Problem 31 - Chemistry The Science in Context

Question

Phosgene \(\left(\mathrm{COCl}_{2}\right)\) is used in the manufacture of foam rubber and bulletproof glass. It is formed from carbon monoxide and chlorine in the following reaction: $$ \mathrm{Cl}_{2}(g)+\mathrm{CO}(g) \rightleftharpoons \mathrm{COCl}_{2}(g) $$ The value of \(K_{\mathrm{c}}\) for the reaction is 5.0 at \(327^{\circ} \mathrm{C} .\) What is the value of \(K_{\mathrm{p}}\) at \(327^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

Question: Determine the value of \(K_{\mathrm{p}}\) for the reaction \(\mathrm{Cl}_{2}(g) + CO(g) \rightleftharpoons COCl_{2}(g)\) at a temperature of \(327^{\circ}\mathrm{C}\) given that \(K_{\mathrm{c}} = 5.0\). Answer: The value of \(K_{\mathrm{p}}\) at \(327^{\circ}\mathrm{C}\) is approximately 0.001.

Step 1: Write the balanced equation

The balanced equation for the reaction is given as: $$ \mathrm{Cl}_{2}(g)+\mathrm{CO}(g) \rightleftharpoons \mathrm{COCl}_{2}(g) $$

Step 2: Determine the change in the number of moles of gas

In the balanced equation, we have 1 mole of \(\mathrm{Cl}_{2}\) and 1 mole of \(\mathrm{CO}\) reacting to form 1 mole of \(\mathrm{COCl}_{2}\). The change in the number of moles of gas for the reaction is given as: $$ \Delta n = \text{moles of products} - \text{moles of reactants} = (1 - 1 - 1) = -1 $$

Step 3: Use the relationship between \(K_{\mathrm{p}}\) and \(K_{\mathrm{c}}\)

The relationship between \(K_{\mathrm{p}}\) and \(K_{\mathrm{c}}\) is given by the equation: $$ K_{\mathrm{p}} = K_{\mathrm{c}}\left(\frac{P}{RT}\right)^{\Delta n} $$ Given that the value of \(K_{\mathrm{c}} = 5.0\), the temperature is \(327^{\circ}\mathrm{C} \Rightarrow 600\,\mathrm{K}\) (adding \(273.15\) to convert to Kelvin), and \(\Delta n = -1\), we can solve for \(K_{\mathrm{p}}\): $$ K_{\mathrm{p}} = 5.0 \left(\frac{P}{(8.314 \,\mathrm{J \cdot K^{-1} \cdot mol^{-1}})(600\,\mathrm{K})}\right)^{-1} $$ Since the pressure P does not affect the relationship between \(K_{\mathrm{p}}\) and \(K_{\mathrm{c}}\), we can leave it out of our final calculation: $$ K_{\mathrm{p}} = 5.0 \left(\frac{1}{(8.314)(600)}\right) $$ Calculating the value of \(K_{\mathrm{p}}\): $$ K_{\mathrm{p}} \approx 0.001 $$ Thus, the value of \(K_{\mathrm{p}}\) at \(327^{\circ}\mathrm{C}\) is approximately \(0.001\).

Key Concepts

Equilibrium Constant

In chemical reactions, the equilibrium constant, represented as either \(K_c\) or \(K_p\), plays a critical role in understanding how reactions reach a state of balance. This constant reflects the ratio of concentrations (\(K_c\)) or pressures (\(K_p\)) of products to reactants when a system is at equilibrium. It is important to note that \(K_c\) is used for reactions in solution, while \(K_p\) is applied to reactions involving gases. Both constants are related, and their values can reveal important information about the direction in which a reaction is favored.

The symbol \(K_c\) is specifically used for reactions in an aqueous phase and is computed based on the concentrations of the reactants and products. On the other hand, \(K_p\) applies to gaseous reactions and is calculated using the partial pressures of the gases involved. The relationship between the two can be expressed with the formula: \[ K_p = K_c (RT)^{\Delta n} \]where \(R\) is the gas constant, \(T\) the temperature in Kelvin, and \(\Delta n\) the change in moles of gas.

Understanding the equilibrium constant allows us to predict the concentrations of reactants and products at equilibrium, and helps in analyzing how changes in conditions can shift equilibrium positions. This is crucial for chemists as it provides insights into reaction applications, including industrial processes like the synthesis of phosgene.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept in chemistry that predicts how a system at equilibrium responds to changes. It asserts that a system disturbed from equilibrium will adjust itself to counteract the change and re-establish equilibrium. This principle is particularly useful in industrial applications, where optimizing reaction conditions is paramount. Let's delve into its practical applications:

• **Changes in Concentration**: When the concentration of either reactants or products in a balanced equation changes, the system will shift to restore balance. If the concentration of a reactant is increased, the equilibrium will shift towards the products.
• **Temperature Adjustments**: Temperature changes can affect equilibrium. For exothermic reactions, increasing temperature will shift equilibrium towards the reactants, while for endothermic reactions, it shifts towards the products.
• **Pressure Variations**: In gaseous systems, changing pressure by altering volume will shift equilibrium towards the side with fewer moles of gas if pressure increases, and towards the side with more moles if pressure decreases.

Le Chatelier's Principle provides a powerful tool for predicting the outcome of operational changes. For the phosgene reaction, an increase in pressure would shift the equilibrium position towards the formation of phosgene, as it decreases the total number of moles of gas.

Thermodynamics

Thermodynamics is the branch of physical science concerned with heat and temperature and their relation to energy and work. When it comes to chemical reactions, thermodynamics helps us understand the energy changes that occur and predict whether a reaction is spontaneous. This is particularly important in studying reactions like the formation of phosgene, where safety and efficiency are crucial.

Thermodynamic concepts involve:

• **Gibbs Free Energy (G)**: This is a key factor in determining whether a reaction is spontaneous. A negative change in Gibbs free energy (\( \Delta G < 0 \)) implies that a reaction will proceed spontaneously.
• **Enthalpy (H) and Entropy (S)**: Enthalpy changes (\( \Delta H \)) denote heat transfer in a reaction; exothermic reactions release heat, while endothermic reactions absorb it. Entropy (\( \Delta S \)) reflects the disorder in the system. Combining these two gives insight into the free energy change:\[ \Delta G = \Delta H - T\Delta S \]

Together, these concepts help us examine the energy landscape of chemical reactions. By allowing us to predict the feasibility and extent of reactions, thermodynamics assists in the design and optimization of industrial chemical processes. Understanding these principles is vital to ensure safety, maximize efficiency, and minimize environmental impacts in chemical manufacturing.