Problem 73 - Chemistry The Molecular Science

Question

The freezing point of \(p\) -dichlorobenzene is \(53.1{ }^{\circ} \mathrm{C},\) and its \(K_{\mathrm{f}}\) is \(-7.10^{\circ} \mathrm{C} \mathrm{kg} / \mathrm{mol}\). A solution of \(1.52 \mathrm{~g}\) of the drug sulfanilamide in \(10.0 \mathrm{~g} p\) -dichlorobenzene freezes at \(46.7^{\circ} \mathrm{C} .\) Calculate the molar mass of sulfanilamide.

Step-by-Step Solution

Answer

The molar mass of sulfanilamide is approximately 168.66 g/mol.

Step 1: Determine Freezing Point Depression

First, calculate the change in freezing point, \( \Delta T_f \), using the formula: \( \Delta T_f = T_f^0 - T_f \). Here, \( T_f^0 \) is the freezing point of pure \( p \)-dichlorobenzene (53.1°C), and \( T_f \) is the freezing point of the solution (46.7°C). \[ \Delta T_f = 53.1 - 46.7 = 6.4 \text{ °C} \]

Step 2: Use Molality Equation

Now, use the freezing point depression formula: \( \Delta T_f = K_f \cdot m \), where \( K_f = -7.10^{\circ} \text{C kg/mol} \) is the cryoscopic constant and \( m \) is the molality of the solution. Rearrange to find molality: \[ m = \frac{\Delta T_f}{K_f} = \frac{6.4}{7.10} = 0.9014 \text{ mol/kg} \]

Step 3: Convert to Moles of Solute

Molality \( m \) is given by \( m = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} \). Use this to find the moles of sulfanilamide: \( \text{mass of p-dichlorobenzene} = 10.0 \text{ g} = 0.0100 \text{ kg} \). Hence, \[ \text{moles of sulfanilamide} = m \times \text{mass of solvent in kg} = 0.9014 \times 0.0100 = 0.009014 \text{ mol} \]

Step 4: Calculate Molar Mass of Sulfanilamide

Molar mass is determined by \( \text{molar mass} = \frac{\text{mass of solute}}{\text{moles of solute}} \). Here, the mass of sulfanilamide is 1.52 g, hence: \[ \text{molar mass} = \frac{1.52 \text{ g}}{0.009014 \text{ mol}} = 168.66 \text{ g/mol} \]

Step 5: Final Verification

Verify the calculations by cross-checking each input value and ensure all units are correctly applied during conversion and in calculations, ensuring precise results.

Key Concepts

Molar Mass Calculation

In chemistry, the molar mass is a fundamental concept. It tells us how much one mole of a substance actually weighs. In our exercise, we calculate the molar mass of sulfanilamide, which involves determining how many moles of the substance are present in a given mass. This measurement is crucial because it links the micro-world of molecules with the macro-world of measurable weights.

The molar mass is calculated using this formula: \[ \text{Molar Mass} = \frac{\text{Mass of Solute (g)}}{\text{Moles of Solute}} \] In the exercise, the mass of sulfanilamide is given as 1.52 grams, and the moles of sulfanilamide, previously calculated, is 0.009014 moles. Plugging these values into the formula gives us a molar mass of approximately 168.66 g/mol. - Understanding molar mass helps in various calculations in chemistry, from stoichiometry to determining concentration.- Molar mass allows for converting between mass and number of particles, interpreting chemical equations quantitatively.

Cryoscopic Constant

The cryoscopic constant, represented as \( K_f \), represents a substance’s tendency to lower the freezing point of a solvent when a solute is added. Each solvent has a unique \( K_f \) value, which is instrumental in calculating freezing point depression.

In our exercise, the cryoscopic constant for \( p\)-dichlorobenzene is given as \(-7.10^{\circ} \text{C kg/mol} \). This negative value indicates that the freezing point of the solution decreases when a solute is added. The usefulness of \( K_f \) lies in its role in the freezing point depression formula:\[ \Delta T_f = K_f \cdot m \] In the given exercise, this relationship allowed us to determine the molality of the solution. Here’s why the cryoscopic constant matters:- It is specific to each solvent: different solvents will have different \( K_f \) values.- It provides insight into molecular interactions between solute and solvent.- It's critical for calculating how much a solute affects a solvent's freezing point.

Molality

Molality is a concentration measurement used in solutions. It is defined as the moles of solute per kilogram of solvent. Unlike molarity, which deals with volume, molality is based on mass, making it particularly useful for temperature-dependent calculations. In our exercise, molality is a key part of finding out how the addition of a solute affects the freezing point of a solvent.

We use the formula: \[ m = \frac{\Delta T_f}{K_f} \]Once the freezing point depression \( \Delta T_f \) was calculated, it was easy to find the molality of the solution by dividing \( \Delta T_f \) by the cryoscopic constant \( K_f \). In the exercise, this came out to be 0.9014 mol/kg, indicating that the solution had this concentration of sulfanilamide.- Molality is ideal for calculations where temperature changes since it remains unaffected by volume changes.- It is liable for calculating other properties like boiling and freezing points due to its mass-based definition.- Learning about molality aids in understanding how solutes affect the properties of solvents.