilibrium, helping chemists predict the behavior of chemical systems under various conditions.

The quadratic formula is a powerful tool used to find the roots of quadratic equations. In the context of chemical equilibrium calculations, it becomes significant when concentrations of species vary quadratically. The general form of a quadratic equation is:\[ax^2 + bx + c = 0\]In our equilibrium problem, we arrived at a quadratic equation while solving for \(x\), the amount of \(\text{COCl}_2\) that dissociates:
\[x^2 + 8.05 \times 10^{-4}x - 3.22 \times 10^{-5} = 0\]To solve it, we use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]**Steps of using the quadratic formula:**

• Identify coefficients: \(a = 1\), \(b = 8.05 \times 10^{-4}\), \(c = -3.22 \times 10^{-5}\).
• Substitute these values into the formula to find \(x\).
• Simplify to find the possible values of \(x\).

This formula provides the positive root for \(x\), which represents a physical, positive concentration in the context of chemistry, simplifying our calculation for the equilibrium scenario.

Equilibrium concentration is a snapshot of reactant and product concentrations when a chemical reaction has reached equilibrium in a closed system. It provides an insight into how far the reaction has proceeded. In the example of phosgene dissociation, we started with 1.00 mol of \(\mathrm{COCl}_2\) in a 25.0 L vessel, leading to an initial concentration of:\[\frac{1.00}{25.0} = 0.040 \text{ mol/L}\]**Determining equilibrium concentrations involves:**

• Writing the balanced equation and identifying initial concentrations.
• Introducing a variable \(x\) to represent the change in concentration as the reaction proceeds to equilibrium.
• Updating concentrations based on stoichiometry and writing expressions for equilibrium concentrations.

For equilibrium concentrations of this reaction, \(\mathrm{CO}\) and \(\mathrm{Cl}_2\) both had concentrations of \(x\) mol/L as they form in a 1:1 ratio.
Finally, by calculating \(x\), we determined how much \(\mathrm{COCl}_2\) dissociated and used this to quantify the other substances' equilibrium concentrations, facilitating the understanding of chemical behavior under thermal conditions.