Calculate the molalities of these aqueous solutions: (a) \(1.22 M\) sugar \(\left(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\right)\) solution (density of solution \(=1.12 \mathrm{~g} / \mathrm{mL}\) ), (b) \(0.87 \mathrm{M} \mathrm{NaOH}\) solution (density of solution \(=1.04 \mathrm{~g} / \mathrm{mL}),(\mathrm{c})\) \(5.24 \mathrm{M} \mathrm{NaHCO}_{3}\) solution (density of solution \(=\) \(1.19 \mathrm{~g} / \mathrm{mL})\)

When dealing with chemical solutions, it's important to understand the difference between molarity and molality, and know how to convert between them. Molarity, denoted as 'M', is the concentration of a solution expressed as the number of moles of solute per liter of solution. Molality, denoted as 'm', however, measures the concentration as the number of moles of solute per kilogram of solvent.

To convert molarity to molality, one must consider the density of the solution and the molar mass of the solute. The steps involve calculating the volume of solution that contains a known moles of solute, finding the mass of the solution using its density, and then determining the amount of the solute and solvent separately. The molality is then calculated using the formula:
\[\begin{equation}\text{molality (m)} = \frac{\text{number of moles of solute}}{\text{mass of solvent in kilograms}}\end{equation}\]
Molality is particularly useful in cases where temperature changes are involved, as it is not affected by changes in temperature, unlike molarity.

Density is a fundamental concept when discussing solutions, denoting the mass per unit volume, typically expressed in grams per milliliter (g/mL) or grams per liter (g/L). When we wish to convert molarity, which relies on volume, to molality, which depends on mass, knowing the density of the solution is crucial. This is because it allows us to relate the volume of the solution to its mass.

To calculate the mass of a solution, you multiply the volume of the solution by its density:\[\begin{equation}Mass_{Solution} = Density \times Volume\end{equation}\]
For any given volume, a solution with higher density contains more mass. In the context of chemistry and stoichiometry, this relationship helps us to calculate the molality of a solution when we are provided with its molarity and density.

Molar mass is a pivotal concept in chemistry, representing the mass of one mole of a substance. It is usually expressed in grams per mole (g/mol) and is calculated by summing the atomic masses of all the atoms in a molecule of the substance. Knowing the molar mass of a solute is essential when converting from molarity to molality.

Here is how it comes into play: to calculate the mass of the solute when you know the number of moles, you use the formula:\[\begin{equation}Mass_{Solute} = Molar mass of solute \times Number of moles\end{equation}\]
For the molarity to molality conversion, after finding the mass of the solute, deducting it from the overall mass of the solution gives us the mass of just the solvent. With the mass of the solvent in kilograms and the number of moles of solute, we can calculate the molality. Understanding the molar mass allows chemists to manipulate and measure the precise amounts of substances needed to create solutions with desired concentrations.