Question

The equilibrium constant \(K_{\mathrm{c}}\) for the decomposition of phosgene, \(\mathrm{COCl}_{2}\), is \(4.63 \times 10^{-3}\) at \(527^{\circ} \mathrm{C}\) : $$ \mathrm{COCl}_{2}(g) \rightleftharpoons \mathrm{CO}(g)+\mathrm{Cl}_{2}(g) $$ Calculate the equilibrium partial pressure of all the components, starting with pure phosgene at 0.760 atm.

Short answer

The equilibrium partial pressures of the components in the reaction are determined by solving a quadratic equation expressing the equilibrium constant of the reaction in terms of changes in the initial pressures of the components. After solving for \(x\), the equilibrium pressures of \(\mathrm{COCl}_{2}\), \(\mathrm{CO}\), and \(\mathrm{Cl}_{2}\) can be determined.

Step-by-step solution

Identify The Initial Conditions

Initially, there is only phosgene, \(\mathrm{COCl}_{2}\), with a partial pressure of 0.760 atm. The initial partial pressures of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) are both 0 since the decomposition has not yet started.

Establish The Chemical Equilibrium Expression

The given equilibrium constant \(\(K_{c}\)\) corresponds to the expression \(\(K_{c}\) = \(\frac{[CO][Cl_{2}]}{[COCl_{2}]}\). Given that the initial concentration of \(\mathrm{COCl}_{2}\) decreases by \(x\) at equilibrium, the concentrations of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\) increase by \(x\). Hence, the equilibrium concentrations can be expressed as: \(\mathrm{[COCl}_{2}]\) = 0.760 - \(x\), \(\mathrm{[CO]}\) = \(x\), \(\mathrm{[Cl}_{2}]\) = \(x\).

Solve For \(x\)

Plug these values into the expression for \(\(K_{c}\)\) to get: \(\frac{x \cdot x}{0.760 - x} = 4.63 \times 10^{-3}\). Solve this quadratic equation for \(x\). The positive root of this quadratic equation gives the equilibrium concentrations of \(\mathrm{CO}\) and \(\mathrm{Cl}_{2}\), and thus, also the partial pressures since these are proportional to the concentrations.

Calculate The Equilibrium Partial Pressures

The equilibrium partial pressures can be calculated by using the relation: \(\(P = n/V \times RT\)\), where \(\(P\)\) is the pressure, \(n\) is the number of moles (concentration after the reaction has reached equilibrium), \(\(V\)\) is the volume, and \(\(R = 0.0821 atm⋅L⋅mol⁻¹⋅K⁻¹\) and \(T\) is the absolute temperature. As the volume and temperature are constant, the pressures at equilibrium will be proportional to the concentrations in moles per liter. Therefore the pressures of \(\mathrm{COCl}_{2}\), \(\mathrm{CO}\), and \(\mathrm{Cl}_{2}\) will be 0.760 atm - \(x\) atm, \(x\) atm, and \(x\) atm respectively.

Key concepts

Equilibrium Constant

The equilibrium constant, often written as \(K_c\) for reactions involving concentrations, is a vital concept in understanding chemical equilibria. It gives us an idea of how far a reaction will proceed at a given temperature. In the context of the phosgene decomposition, the equilibrium constant \(K_c = 4.63 \times 10^{-3}\) at 527°C tells us that at equilibrium, the concentration of products is much lower than the concentration of reactants.
This small value suggests that the reaction heavily favors the formation of phosgene, \(\mathrm{COCl}_2\).

• The expression for \(K_c\) is derived from the balanced chemical equation for the reaction.
• It includes only the concentrations of gases involved in the equilibrium state.
• The reaction quotient \(Q\) changes as the reaction progresses towards equilibrium, and when \(Q = K_c\), the system is at equilibrium.

Understanding \(K_c\) allows chemists to predict the concentrations of different species at equilibrium, which is crucial in industrial and laboratory settings.

Partial Pressure

Partial pressure is a concept that helps us understand how different gases behave when they are part of a mixture. Each gas in a mixture exerts pressure independently, known as its partial pressure. For the decomposition of phosgene, starting with pure phosgene at 0.760 atm means it initially contributes its entire partial pressure.
As the decomposition proceeds, equilibrium is established, and partial pressures adjust to reflect the new balance.

• At equilibrium, the partial pressure of \(\mathrm{COCl}_2\) drops by \(x\) atm, while \(\mathrm{CO}\) and \(\mathrm{Cl}_2\) each gain \(x\) atm.
• The sum of these partial pressures remains constant, due to the proportional relationship between pressure and concentration.
• By using the equilibrium constant expression and solving for \(x\), we determine the partial pressures at equilibrium.

Partial pressures are important for calculations in gas reactions, as they directly relate to the number of moles and conditions like volume and temperature.

Phosgene Decomposition

Phosgene, \(\mathrm{COCl}_{2}\), is a toxic gas that decomposes into carbon monoxide \(\mathrm{CO}\) and chlorine gas \(\mathrm{Cl}_2\), both of which are also harmful. The decomposition reaction can be summarized as:\[\mathrm{COCl}_{2}(g) \rightleftharpoons \mathrm{CO}(g) + \mathrm{Cl}_{2}(g)\]This reaction moves towards equilibrium whereby some of the phosgene breaks down into its components.

• Initially, there is only phosgene, so as it decomposes, the products start to form.
• With time, the reaction slows and reaches a point where the rate of decomposition equals the rate of recombination, called equilibrium.
• Calculating the partial pressures of all components at this state is crucial to determine how much phosgene remains and how much \(\mathrm{CO}\) and \(\mathrm{Cl}_2\) are formed.

This understanding is important for handling phosgene safely, especially in industrial scenarios where control of such reactions is necessary to ensure safety.