Problem 45 - Chemistry: The Central Science in SI Units, Global Edition

Question

Calculate the molality of each of the following solutions: (a) \(10.0 \mathrm{~g}\) of benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) dissolved in \(50.0 \mathrm{~g}\) of carbon tetrachloride \(\left(\mathrm{CCl}_{4}\right),(\mathbf{b}) 5.00 \mathrm{~g}\) of \(\mathrm{NaCl}\) dissolved in \(0.100 \mathrm{~L}\) of water.

Step-by-Step Solution

Answer

In conclusion, the molality of solution (a) containing \(10.0\,g\) of benzene dissolved in \(50.0\,g\) of carbon tetrachloride is \(2.56\,m\), and the molality of solution (b) containing \(5.00\,g\) of NaCl dissolved in \(0.100\,L\) of water is \(0.86\,m\).

Step 1: Calculate moles of benzene

The molecular formula of benzene is \(C_6H_6\), so the molar mass of benzene is \((6 × 12.01)+(6 × 1.01)=78.12 g/mol\). Given 10.0 g of benzene, we can calculate the number of moles by: Moles of benzene = \(\frac{10.0\,\text{g}}{78.12\,\text{g/mol}} = 0.128\, \text{moles}\)

Step 2: Convert mass of carbon tetrachloride

We must convert the mass of carbon tetrachloride from grams to kilograms: 50.0 g = \(50.0\, \times 10^{-3}\) kg = 0.050 kg

Step 3: Calculate molality for (a)

Now, we can calculate the molality of the solution (a) as follows: Molality = \(\frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.128\,\text{moles}}{0.050\,\text{kg}} = 2.56\,m\) For (b):

Step 4: Calculate moles of NaCl

The molecular formula of NaCl; the molar mass of NaCl is \((1 × 22.99)+(1 × 35.45)=58.44\,g/mol\). Given 5.00 g of NaCl, we can calculate the number of moles by: Moles of NaCl = \(\frac{5.00\,\text{g}}{58.44\,\text{g/mol}} = 0.086\,\text{moles}\)

Step 5: Convert mass of water

We know that the density of water is about \(1\,\text{g/mL}\), so the mass of 0.100 L of water is: 0.100 L = 100.0 mL = 100.0 g 100.0 g = \(100.0\, \times 10^{-3}\) kg = 0.100 kg

Step 6: Calculate molality for (b)

Now, we can calculate the molality of the solution (b) as follows: Molality = \(\frac{\text{moles of solute}}{\text{mass of solvent in kg}} = \frac{0.086\,\text{moles}}{0.100\,\text{kg}} = 0.86\,m\) In conclusion, the molality of solution (a) is \(2.56\,m\()), and the molality of solution (b) is \(0.86\,m\().

Key Concepts

Moles Calculation

To calculate the molality of a solution, one must first determine the number of moles of the solute present. The concept of moles is central in chemistry because it allows us to relate the mass of a substance to the number of particles contained within it. The number of moles () can be calculated using the formula:
\[\text{Moles} = \frac{\text{Given Mass}}{\text{Molar Mass}}\]
For example, in calculating the moles of benzene (C_6H_6), given that we have 10.0 grams of benzene, we will first need its molar mass. As each carbon atom contributes 12.01 g/mol and each hydrogen atom 1.01 g/mol, benzene's molar mass totals to 78.12 g/mol. To find the moles of benzene:

• Use the molar mass of benzene (78.12 g/mol)
• Plug into the formula: \( \frac{10.0\, \text{g}}{78.12\, \text{g/mol}} = 0.128\; \text{moles} \)

This same process is used for NaCl, where 5.00 grams is given, and using its molar mass (58.44 g/mol), we calculate 0.086 moles of NaCl. Calculating moles is a straightforward process once you know the molar mass, enabling the rest of your calculations.

Molar Mass

Molar mass is a fundamental concept as it connects the mass of a substance with the amount of particles, or the number of moles, it contains. It is defined as the mass of one mole of a given substance in grams and is expressed in g/mol. To calculate molar mass, sum the average atomic masses of all atoms in the molecular formula.
Take benzene ( C_6H_6 ) as an example. It consists of 6 carbon atoms and 6 hydrogen atoms:

• Carbon contributes: 6 atoms × 12.01 g/mol = 72.06 g/mol
• Hydrogen contributes: 6 atoms × 1.01 g/mol = 6.06 g/mol

Thus, benzene's molar mass is 78.12 g/mol.
For NaCl, you have:

• Sodium (Na): 1 atom × 22.99 g/mol = 22.99 g/mol
• Chlorine (Cl): 1 atom × 35.45 g/mol = 35.45 g/mol

Thus, the molar mass of NaCl is 58.44 g/mol.
Understanding molar mass is crucial for converting between grams and moles, which is necessary for calculating other solution properties like molality.

Solvent Mass Conversion

Solvent mass conversion becomes crucial, particularly in chemistry solutions, where the mass of the solvent influences concentration calculations such as molality. For precise molality calculations, solvent mass must be expressed in kilograms.
In many experimental setups, it's common to begin with mass in grams. For instance, in solution (a), the mass of carbon tetrachloride is initially 50.0 grams. To convert:

• Start by dividing the mass by 1000 to convert to kilograms, since 1 kg = 1000 g.
• Thus, 50.0 g becomes 0.050 kg.

The same conversion applies to water in solution (b), where 0.100 L of water is equivalent to 100 g due to water's density of 1 g/mL:

• Convert 100 g to kilograms: 100 g = 0.100 kg

Converting mass correctly is essential for performing accurate concentration calculations, ensuring that the molality represents the number of moles per kilogram of solvent, which reflects the true nature of the solution.