Calculate the molalities of the following 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 \(=1.04 \mathrm{~g} / \mathrm{mL})\) (c) \(5.24 \mathrm{M} \mathrm{NaHCO}_{3}\) solution (density of solution \(=\) \(1.19 \mathrm{~g} / \mathrm{mL}\)

Understanding how to convert molarity to molality is a fundamental skill in chemistry, particularly when dealing with solution concentration. Molarity, denoted as 'M', is the number of moles of solute per liter of solution. This concentration measurement is volume-based. On the other hand, molality, represented by 'm', is the number of moles of solute per kilogram of solvent, which is a weight-based measure.

To convert molarity to molality, we need to take into account the density of the solution and the molar mass of the solute. The formula for this conversion is: \[ Molality = \frac{Molarity \times Density\ of\ solution}{Molar\ mass\ of\ solute} \]
Let's look at one example from the exercise to see this calculation in action. The molarity of a sugar solution is given as 1.22 M. After converting the density from g/mL to kg/L, and knowing the molar mass of sugar, the molality is computed using the formula above. This process reveals the relationship between molarity, density, and the molar mass of the solute in determining the molality of a solution.

The density of a solution is a critical factor in various chemical calculations, including the conversion from molarity to molality. Density is defined as the mass per unit volume of a substance and is typically expressed in grams per milliliter (g/mL) or kilograms per liter (kg/L). For the conversion of molarity to molality, it's essential to express density in kg/L, as molality involves the weight of the solvent (kg).

Density can impact the concentration and physical behavior of solutions. When solving for molality, you must first confidently convert the given density into the appropriate units. For example, if the density is provided in g/mL, as in the original exercise, multiply by 1000 to convert to kg/L. This subtle but crucial step ensures that the units in the molality formula are consistent and that the calculation will yield an accurate result.

Molar mass is another vital concept integral to chemistry calculations, including determining molality. It is the mass of one mole of a given substance, typically expressed in grams per mole (g/mol). The molar mass is equivalent to the atomic or molecular weight of a substance, as listed on the periodic table, but expressed in grams per mole for practical purposes.

The molar mass serves as a bridge between the microscopic particles of a substance (atoms or molecules) and macroscopic quantities that can be measured in the laboratory. To find the molality of a solution, as seen in the exercise we are discussing, you must know the molar mass of the solute to use in conjunction with the solution's molarity and density. With this information, a proper conversion from molarity to molality can be achieved, allowing for the accurate determination of concentration in a weight-based measure.