Question

At \(395 \mathrm{~K}\), chlorine and carbon monoxide react to form phosgene, \(\mathrm{COCI}_{2}\), according to the equation $$ \mathrm{CO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftarrows \mathrm{COCI}_{2}(\mathrm{~g}) $$ The equilibrium partial pressures are \(\mathrm{PCL}_{2}=0.128 \mathrm{~atm}\), \(\mathrm{P}_{\mathrm{CO}}=0.116 \mathrm{~atm}\), and \(\mathrm{P}_{(\mathrm{COCl}) 2}=0.334 \mathrm{~atm} .\) Determine the equilibrium constant \(\mathrm{K}_{\mathrm{p}}\) for the dissociation of phosgene and the degree of dissociation at \(395 \mathrm{~K}\) under a pressure of \(1 \mathrm{~atm} .\)

Short answer

The equilibrium constant \(K_p \approx 22.4\), and the degree of dissociation of phosgene at 395 K under a pressure of 1 atm is approximately 0.042.

Step-by-step solution

Write the equilibrium expression

For the reaction, \[CO(g) + Cl_2(g) \rightleftarrows COCl_2(g)\] The equilibrium constant expression can be written as: \[K_p = \frac{P_{COCl_2}}{P_{CO} \cdot P_{Cl2}}\]

Use the given equilibrium partial pressures to find Kp

The given equilibrium partial pressures are: \[P_{CO} = 0.116 \, atm\] \[P_{Cl_2} = 0.128 \, atm\] \[P_{COCl_2} = 0.334 \, atm\] Substitute these values into the Kp expression: \[K_p = \frac{0.334}{(0.116) \cdot (0.128)}\] Now simplify to find K_p: \[K_p \approx 22.4\]

Determine the degree of dissociation using the pressure of 1 atm

We know that the partial pressure of phosgene at equilibrium at 1 atm is given as: \[P_{COCl_2} = 1 - \alpha\] where α is the degree of dissociation of phosgene. From the balanced equation, the partial pressure of CO is equal to the degree of dissociation α: \[P_{CO} = \alpha\] and the partial pressure of Cl_2 is also equal to the degree of dissociation α: \[P_{Cl_2} = \alpha\] Substitute these expressions into the equilibrium constant expression: \[K_p = \frac{1 - \alpha}{\alpha^2}\] Now, substitute the value of K_p and solve for α: \[22.4 = \frac{1 - \alpha}{\alpha^2}\] To find the degree of dissociation α, rearrange and solve the quadratic equation: \[\alpha^2 + 22.4\alpha - 1 = 0\] Solving this quadratic equation using the quadratic formula, we get two possible values for α: \[\alpha \approx 0.042 \,\, \text{or} \,\, \alpha \approx -23.6\] Since the degree of dissociation has to be between 0 and 1, our answer is: \[\alpha \approx 0.042\] So, the equilibrium constant \(K_p \approx 22.4\), and the degree of dissociation of phosgene at 395 K under a pressure of 1 atm is approximately 0.042.

Key concepts

Chemical Equilibrium

When we talk about chemical equilibrium, we refer to the condition within a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. As a result, the concentrations, or in the case of gases, the partial pressures of the reactants and products remain constant over time.

For the reaction between chlorine and carbon monoxide to form phosgene, we can apply this principle. At equilibrium, while the reactions continue to occur, there is no net change in the system's composition. This state is dynamic, meaning that molecules are constantly reacting, but the overall quantities of reactants and products do not change.

Understanding equilibrium is crucial for predicting how the system will respond to changes, such as variations in temperature or pressure. This concept is quantitatively expressed by the equilibrium constant, represented by K or K_p when dealing with gases and partial pressures.

Partial Pressure

The term partial pressure is used to describe the pressure that a gas would exert if it alone occupied the whole volume of the mixture at the same temperature. It's an essential concept when we are dealing with gases in chemistry, particularly when gases are involved in reactions.

In the context of our problem with chlorine, carbon monoxide, and phosgene, the equilibrium condition is described using the partial pressures of these gases. To calculate the equilibrium constant (K_p), we use the partial pressures of the products and reactants at equilibrium. The higher the partial pressure of a product, the more of that product is present at equilibrium.

Degree of Dissociation

The degree of dissociation, symbolized by α (alpha), is a measure of the extent to which a compound separates into its constituent elements or simpler compounds. It is a dimensionless number, usually expressed as a fraction or a percentage.

In the given problem, α represents the fraction of phosgene that dissociates into chlorine and carbon monoxide at a certain pressure and temperature. The degree of dissociation is crucial for understanding the composition of the equilibrium mixture and can also be used to calculate partial pressures when the total pressure is known. By knowing α, we have insight into the efficiency of the reaction and how much product can be expected from a given amount of reactant.

Dynamics of Reversible Reactions

The dynamics of reversible reactions, is centered around the concept that reactions can proceed in both directions, forward and reverse. This means that as reactants are converted to products, products can also revert back into reactants.

In our example of the formation of phosgene, the reaction can reach a point where the rate of the forward reaction (forming COCl₂) is equal to the rate of the backward reaction (breaking down COCl₂ into CO and Cl₂). This is when chemical equilibrium is achieved. The concept emphasizes that equilibrium does not mean the reactions have stopped; rather, there is a continuous and dynamic exchange between reactants and products, with their concentrations remaining steady in the long term.