Problem 33 - Chemistry and Chemical Reactivity

Question

\(K_{\mathrm{p}}\) for the formation of phosgene, \(\mathrm{COCl}_{2},\) is \(6.5 \times 10^{11}\) at \(25^{\circ} \mathrm{C}\) $$ \mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{COCl}_{2}(\mathrm{g}) $$ What is the value of \(K_{p}\) for the dissociation of phosgene? $$ \mathrm{COCl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) $$

Step-by-Step Solution

Answer

The value of \(K_{p}\) for dissociation is \(1.54 \times 10^{-12}\).

Step 1: Understanding the Reaction

The problem involves the equilibrium constant, known as the equilibrium constant expression ( K_p ), for two different reactions: the formation of phosgene and the dissociation of phosgene. We must calculate K_p for the dissociation reaction from the given information.

Step 2: Expression for Formation Reaction

For the first reaction where phosgene forms, the equilibrium constant expression is given by: \[ K_{p1} = \frac{P_{\text{COCl}_{2}}}{P_{\text{CO}} \times P_{\text{Cl}_{2}}} = 6.5 \times 10^{11} \]where P denotes partial pressures of the gases.

Step 3: Expression for Dissociation Reaction

For the second reaction, which is the reverse of the first, the expression is: \[ K_{p2} = \frac{P_{\text{CO}} \times P_{\text{Cl}_{2}}}{P_{\text{COCl}_{2}}} \].Since this is the reverse reaction of the formation reaction, K_{p2} is the reciprocal of K_{p1}.

Step 4: Calculate Kp for Dissociation

Apply the reciprocal relationship:\[ K_{p2} = \frac{1}{K_{p1}} = \frac{1}{6.5 \times 10^{11}} \].Calculate the value to find K_{p2}.

Step 5: Final Calculation and Answer

Perform the reciprocal calculation:\[ K_{p2} = 1.54 \times 10^{-12} \].Thus, the equilibrium constant K_p for the dissociation of phosgene is calculated.

Key Concepts

Phosgene Formation

Phosgene, a chemical compound with the formula \( \text{COCl}_2 \), forms through a simple yet essential reaction. This reaction is between carbon monoxide (\( \text{CO} \)) and chlorine gas (\( \text{Cl}_2 \)). These two gases come together to produce phosgene. At the molecular level, the carbon atom in carbon monoxide bonds with two chlorine atoms, resulting in the formation of phosgene as a single molecule. The reaction can be represented as follows: \[ \text{CO}(\text{g}) + \text{Cl}_2(\text{g}) \rightleftharpoons \text{COCl}_2(\text{g}) \]. Phosgene is a valuable industrial chemical. It was historically used in chemical warfare, but today it is primarily utilized to manufacture pharmaceuticals, agrochemicals, and dyes. It is paramount to understand the formation of phosgene to appreciate how this reaction is controlled and applied in various chemical industries.

Chemical Equilibrium

Chemical equilibrium forms the basis for understanding reactions like phosgene formation. At equilibrium, a chemical reaction has reached a state where the rates of the forward and reverse reactions are equal. Thus, the concentrations of the reactants and products no longer change over time. Equilibrium does not mean the reactants and products are equal in concentration, but rather that their proportions remain constant. For the phosgene formation reaction, the equilibrium position is governed by the equilibrium constant \( K_p \). This constant is temperature-dependent and reflects the relative energies of the reactants and products. A large \( K_p \) value indicates the reaction heavily favors the formation of products, hence, for phosgene, this value is extremely high \( 6.5 \times 10^{11} \), suggesting that phosgene forms very readily under these conditions.

Reversible Reactions

Reversible reactions are central to the concept of equilibrium. A reversible reaction is one where the reactants can convert to products and vice versa. The reaction doesn't go to completion, meaning not all reactants are necessarily converted to products. This back and forth creates a dynamic situation where both sides of the reaction are constantly changing. In a closed system, these reactions can achieve equilibrium. The formation and dissociation of phosgene exemplifies a reversible reaction: - **Forward Reaction:** \( \text{CO}(\text{g}) + \text{Cl}_2(\text{g}) \rightarrow \text{COCl}_2(\text{g}) \) - **Reverse Reaction:** \( \text{COCl}_2(\text{g}) \rightarrow \text{CO}(\text{g}) + \text{Cl}_2(\text{g}) \) At equilibrium, both the formation of phosgene and its breakdown occur at the same rate, maintaining a balance.

Partial Pressures

In gas-phase reactions like the formation of phosgene, understanding partial pressures is crucial. Each gas in a mixture exerts a partial pressure, which is the pressure it would have if it alone filled the container. Partial pressures are additive, and the total pressure of the system is the sum of the partial pressures of each gas present. In the equilibrium expression for the phosgene reaction, partial pressures are used. This can be written as: \[ K_p = \frac{P_{\text{COCl}_2}}{P_{\text{CO}} \times P_{\text{Cl}_2}} \] Here, \( P_{\text{CO}} \), \( P_{\text{Cl}_2} \), and \( P_{\text{COCl}_2} \) represent the partial pressures of carbon monoxide, chlorine gas, and phosgene, respectively. The precise balance of these pressures determines the position of equilibrium. Therefore, understanding how to manipulate and calculate partial pressures is essential for controlling such reactions.