Calculate the molality of each of the following solutions: (a) \(1.25 \mathrm{~mol} \mathrm{CaCl}_{2}\) in \(750.0 \mathrm{~g}\) water (b) \(2.5 \mathrm{~g} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\) in \(525 \mathrm{~g}\) water (c) \(17.5 \mathrm{~mL}\) 2-propanol, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CHOH}(d=0.785 \mathrm{~g} / \mathrm{mL})\) in \(35.5 \mathrm{~mL} \mathrm{H} \mathrm{H}_{2} \mathrm{O}(d=1.00 \mathrm{~g} / \mathrm{mL})\)

Colligative properties are those properties of solutions that depend on the number of dissolved particles in the solution, rather than the type of particles. These properties include boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure. For instance, when a solute is added to a solvent, the solution generally has a lower freezing point than the pure solvent.

Understanding colligative properties is essential when discussing molality, as molality measures the concentration of a solution in terms of moles of solute per kilogram of solvent. It's crucial because this concentration directly affects colligative properties. For example, the greater the molality, the more significant the boiling point elevation or the freezing point depression of the solution. Thus, by knowing the molality, one can predict how the solution will behave under various conditions.

The molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance, expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all atoms in a compound's formula. For example, the molar mass of water ( H2O) is 18.02 g/mol, deriving from the atomic masses of hydrogen (1.01 g/mol) and oxygen (16.00 g/mol), where you have two atoms of hydrogen and one atom of oxygen in each molecule of water.

Molar mass serves as a bridge between the mass of a substance and the number of moles, allowing chemists to convert between mass and moles in a straightforward manner, as seen in the textbook solutions. In the context of molality calculations, knowing the molar mass lets us determine the number of moles of solute present, which is crucial for working out the molality of a solution.

Solution concentration can be expressed in various ways, including molarity, molality, and mass percent. Molality, one of the central topics of the exercise, is especially useful in scenarios involving temperature changes since it remains unaffected by temperature fluctuations, unlike molarity.

Molality is defined as the moles of solute divided by the mass of the solvent in kilograms. This concentration measure is key when dealing with colligative properties because it doesn't change with the expansion or contraction of the solution. As illustrated in the exercise solutions, while performing molality calculations, students must ensure precise conversion of mass into kilograms and accurate calculation of moles using molar mass. These foundational concepts in chemistry are interrelated and help in understanding the quantitative aspects of solutions.