Question

A solution of \(1.50 \mathrm{~g}\) of solute dissolved in \(25.0 \mathrm{~mL}\) of \(\mathrm{H}_{2} \mathrm{O}\) at \(25^{\circ} \mathrm{C}\) has a boiling point of \(100.45^{\circ} \mathrm{C}\). (a) What is the molar mass of the solute if it is a nonvolatile nonelectrolyte and the solution behaves ideally \(\left(d\right.\) of \(\mathrm{H}_{2} \mathrm{O}\) at \(\left.25^{\circ} \mathrm{C}=0.997 \mathrm{~g} / \mathrm{mL}\right) ?\) (b) Conductivity measurements show that the solute is ionic with general formula \(\mathrm{AB}_{2}\) or \(\mathrm{A}_{2} \mathrm{~B}\). What is the molar mass if the solution behaves ideally? (c) Analysis indicates that the solute has an empirical formula of \(\mathrm{CaN}_{2} \mathrm{O}_{6}\). Explain the difference between the actual formula mass and that calculated from the boiling point elevation. (d) Find the van't Hoff factor ( \(i\) ) for this solution.

Short answer

Step 1 boiling calculation -> Determine molality, Step 2: calculate molar mass logic -> 68.0 g/mol . ionic adjustment calculation --> 164.10 g/mol correcting i.

Step-by-step solution

Determine the Molar Mass using Boiling Point Elevation

First, use the boiling point elevation formula: \[ \text{ΔT}_{b} = i \times K_{b} \times m \]where \( \text{ΔT}_{b} \) is the boiling point elevation (\(100.45^\text{∘}\text{C} - 100.00^\text{∘}\text{C} = 0.45^\text{∘}\text{C}\)), \( i \) is the Van't Hoff factor (for nonvolatile nonelectrolyte, \( i = 1 \)), \( K_{b} \) is the ebullioscopic constant for water (\(0.512 \text{∘}\text{C m}^{-1}\)), and \( m \) is the molality. Calculate molality by first finding the mass of water (\(25.0 \text{ mL} \times 0.997 \text{ g/mL} = 24.925 \text{ g}\)) and converting it to kg (\( 24.925 \text{ g} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 0.024925 \text{ kg} \)). Then solve for molality (\( m \)).

Calculate Molality

Calculate molality, \( m \): \[ m = \frac{1.50 \text{ g} \text{ of solute}}{M \text{ mol}} \times \frac{1}{0.024925 \text{ kg of solvent}} \]Rearrange the boiling point elevation formula to solve for molar mass (\( M \)):\[ \text{ΔT}_{b} = K_{b} \times m \rightarrow m = \frac{\text{ΔT}_{b}}{K_{b}} = \frac{0.45^\text{∘}\text{C}}{0.512^\text{∘}\text{C m}^{-1}} = 0.8789 \text{ m} \]\[ m = \frac{1.50}{M \times 0.024925} = 0.8789 \]Solve for \( M \):\[ 1.50 = 0.8789M \times 0.024925 \rightarrow M = \frac{1.50}{0.8789 \times 0.024925} = 68.0 \text{ g/mol} \]

Determine the Molar Mass for Ionic Compound

Given the ionic compound formula \( \text{AB}_2 \) or \( \text{A}_2\text{B} \), the Van't Hoff factor, \(i\), would not be 1. Using the previous molality, adjust the molar mass calculation for the ionic nature.First, determine Zeff (effective particles): \[ 0.8789 \text{m} \times i \rightarrow i = \frac{0.45}{0.512 \times 0.8789} = 0.995 \rightarrow i \text{ approx.} 1 \]Then solve for the new molar mass if it were an ion: \[ M_{\text{actual}} \rightarrow M_{\text{ion}} = 2iM \rightarrow 2(1 \text{ approx}.)(68.0) = 0.68.0 \text{ g/mol} \]

Analyze the Empirical Formula and Compare Molar Mass

Based on the empirical formula \( \text{CaN}_2 \text{O}_6 \), calculate its molar mass: \[ 40.08 \text{ (Ca)} + 2(14.01 \text{ (N)}) + 6(16.00 \text{ (O)}) = 40.08 + 28.02 + 96.00 = 164.10 \text{ g/mol} \].The actual calculated mass using boiling points is around 68.0 g/mol likely implying subunits or incorrect factor assumption.

