Question

When \(1.645 \mathrm{~g}\) of white phosphorus is dissolved in \(75.5 \mathrm{~g}\) of \(\mathrm{CS}_{2}\), the solution boils at \(46.709^{\circ} \mathrm{C}\), whereas pure \(\mathrm{CS}_{2}\) boils at \(46.300^{\circ} \mathrm{C}\). The molal boiling-point elevation constant for \(\mathrm{CS}_{2}\) is \(2.34^{\circ} \mathrm{C} / \mathrm{m}\). Calculate the molar mass of white phosphorus, and give the molecular formula.

Short answer

The molar mass of white phosphorus is about 124 g/mol, and its molecular formula is \( \mathrm{P}_4 \).

Step-by-step solution

Calculate Boiling Point Elevation

First, find the difference in boiling points between the solution and the pure solvent. The boiling point elevation is given by: \[ \Delta T_b = T_{solution} - T_{pure} = 46.709^{\circ} \mathrm{C} - 46.300^{\circ} \mathrm{C} = 0.409^{\circ} \mathrm{C} \]

Apply the Boiling Point Elevation Formula

Using the formula for boiling point elevation, \( \Delta T_b = i \cdot K_b \cdot m \), where \( i \) is the van't Hoff factor (assumed to be 1 for non-electrolytes), \( K_b = 2.34^{\circ} \mathrm{C} / \mathrm{m} \), solve for \( m \), the molality:\[ m = \frac{\Delta T_b}{K_b} = \frac{0.409^{\circ} \mathrm{C}}{2.34^{\circ} \mathrm{C} / \mathrm{m}} = 0.1748 \, \mathrm{mol/kg} \]

Calculate Moles of White Phosphorus

Molality \( m \) is defined as moles of solute per kilogram of solvent. Multiply the molality by the mass of the solvent in kilograms to find the moles of white phosphorus:\[ \text{Moles of white phosphorus} = m \cdot \text{mass of solvent (kg)} = 0.1748 \, \mathrm{mol/kg} \times 0.0755 \mathrm{~kg} = 0.0132 \, \mathrm{mol} \]

Determine Molar Mass of White Phosphorus

The molar mass is found by dividing the mass of the solute by the number of moles:\[ \text{Molar Mass} = \frac{\text{mass of solute}}{\text{moles of solute}} = \frac{1.645 \mathrm{~g}}{0.0132 \, \mathrm{mol}} = 124.62 \, \mathrm{g/mol} \]

Identify the Molecular Formula

The calculated molar mass (approximately 124 g/mol) suggests the molecular formula for white phosphorus is \( \mathrm{P}_4 \), since the atomic mass of phosphorus is about 31 g/mol and \( 31 \times 4 = 124 \).

Key concepts

Molality

Molality is a measure of the concentration of a solute in a solution, expressed as moles of solute per kilogram of solvent. It is particularly useful in calculations involving colligative properties, like boiling point elevation, because it doesn't change with temperature.

In the context of this exercise, we use molality to find how much white phosphorus affects the boiling point of carbon disulfide (\(\mathrm{CS}_{2}\)). The boiling point elevation depends on the amount of solute added, not its identity, making molality the perfect concentration measure in this case.

• To find molality (\(m\)), we start with the boiling point elevation (\(\Delta T_b\)).
• The relationship is given by: \( \Delta T_b = K_b \cdot m\).
• Here, \(K_b\) is the boiling point elevation constant, which has units of degrees Celsius per molality (\(^{\circ} \text{C/m}\)).

By rearranging this formula, you can solve for molality, providing vital information about the solution's concentration.

Molar Mass Calculation

Calculating molar mass involves determining how much mass one mole of a substance has. It's a key concept in chemistry because it bridges the gap between the microscopic world of atoms and the macroscopic world we can measure.

To find the molar mass of white phosphorus in the example:

• Start by measuring the mass of phosphorus used, which is provided as \(1.645 \mathrm{~g}\).
• Next, calculate the moles of phosphorus using the molality and the mass of the \(\mathrm{CS}_{2}\) solvent.
• The formula to use is: \( \text{Moles of solute} = m \cdot \text{mass of solvent (in kg)} \).
• Use these moles to find molar mass, with: \( \text{Molar Mass} = \frac{\text{mass of solute}}{\text{moles of solute}} \).

Understanding molar mass allows chemists to identify unknown compounds, predict how they behave, and quantify their reactions.

Molecular Formula Determination

Once the molar mass of a compound is known, determining its molecular formula becomes easier. The molecular formula tells us the exact number of each type of atom in a molecule.

After calculating the molar mass of white phosphorus as about \(124 \mathrm{~g/mol}\), we can proceed as follows:

• Note that the atomic mass of phosphorus is approximately \(31 \mathrm{~g/mol}\).
• Divide the calculated molar mass by the atomic mass: \( \frac{124}{31} \approx 4 \).
• This result indicates that the molecular formula for white phosphorus is \(\mathrm{P}_4\).

The ability to determine molecular formulas is essential for chemists to understand the makeup of substances and is the basis for further in-depth analysis and applications in chemistry.