Problem 104 - Chemistry The Central Science

Question

Carbon disulfide \(\left(\mathrm{CS}_{2}\right)\) boils at \(46.30^{\circ} \mathrm{C}\) and has a density of 1.261 \(\mathrm{g} / \mathrm{mL}\) . (a) When 0.250 \(\mathrm{mol}\) of a nondissociating solute is dissolved in 400.0 \(\mathrm{mL}\) of \(\mathrm{CS}_{2},\) the solution boils at \(47.46^{\circ} \mathrm{C} .\) What is the molal boiling-point-elevation constant for \(\mathrm{CS}_{2}\) ? (b) When 5.39 \(\mathrm{g}\) of a nondissociating unknown is dissolved in 50.0 \(\mathrm{mL}\) of \(\mathrm{CS}_{2},\) the solution boils at \(47.08^{\circ} \mathrm{C} .\) What is the molar mass of the unknown?

Step-by-Step Solution

Answer

(a) The molal boiling-point-elevation constant for CS2 is 2.34 ℃/(mol/kg). (b) The molar mass of the unknown solute is 256.67 g/mol.

Step 1: (a) Find the boiling-point elevation and molality

First, we find the boiling-point elevation (ΔTb). It is the difference between the boiling point of the solution and the boiling point of the pure solvent. ΔTb = 47.46℃ - 46.30℃ = 1.16℃ Next, we find the molality (m) of the solution. Molality is defined as the moles of solute per kilogram of solvent. We are given that there are 0.250 moles of solute dissolved in 400.0 mL of CS2. To find the molality, we must convert the volume of the CS2 to mass and then convert it to kilograms. Mass of CS2 = 400.0 mL × 1.261 g/mL = 504.4 g Mass of CS2 in kg = 504.4 g ÷ 1000 = 0.5044 kg Molality (m) = 0.250 mol ÷ 0.5044 kg = 0.4956 mol/kg

Step 2: (a) Calculate Kb for CS2

Now, we can use the boiling-point elevation formula to find the value of Kb: ΔTb = Kb * m Rearrange the formula to solve for Kb: Kb = ΔTb / m Kb = 1.16℃ / 0.4956 mol/kg = 2.34 ℃/(mol/kg)

Step 3: (b) Find the molality of the unknown solute

We are given that the boiling-point elevation for the new solution is 47.08℃. Calculate the boiling-point elevation (ΔTb) for the unknown solute with this new boiling point: ΔTb = 47.08℃ - 46.30℃ = 0.78℃ Now, we can use the known boiling-point elevation constant (Kb = 2.34 ℃/(mol/kg)) to find the molality of the unknown solute: ΔTb = Kb * m Rearrange the formula to solve for m: m = ΔTb / Kb m = 0.78℃ / 2.34 ℃/(mol/kg) = 0.333 mol/kg

Step 4: (b) Calculate the molar mass of the unknown solute

We are given that 5.39 g of an unknown solute is dissolved in 50.0 mL of CS2. Calculate the mass of the solvent in kg: Mass of CS2 = 50.0 mL × 1.261 g/mL = 63.05 g Mass of CS2 in kg = 63.05 g ÷ 1000 = 0.06305 kg Using the molality (m) and the mass of the solvent (in kg), we calculate the moles of solute in the solution: m = moles of solute / kg of solvent moles of solute = m × kg of solvent moles of solute = 0.333 mol/kg × 0.06305 kg = 0.0210 mol Now, using the mass and moles of the unknown solute, we can find its molar mass: molar mass = mass of solute / moles of solute molar mass = 5.39 g / 0.0210 mol = 256.67 g/mol The molar mass of the unknown solute is 256.67 g/mol.

Key Concepts

Boiling-Point Elevation

Understanding the concept of boiling-point elevation is essential for students studying solutions and their colligative properties. The phenomenon of boiling-point elevation occurs when a solute is dissolved in a solvent, resulting in the increase of the boiling point of the solution compared to the pure solvent. This effect is observed because the presence of solute particles impedes the ability of solvent molecules to escape into the vapor phase, which requires additional energy, and thereby increases the temperature at which the liquid will boil.

For instance, in the original exercise, carbon disulfide (CS2) boiled at a higher temperature when a solute was added. Mathematically, this can be described with the equation \( \Delta T_b = K_b \cdot m \) where \( \Delta T_b \) is the boiling-point elevation, \( K_b \) is the molal boiling-point-elevation constant, and \( m \) is the molality of the solution. By knowing any two of these values, one can easily calculate the third. This equation reflects a directly proportional relationship; as the molality of the solution increases, the boiling point elevation also increases.

Molality

Molality plays a pivotal role in calculations related to boiling-point elevation. Unlike molarity, which is dependent on the volume of the solution, molality is defined by the number of moles of solute per kilogram of solvent, making it a temperature-independent concentration unit. This is particularly beneficial in situations where temperature can cause significant volume changes.

To calculate molality, as seen in the exercise, one has to first find the mass of the solvent in kilograms and then use the formula \( m = \frac{\text{moles of solute}}{\text{kg of solvent}} \) where \( m \) stands for molality. In this case, we converted the volume of CS2 into mass using its given density, then converted the mass into kilograms to finally find the molality of the solution. It is crucial to remember this method for converting volumes to masses and subsequently to molalities when dealing with boiling-point elevation problems.

Molar Mass Calculation

Students often come across the need to calculate the molar mass of a compound, most typically in chemistry when dealing with reactions and calculations involving quantities of substances. The molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). It plays an integral part in various chemical calculations and can be determined from the periodic table as the sum of the atomic masses of the elements in the compound.

In the context of our exercise, the molar mass of an unknown substance was found by rearranging the relationship between molality, the mass of the solvent, and the moles of solute. To compute the molar mass, you can use the formula \( \text{molar mass} = \frac{\text{mass of solute}}{\text{moles of solute}} \) where the mass of solute is given and the moles of solute are determined by using the previously calculated molality and mass of the solvent. This step-by-step approach helps students connect the concept of molality to practical applications such as determining the molar mass of an unknown compound based on its effect on the boiling point of a solution.