Problem 90 - Chemistry Matter and Change

Question

Synthesis of Phosgene Phosgene \(\left(\mathrm{COCl}_{2}\right)\) is a toxic gas that is used in the manufacture of certain dyes, pharma- ceuticals, and pesticides. Phosgene can be produced by the reaction between carbon monoxide and chlorine described by the equation \(\mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{COCl}_{2}(\mathrm{g})\) . Initially 1.0000 \(\mathrm{mol} \mathrm{CO}\) and 1.0000 \(\mathrm{mol} \mathrm{Cl}_{2}\) are intro- duced into a 10.00 -L reaction vessel. When equilibrium is established, both of their molar concentrations are found to be 0.0086 \(\mathrm{mol} / \mathrm{L}\) . What is the molar concentration of phosgene at equilibrium? What is \(K_{\mathrm{eq}}\) for the system?

Step-by-Step Solution

Answer

At equilibrium, the molar concentration of phosgene (COCl₂) is 0.0914 M, and the equilibrium constant (Keq) for the system is approximately 1239.

Step 1: Calculate the initial concentrations of CO and Cl₂

We can calculate the initial concentrations by dividing the number of moles by the volume of the reaction vessel: Initial concentration of CO: \( \frac{1.0000 \, mol}{10.00 \, L} = 0.1000 \, M \) Initial concentration of Cl₂: \( \frac{1.0000 \, mol}{10.00 \, L} = 0.1000 \, M \)

Step 2: Calculate the change in concentration for CO and Cl₂ at equilibrium

The concentrations of CO and Cl₂ at equilibrium are 0.0086 M. So, the change in concentrations for CO and Cl₂ can be calculated as: Change in concentration of CO: \( 0.1000 \, M - 0.0086 \, M = 0.0914 \, M \) Change in concentration of Cl₂: \( 0.1000 \, M - 0.0086 \, M = 0.0914 \, M \)

Step 3: Calculate the molar concentration of Phosgene (COCl₂) at equilibrium

Since the reaction produces 1 mol of COCl₂ for every mol of CO and Cl₂ used, the increase in concentration of COCl₂ at equilibrium must be equal to the change in concentration of CO and Cl₂. Therefore: Equilibrium concentration of COCl₂ = 0.0914 M

Step 4: Calculate the equilibrium constant, Keq

The equilibrium constant for the reaction is given using the formula: $$ K_{eq} = \frac{[COCl_2]}{[CO][Cl_2]} $$ Plugging in the equilibrium molar concentrations of CO, Cl₂, and COCl₂ we found earlier: $$ K_{eq} = \frac{0.0914}{(0.0086)(0.0086)} \approx 1239 $$ #Conclusion# At equilibrium, the molar concentration of phosgene (COCl₂) is 0.0914 M, and the equilibrium constant (Keq) for the system is approximately 1239.

Key Concepts

Phosgene Synthesis

Phosgene, with the chemical formula \(\text{COCl}_2\), is a hazardous but important chemical used mainly in producing dyes, pharmaceuticals, and pesticides. It is formed via a straightforward process, reacting carbon monoxide \(\text{CO}\) with chlorine gas \(\text{Cl}_2\). The reaction equation is: \(\mathrm{CO}(\mathrm{g}) + \mathrm{Cl}_2(\mathrm{g}) \rightleftharpoons \mathrm{COCl}_2(\mathrm{g})\). At the molecular level, this synthesis is quite direct. For every molecule of carbon monoxide that reacts, a molecule of chlorine gas combines with it to form one molecule of phosgene.
In a controlled environment such as a reaction vessel, this straightforward mechanism makes it easier to predict and calculate the production of phosgene when given the initial amounts of \(\text{CO}\) and \(\text{Cl}_2\). This reaction reaches a point where the rate of the forward reaction (producing phosgene) equals the rate of the reverse reaction (decomposition back to CO and Cl₂), signifying chemical equilibrium.

Equilibrium Constant (K_eq)

The equilibrium constant, represented as \(K_{eq}\), provides insights into the extent of a reaction at equilibrium. Simply put, it tells us whether the reaction favors the production of products or reactants. For the reaction synthesizing phosgene, \(K_{eq}\) is determined using the concentrations of the components at equilibrium: \[ K_{eq} = \frac{[COCl_2]}{[CO][Cl_2]} \] A high \(K_{eq}\) value, like the one calculated to be approximately 1239, indicates a strong tendency towards forming the product, COCl₂, rather than retaining CO and Cl₂.
This makes sense considering industrial processes aim for efficient phosgene production.

• When \(K_{eq} > 1\), the products dominate at equilibrium.
• When \(K_{eq} < 1\), the reactants dominate at equilibrium.

For students seeking to understand chemical equilibria, the \(K_{eq}\) value is pivotal as it provides a quantitative understanding of the balance between products and reactants once the reaction system has reached equilibrium.

Molar Concentration Calculation

Calculating molar concentrations is a fundamental skill in chemistry, particularly when dealing with reactions involving gases. The concentration, often expressed in moles per liter (M), provides an understanding of how much of a substance is present in a certain volume. For instance, the reaction of phosgene synthesis starts with initial concentrations: \[ \text{Initial concentration of CO and Cl}_2: \frac{1.0000 \text{ mol}}{10.00 \text{ L}} = 0.1000 \text{ M} \] Changes occur as the system approaches equilibrium. Given the equilibrium concentration for CO and Cl₂ is 0.0086 M, the change is: \[ \text{Change} = 0.1000 \text{ M} - 0.0086 \text{ M} = 0.0914 \text{ M} \] This change impacts the phosgene concentration because phosgene is produced directly from these reactants. Therefore, the equilibrium concentration of phosgene mirrors this amount: \[ \text{Equilibrium concentration of COCl}_2 = 0.0914 \text{ M} \] Understanding these steps is crucial for correctly setting up reaction calculations and successfully determining the outcome in terms of reactants converted into products.