Question

An aqueous solution containing \(288 \mathrm{~g}\) of a nonelectrolyte compound having the stoichiometric composition \(\mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}} \mathrm{O}_{\mathrm{n}}\) in \(90 \mathrm{~g}\) water boils at \(101.24^{\circ} \mathrm{C}\) and 1 atm pressure. \(K_{b}=0.512 \mathrm{~K} \mathrm{~mol}^{-1} \mathrm{~kg}\). Which of the following statements are correct? (i) molecular mass of \(\mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}} \mathrm{O}_{\mathrm{n}}=1321.2 \mathrm{~g}\) (ii) molecular formula of \(\mathrm{C}_{\mathrm{a}} \mathrm{H}_{2 \mathrm{n}} \mathrm{O}_{\mathrm{n}}=\mathrm{C}_{44} \mathrm{H}_{88} \mathrm{O}_{44}\) (iii) molecular mass of \(\mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}} \mathrm{O}_{\mathrm{a}}=104.2 \mathrm{~g}\) (iv) empirical formula of \(\mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}} \mathrm{O}_{\mathrm{n}}\) is \(\mathrm{CH}_{2} \mathrm{O}\) a. (i) \& (iv) b. (ii) \& (iv) c. (i) \& (ii) d. (iv) \& (ii)

Short answer

Option (a): (i) & (iv) are correct.

Step-by-step solution

Understand the Boiling Point Elevation

Boiling point elevation is a colligative property that depends on the concentration of solute particles in a solution. It can be calculated using the formula: \[ \Delta T_b = i K_b m \] where \( \Delta T_b \) is the boiling point elevation, \( i \) is the van't Hoff factor (equals 1 for nonelectrolytes), \( K_b \) is the ebullioscopic constant, and \( m \) is the molality of the solution.

Calculate the Boiling Point Elevation

Given that the boiling point of the solution is \( 101.24^{\circ}C \), and the normal boiling point of water is \( 100^{\circ}C \), the boiling point elevation is: \[ \Delta T_b = 101.24 - 100 = 1.24^{\circ}C \]

Find the Molality of the Solution

Using the boiling point elevation formula, we have \( \Delta T_b = K_b \times m \). Substituting the values gives us: \[ 1.24 = 0.512 \times m \] Solving for \( m \) (molality), we find: \[ m = \frac{1.24}{0.512} \approx 2.42 \text{ mol/kg} \]

Calculate the Moles of Solute

Since \( m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \) and the mass of water (solvent) is 90 g or 0.090 kg, we can calculate the moles of solute: \[ \text{moles of solute} = m \times \text{mass of solvent in kg} = 2.42 \times 0.090 \approx 0.2178 \text{ moles} \]

Determine the Molar Mass of the Solute

The molar mass can be calculated as: \[ \text{Molar Mass} = \frac{\text{mass of solute}}{\text{moles of solute}} = \frac{288}{0.2178} \approx 1322 \text{ g/mol} \] This suggests option (i) is essentially correct, considering rounding differences.

Assess the Validity of Structures

For the compound \( \text{C}_n\text{H}_{2n}\text{O}_n \), an empirical formula could indeed be \( \text{CH}_2\text{O} \) since each has the same ratio, suggesting statement (iv) is correct as it matches the empirical formula. Therefore, option (a) holds true given the elements in the formula.

Conclusion on Molecular and Empirical Formulas

The empirical formula \( \text{CH}_2\text{O} \), matching statement (iv), aligns with the given compound composition. Moreover, the calculated molar mass approximately validates statement (i). Thus, options (i) and (iv) are indeed the correct statements.

Key concepts

Boiling Point Elevation

Boiling point elevation is a concept under colligative properties, where the addition of a solute to a solvent results in an increase in the boiling point of that solvent. This phenomenon occurs because the solute particles disrupt the solvent's ability to enter the gas phase, thus requiring more energy (heat) for the solvent molecules to overcome this new level of instability brought about by the presence of solute particles.
To determine the boiling point elevation quantitatively, we use the formula:\[ \Delta T_b = i K_b m \]- \(\Delta T_b\) is the boiling point elevation (change in boiling point).- \(i\) is the van't Hoff factor, which is 1 for nonelectrolyte solutions since they do not ionize in solution.- \(K_b\) is the ebullioscopic constant, which is specific to each solvent.- \(m\) is the molality of the solution, measuring how many moles of solute are present per kilogram of solvent.
This principle is applied to understand how solutions behave differently than pure solvents and to perform molecular mass determinations.

Molecular Mass Determination

Determining the molecular mass of a compound is crucial in chemistry, providing valuable information about the substance's identity and chemical properties. In the context of colligative properties, such as boiling point elevation, the molecular mass can be calculated using experimental data on the change in boiling point.
By rearranging the boiling point elevation formula to find molality, we can derive the number of moles of solute from the fatty solution. Given:\[ m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \]When we know the mass of the solute and the calculated moles of solute, we can find the molar mass of the compound:\[ \text{Molar Mass} = \frac{\text{mass of solute}}{\text{moles of solute}} \]
Using colligative properties allows chemists to determine the molecular mass without requiring the substance's pure state, which may not always be feasible. This method is beneficial for analyzing substances in solution form.

Empirical and Molecular Formulas

Empirical formulas represent the simplest whole-number ratio of elements in a compound. They provide an essential basis for understanding molecular structures and reactions in chemistry.
For example, in the case of a compound with the stoichiometric composition \( \text{C}_n\text{H}_{2n}\text{O}_n \), where the empirical formula \( \text{CH}_2\text{O} \) indicates the simplest ratio, it connects to the larger molecular formula, which signifies the actual number of atoms in a molecule but in a multiplied form.To identify the molecular formula from the empirical formula, additional information such as the molecular mass is necessary. The structure's scaling from empirical to molecular is determined by comparing the known empirical formula mass to the calculated molecular mass of the compound.
Understanding both empirical and molecular formulas is pivotal for interpreting chemical reactions and for further computations in chemistry.

Nonelectrolyte Solutions

Nonelectrolyte solutions are mixtures where the solute does not dissociate into ions when dissolved in a solvent. This means that the solute exists in a molecular form, not contributing to electrical conductivity in the solution.
The characteristic of not producing ions is crucial because nonelectrolyte solutions influence colligative properties differently than electrolyte solutions. For example, in the boiling point elevation equation, the van't Hoff factor \(i\) is 1 for nonelectrolyte solutions, as there is no dissociation to factor in.This behavior is instrumental in applications such as calculating molecular masses using properties like boiling point elevation, where accurate predictions depend on knowing that there are no additional particles introduced due to dissociation. Understanding the nature of nonelectrolyte solutions helps in designing experiments and interpreting results involving solution chemistry.