Problem 44 - Chemistry and Chemical Reactivity

Question

The organic compound called aluminon is used as a reagent to test for the presence of the aluminum ion in aqueous solution. A solution of \(2.50 \mathrm{g}\) of aluminon in \(50.0 \mathrm{g}\) of water freezes at \(-0.197^{\circ} \mathrm{C} .\) What is the molar mass of aluminon?

Step-by-Step Solution

Answer

The molar mass of aluminon is approximately 471 g/mol.

Step 1: Understand the Freezing Point Depression Formula

The freezing point depression can be calculated using the formula: \( \Delta T_f = i \cdot K_f \cdot m \), where \( \Delta T_f \) is the decrease in freezing point, \( i \) is the van't Hoff factor (which is 1 for non-electrolytes like aluminon), \( K_f \) is the freezing point depression constant for water (1.86 °C·kg/mol), and \( m \) is the molality of the solution.

Step 2: Calculate the Freezing Point Depression

Given that the freezing point of pure water is 0 °C and the solution freezes at -0.197 °C, you can calculate \( \Delta T_f = 0 - (-0.197) = 0.197 \; ^\circ \mathrm{C}.\)

Step 3: Rearrange Formula to Find Molality

Rearrange the freezing point depression formula to solve for molality: \( m = \frac{\Delta T_f}{K_f} = \frac{0.197 \; ^\circ \mathrm{C}}{1.86 \; ^\circ \mathrm{C}\cdot \mathrm{kg/mol}} \approx 0.106 \; \mathrm{mol/kg}.\)

Step 4: Determine moles of Solute Present

Molality \( m \) is defined as moles of solute per kilogram of solvent. Therefore, \( 0.106 \; \mathrm{mol/kg} \times 0.0500 \; \mathrm{kg} = 0.0053 \; \mathrm{mol} \) of the solute (aluminon).

Step 5: Calculate Molar Mass of Aluminon

The molar mass \( M \) is the mass of the solute divided by the number of moles of the solute: \( M = \frac{2.50 \; \mathrm{g}}{0.0053 \; \mathrm{mol}} \approx 471 \; \mathrm{g/mol}.\)

Key Concepts

Molar Mass Calculation

Calculating the molar mass of a compound means finding out how much one mole of that substance weighs in grams. It is crucial for understanding various chemical reactions and processes in chemistry. For instance, in the problem involving aluminon, the molar mass is determined as part of identifying the substance based on its freezing point depression in a solvent.
To do this, first, calculate the number of moles (\(n\)) of the compound using the formula:

• \(m = \frac{n}{mass\, in\, kg}\)where \(m\) is the molality and \(mass\, in\, kg\) is the mass of the solvent.

After finding the number of moles, use the equation:

• \(M = \frac{mass\, in\, g}{n}\)

where \(M\) is the molar mass, \(mass\, in\, g\) is the mass of the solute. This gives the weight of one mole of the compound, allowing you to determine its molar mass in \(g/mol\).

Molality

Molality is a way of expressing the concentration of a solution and is used particularly in colligative properties like freezing point depression. Unlike molarity, which depends on the volume of the solution, molality depends on the mass of the solvent.
It is determined using the following formula:

• \(m = \frac{moles\, of\, solute}{kilograms\, of\, solvent}\)

In the aluminon problem, for example, the molality helps calculate the change in freezing point. Knowing that the solvent is water and weighing it precisely allows you to find the molality accurately without changes due to temperature or volume.
This measure is particularly useful in experiments where temperature might vary, as molality remains constant across those conditions.

Van't Hoff Factor

The Van't Hoff factor, denoted as \(i\), plays an essential part in calculating colligative properties such as boiling point elevation and freezing point depression. It accounts for the number of particles into which a solute dissociates in solution.
For non-electrolytes, substances that do not dissociate into ions, the Van't Hoff factor is simply 1. In the case of aluminon, which is a non-electrolyte, \(i\) is 1.
However, if aluminon were electrolytic, the factor could be greater than 1, affecting both calculations and the actual freezing point depression. For example:

• If a solute breaks into three ions, \(i\) would be 3.
• This means, the formula \( \Delta T_f = i \cdot K_f \cdot m \) would have a more significant \(\Delta T_f\) for the same solute mass.

Thus, \(i\) is crucial not only for ice-cream making but also for understanding how compounds behave in solutions.

Aluminum Ion Testing

Aluminum ion testing involves detecting the presence of aluminum in an aqueous solution. This process is important in various industrial and environmental contexts where aluminum levels must be monitored, such as water purification and food safety.
Aluminon is a chemical reagent often used for this purpose. When aluminon is added to a solution containing aluminum ions, it forms a pink-colored complex. This coloration serves as a qualitative confirmation of aluminum presence.

• It becomes essential in scenarios where visual identification needs to be quick and straightforward.
• Testing with aluminon is usually accompanied by other methods to quantify the aluminum concentration more precisely.

Understanding the specifics of this reaction helps in better designing experiments and in ensuring accuracy in identifying aluminum ions in different contexts.