Calculate the molality of a solution containing \(257 \mathrm{~g}\) glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) dissolved in \(1.62 \mathrm{~L}\) of water. (Assume a density of \(1.00 \mathrm{~g} / \mathrm{mL}\) for water.)

When calculating the molar mass of a substance, the first step is identifying the molecular formula. In our case, glucose is represented by \( \mathrm{C}_{6}\mathrm{H}_{12}\mathrm{O}_{6} \). You need to know the atomic masses of carbon (C), hydrogen (H), and oxygen (O) to perform the calculation.

In glucose, there are 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. The atomic mass of these elements (approximately) are 12.01 g/mol for carbon, 1.008 g/mol for hydrogen, and 16.00 g/mol for oxygen. Molar mass is calculated by adding the total mass of all atoms in the molecule:

• \[ Molar \, mass \, of \, glucose = 6 \times 12.01 \, g/mol \, + \, 12 \times 1.008 \, g/mol \, + \, 6 \times 16.00 \, g/mol \]
• \[ = (6 \times 12.01) \, g/mol \, + \, (12 \times 1.008) \, g/mol \, + \, (6 \times 16.00) \, g/mol \]
• \[ = 72.06 \, g/mol \, + \, 12.096 \, g/mol \, + \, 96.00 \, g/mol \]
• \[ = 180.156 \, g/mol \]

Calculating the molar mass correctly is crucial for the subsequent steps, as it will dictate the accuracy of molality and solution concentration calculations.

The number of moles of a solute in a solution is a fundamental measure in chemistry, indicating the amount of substance present. After calculating the molar mass of the solute, we use it to convert from grams to moles.

To find the moles of solute (glucose in this case), divide the mass of glucose by its molar mass:

• \[ Moles \, of \, solute = \, \frac{mass \, of \, glucose}{molar \, mass \, of \, glucose} \]
• \[ = \, \frac{257 \, g}{180.156 \, g/mol} \]
• \[ = 1.426 \, moles \]

This step establishes exactly how much of the glucose substance is dissolved in the solution, which is necessary for calculating the molality.

Converting mass to moles is a key skill in chemistry as it allows you to transition from a tangible mass measurement to the more conceptual moles which chemists use to quantify substances. You already know the mass of glucose (257 g) and the molar mass (180.156 g/mol), so converting is straightforward:

\[ Moles \, = \, \frac{mass}{molar \, mass} \]

• \[ = \, \frac{257 \, g}{180.156 \, g/mol} \]
• \[ = 1.426 \, moles \]

It is important to understand that moles link the mass of a substance to the number of particles (atoms, molecules, or ions) it contains, a vital relationship in various chemical calculations.

Solution concentration can be measured in several ways, and molality is one such measure. Molality is defined as the moles of solute per kilogram of solvent. It differs from molarity, which is moles of solute per liter of solution. To calculate molality, you take the number of moles of solute and divide by the mass in kilograms of the solvent (water, in this case):

• \[ Molality\, (m) \, = \, \frac{moles \, of \, solute}{kg \, of \, solvent} \]
• \[ = \, \frac{1.426 \, moles}{1.62 \, kg} \]
• \[ = 0.880 \, mol/kg \]

Remember to always use kilograms for the solvent mass when calculating molality, as this is a standard convention in chemistry, allowing for consistent results and comparisons.