Problem 8 - Chemistry in 30 days

Question

The vapour density of \(\mathrm{PCl}_{5}\) is \(104.16\) but when heated to \(230^{\circ}\) vapour density is reduced to 62. The degree of dissociation of \(\mathrm{PCl}_{5}\) at this temperature will be (a) \(6.8 \%\) (b) \(68 \%\) (c) \(46 \%\) (d) \(64 \%\)

Step-by-Step Solution

Answer

None of the given options match exactly, as the degree of dissociation is approximately 40.49%.

Step 1: Understand the Concept

Vapor density is defined as the mass of a certain volume of a gas compared to the mass of an equal volume of hydrogen. It is related to molar mass by the formula: \[\text{Vapor Density} = \frac{\text{Molar Mass}}{2}.\] Understanding this will help in calculating the molar mass and finding the degree of dissociation.

Step 2: Calculate Molar Mass from Initial Vapor Density

Given the initial vapor density of \(PCl_5\) is 104.16, we can find the molar mass: \[\text{Molar Mass} = 2 \times 104.16 = 208.32\, \text{g/mol}.\]

Step 3: Set Up the Dissociation Equation

When heated, \(PCl_5\) dissociates into \(PCl_3\) and \(Cl_2\): \[PCl_5 \rightleftharpoons PCl_3 + Cl_2.\] This affects the vapor density, decreasing it.

Step 4: Determine New Molar Mass Using Reduced Vapor Density

The vapor density changes to 62 at \(230^{\circ}\). Calculate the new molar mass: \[\text{New Molar Mass} = 2 \times 62 = 124\, \text{g/mol}.\]

Step 5: Calculate the Degree of Dissociation

The degree of dissociation, \(\alpha\), can be found using the formula: \[\alpha = \frac{\text{Initial Molar Mass} - \text{New Molar Mass}}{\text{New Molar Mass}}.\]Substituting the values, \[\alpha = \frac{208.32 - 124}{208.32} = \frac{84.32}{208.32} \approx 0.4049.\]Convert this to a percentage: \[\alpha \approx 40.49\%\].

Key Concepts

Molar Mass

Molar mass is a crucial concept in chemistry, as it represents the mass of one mole of a substance. It allows you to quantify the amount of a substance based on its mass, linking the microscopic world of atoms and molecules to measurable quantities. To find molar mass, add up the atomic masses of all atoms in a molecule. For instance, to calculate the molar mass of \( \mathrm{PCl}_5 \), sum the atomic masses of phosphorus (\( \approx 31 \, \text{g/mol} \)) and chlorine (\( \approx 35.5 \, \text{g/mol} \), and there are five chlorine atoms) to get \( 208.32 \, \text{g/mol} \). This value was used to understand the initial vapor density, noting how closely related vapor density and molar mass are.

Degree of Dissociation

The degree of dissociation is a measure of the extent to which a compound splits into its constituents in a chemical reaction. It helps evaluate dynamic processes in reactions, particularly where complex molecules break into simpler ones. Lowering vapor density, as a result, indicates increased dissociation in gases like \( \mathrm{PCl}_5 \). When \( \mathrm{PCl}_5 \) dissociates into \( \mathrm{PCl}_3 \) and \( \mathrm{Cl}_2 \), the vapor density decreases, showing a change in the molar mass observable as altered vapor density. Calculating degree of dissociation \( \alpha \) involves comparing initial and new molar mass:

• Initial Molar Mass = \( 208.32 \, \text{g/mol} \)
• New Molar Mass = \( 124 \, \text{g/mol} \)

The decrease provides insight into the extent of reaction dissociation, where a calculated \( \alpha \) of 40.49\% indicates the dissociation proportion at the given temperature.

Chemical Equilibrium

Chemical equilibrium occurs when a reaction proceeds at such a rate that the concentrations of reactants and products remain constant, though not necessarily equal. In the dissociation of \( \mathrm{PCl}_5 \), equilibrium is achieved as the forward reaction of \( \mathrm{PCl}_5 \) forming \( \mathrm{PCl}_3 \) and \( \mathrm{Cl}_2 \) matches the reverse reaction. Even as heat increases the dissociation, equilibrium balances these components. This kind of chemical equilibrium is a dynamic process, where reactions continue to occur, but with no net change in concentrations at equilibrium. Understanding these principles is crucial for predicting reaction direction and amounts at different conditions.

Gaseous State Calculations

Gaseous state calculations involve using relationships between measurable gas properties to understand behavior under different conditions. In the case of the \( \mathrm{PCl}_5 \) dissociation, calculations hinge on vapor density, providing insights into changes due to temperature-induced reactions. Molar mass is a primary factor in determining vapor density, and using equations like \[ \text{Vapor Density} = \frac{\text{Molar Mass}}{2} \], one can predict these changes. The steps include calculating initial and new vapor densities, deducing molar mass, and applying these to find the degree of dissociation \( \alpha \). These calculations help clarify how gaseous reactions progress with conditions, informing about reactive behaviors and properties, and they highlight the practical impact of fundamental chemical principles on tangible gas properties.