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The degree of dissociation is a concept used in chemistry to describe how much of a compound breaks down into its components. It's crucial for understanding how substances interact in a reaction, especially when dealing with reversible reactions, as in the case of \( \mathrm{PCl}_5 \rightleftharpoons \mathrm{PCl}_3 + \mathrm{Cl}_2 \).
In this exercise, the degree of dissociation (\( \alpha \)) is given as 0.4, meaning that 40% of the original \( \mathrm{PCl}_5 \) has dissociated into \( \mathrm{PCl}_3 \) and \( \mathrm{Cl}_2 \).
This can be visualized as follows: if you start with 1 mole of \( \mathrm{PCl}_5 \), at equilibrium, 0.4 moles dissociate. This process affects how we calculate quantities at equilibrium.

• Initial moles of \( \mathrm{PCl}_5 \): 1 mole
• Equilibrium moles of \( \mathrm{PCl}_5 \): \(1 - 0.4 = 0.6 \text{ moles}\)
• Produced moles of \( \mathrm{PCl}_3 \): 0.4 moles
• Produced moles of \( \mathrm{Cl}_2 \): 0.4 moles

Overall, understanding the degree of dissociation helps predict the concentration of different species in a chemical equilibrium, which is essential for further calculations like density or reaction dynamics.

The Ideal Gas Law is a fundamental principle in chemistry and physics used to describe the behavior of gases. It is expressed as \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for number of moles, \( R \) for the ideal gas constant, and \( T \) for temperature in Kelvin.
In this problem, the Ideal Gas Law helps in calculating the volume of the gas mixture at equilibrium. We know the following:

• Pressure (\( P \)) = 1.0 atm
• Total moles at equilibrium (\( n \)) = 1.4 moles
• Temperature (\( T \)) = 400 K
• Gas constant (\( R \)) = 0.0821 L·atm/mol·K

Using the Ideal Gas Law, we find: \[ V = \frac{nRT}{P} = \frac{1.4 \times 0.0821 \times 400}{1} = 45.96 \text{ L} \]So, all components combined occupy 45.96 liters in an ideal condition. By using the Ideal Gas Law, we assume no interactions between gas particles, making calculations straightforward when the system conditions are close to ideal.

Molar mass is defined as the mass of one mole of a substance (usually in g/mol). It is a critical figure in stoichiometry, allowing you to convert between moles and grams.
In the given reaction, we calculate the molar masses for each participating gas:

• \( \mathrm{PCl}_5 \): \[ M(\mathrm{PCl}_5) = 31.0 + 5 \times 35.5 = 208.5 \text{ g/mol} \]
• \( \mathrm{PCl}_3 \): \[ M(\mathrm{PCl}_3) = 31.0 + 3 \times 35.5 = 137.5 \text{ g/mol} \]
• \( \mathrm{Cl}_2 \): \[ M(\mathrm{Cl}_2) = 2 \times 35.5 = 71.0 \text{ g/mol} \]

For a practical example, at equilibrium, we determine the mass contributions from each component. This involves multiplying the number of moles by their respective molar masses:

• \( \mathrm{PCl}_5 \): \[ 0.6 \times 208.5 = 125.1 \text{ g} \]
• \( \mathrm{PCl}_3 \): \[ 0.4 \times 137.5 = 55.0 \text{ g} \]
• \( \mathrm{Cl}_2 \): \[ 0.4 \times 71.0 = 28.4 \text{ g} \]

Summing these gives a total mass of 208.5 grams for the mixture, which can then be used to calculate the density or other properties. Understanding molar mass is foundational to translating theoretical chemistry into practical results.