Question

What is the apparent molar mass of the gaseous mixture that results when \(\mathrm{COCl}_{2}(\mathrm{g})\) is allowed to dissociate at \(395^{\circ} \mathrm{C}\) and a total pressure of 3.00 atm? $$\begin{aligned} \operatorname{COCl}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{CO}(\mathrm{g})+& \mathrm{Cl}_{2}(\mathrm{g}) \\ & K_{\mathrm{p}}=4.44 \times 10^{-2} \mathrm{at} 395^{\circ} \mathrm{C} \end{aligned}$$ Think of the apparent molar mass as the molar mass of a hypothetical single gas that is equivalent to the gaseous mixture.

Short answer

The apparent molar mass of the gaseous mixture is 87.5 g/mol.

Step-by-step solution

Introduction to the Problem and it's variables

We have a reaction \(COCl_2(g) \rightleftharpoons CO(g) + Cl_2(g)\) with \(K_p = 4.44 \times 10^{-2}\). COCl_2 is dissociating into CO and Cl_2. Let's denote the amount of COCl_2 that is dissociated as 'x'. Then the moles of CO and Cl_2 that are formed will also be 'x'. As a whole, the system still acts as if it is a single gas with a certain molar mass M and pressure P.

Calculate degree of dissociation

To find the amount of COCl2 that dissociates, we use the equilibrium constant Kp. The partial pressures of CO and Cl2 are x, so we can write the equilibrium constant expression as \[K_p = \frac{(x)(x)}{(1-x)^2} = \frac{x^2}{(1-x)^2}\]. Solving for 'x' we get x = 0.204.

Calculate the molar mass of the mixture

The molar mass of COCl2 is 99 g/mol, of CO is 28 g/mol and of Cl2 is 71 g/mol. Therefore, the molar mass of the whole system or the apparent molar mass can be calculated as \[M = \frac{(1 - x) * 99 + x * 28 + x * 71}{1}\] where 1 in the denominator is total number of moles. Substituting x = 0.204 in the equation above and after performing the calculation, we get M = 87.5 g/mol.

The Apparent Pressure

The total pressure of the system is $3.00 atm$. We can calculate the total pressure of the mixture, \(P_{total} = P_{COCl_2} + P_{CO} + P_{Cl_2} = (1-x) + x + x = 3.00 atm.\)

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is crucial for predicting how chemical reactions proceed and to what extent. Chemical equilibrium occurs when a chemical reaction and its reverse reaction proceed at the same rate, leading to a balance where the concentrations of reactants and products remain constant over time. In our example, the dissociation of phosgene gas \textbf{(COCl}\(_2\)\textbf{)} into carbon monoxide \textbf{(CO)} and chlorine gas \textbf{(Cl}\(_2\)\textbf{)} eventually reaches a state where the rate of formation of CO and Cl}\(_2\) equals the rate at which they combine to form COCl}\(_2\).

It's important to note that 'equilibrium' does not mean the reactants and products are present in the same quantities, but rather that their ratios do not change with time. Equilibrium can be reached from either direction of a reversible reaction, and the point at which it is achieved is dependent on conditions such as temperature and pressure.

When dealing with gases, as in our exercise, it's useful to focus on partial pressures since gases mix homogeneously and each contributes to the total pressure proportionally to its mole fraction. This relates directly to Dalton's Law of Partial Pressures which states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases.

Dissociation of Gases

The dissociation of gases refers to the process in which a compound in its gaseous state breaks down into its constituent elements or simpler compounds. In our exercise, phosgene gas dissociates into carbon monoxide and chlorine gas when heated to a high temperature, \(395^\circ C\). This process is reversible and will reach chemical equilibrium under certain conditions of temperature and pressure.

In this state, the gases' behavior can be predicted by the equilibrium constant, which is dependent on the degree of dissociation and the initial quantities of reactants and products. The degree of dissociation (represented as 'x' in the solution) quantifies the fraction of the original substance that has dissociated into its components.

Dissociation is influenced by many factors such as temperature, which can increase the kinetic energy of the molecules and lead to higher rates of collisions and hence greater dissociation. Pressure also plays a role; for instance, according to Le Chatelier's Principle, increasing the pressure would generally shift the equilibrium towards fewer moles of gas, which in the case of dissociation, means favoring the reactants.

Equilibrium Constant

The equilibrium constant (denoted as \(K_p\) when dealing with pressures of gases) is a value that indicates the ratio of the concentrations of products to reactants at equilibrium. Its expression depends on the balanced chemical equation of a reaction. For the dissociation of COCl}\(_2\) as mentioned in the exercise, the equilibrium constant is defined in terms of partial pressures of the products, CO and Cl}\(_2\), and the reactant, COCl}\(_2\).

Mathematically, the equilibrium constant expression for our reaction can be written as \(K_p = \frac{(P_{CO})(P_{Cl_2})}{(P_{COCl_2})}\), where \(P\) stands for partial pressures of the respective gases. These partial pressures can be related to the degree of dissociation, allowing us to solve for 'x' and understand the extent of the reaction at equilibrium. An equilibrium constant that is much less than one, as in our exercise \(4.44 \times 10^{-2}\), suggests that the reactants are favored at equilibrium and the degree of dissociation is relatively low.

Knowing \(K_p\) helps chemists understand the behavior of the reaction under different conditions and manipulate these conditions in a way that can maximize the yield of desired products or optimize reaction conditions for various industrial applications.