Determine the Van't Hoff Factor

Finally, determine the van't Hoff factor, using installations: \[ i = 1 + 2, \text{ hence the assumption, correcting} \]

Key concepts

boiling point elevation

Boiling point elevation occurs when a solute is added to a solvent, causing the boiling point of the solution to be higher than that of the pure solvent. The formula used to calculate this phenomenon is given by: \[ \text{ΔT}_{b} = i \times K_{b} \times m \] Here:
- \( \text{ΔT}_{b} \) is the boiling point elevation.
- \( i \) is the Van't Hoff factor, which represents the number of particles the solute breaks into when dissolved.
- \( K_{b} \) is the ebullioscopic constant of the solvent (for water, \( K_{b} = 0.512^\text{∘}\text{C m}^{-1} \)).
- \( m \) is the molality of the solution.
Using these parameters, we can determine the extent to which the boiling point is elevated.

molar mass calculation

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). To determine the molar mass using boiling point elevation, we rearrange the boiling point elevation formula: \[ \text{ΔT}_{b} = i \times K_{b} \times m \] First, we calculate the molality (\( m \)) by using the given mass of the solute and the mass of the solvent in kg: \[ m = \frac{\text{mass of solute}}{\text{molar mass} \times \text{mass of solvent in kg}} \] Then, we rearrange to solve for the molar mass: \[ \text{molar mass} = \frac{\text{mass of solute}}{m \times \text{mass of solvent in kg}} \] This formula allows us to find the molar mass of an unknown solute when we know the change in boiling point, the Van't Hoff factor, the ebullioscopic constant, and the mass of both solute and solvent.

Van't Hoff factor

The Van't Hoff factor (\( i \)) indicates how many particles a solute forms when dissolved in a solvent. For non-electrolytes, \( i \) is typically 1 because they do not dissociate in solution. For electrolytes, the factor is determined by how many ions the compound dissociates into. For instance: - NaCl dissociates into Na⁺ and Cl⁻, so \( i = 2 \) - \( \text{CaCl}_2 \) dissociates into Ca²⁺ and 2 Cl⁻, so \( i = 3 \) The Van't Hoff factor is used in colligative property calculations like boiling point elevation and freezing point depression to account for the number of particles in the solution.

nonvolatile nonelectrolyte

A nonvolatile nonelectrolyte is a substance that does not evaporate easily and does not dissociate into ions in solution. Examples include sugar and urea. In the context of boiling point elevation, these substances generally have a Van't Hoff factor (\( i \)) of 1 because they do not break up into multiple particles upon dissolution.
Since they remain as whole molecules in solution, their colligative effects are predictable and straightforward to calculate.

molality

Molality \( (m) \) is a measure of the concentration of a solute in a solvent, expressed as moles of solute per kilogram of solvent: \[ m = \frac{\text{moles of solute}}{\text{kg of solvent}} \] It is useful in boiling point elevation and freezing point depression calculations because it remains unaffected by temperature changes. To find the molality, we first need to calculate the moles of solute using its mass and molar mass, and then divide by the mass of the solvent in kilograms.

empirical formula

The empirical formula of a compound shows the simplest whole-number ratio of its elements. To determine the empirical formula: - Find the mass of each element in the compound.
- Convert these masses to moles by using the atomic masses of the elements.
- Divide all the moles by the smallest number of moles obtained.

If necessary, multiply these ratios to get whole numbers. The empirical formula is a crucial first step in determining the compound's molecular formula, which gives the actual number of atoms of each element in a molecule.

ionic compounds

Ionic compounds are composed of ions held together by electrostatic forces known as ionic bonds. They typically form between metals and nonmetals, such as NaCl (sodium chloride) or \( \text{CaCl}_2 \) (calcium chloride).
What makes ionic compounds significant in colligative properties is their dissociation into multiple particles when dissolved in water. This dissociation effectively increases the number of dissolved particles, affecting properties like boiling point elevation and freezing point depression. The degree of dissociation is also reflected in the Van't Hoff factor (\( i \)), which is used in determining the colligative properties for solutions containing ionic compounds.