A sample of sulfur weighing 0.210 g was dissolved in 17.8 g of carbon disulfide, \(C S_{2}\left(K_{b}=2.43^{\circ} \mathrm{C} / m\right)\). If the boiling point elevation was \(0.107^{\circ} \mathrm{C},\) what is the formula of a sulfur molecule in carbon disulfide (assuming ideal solution behavior)?

When we discuss colligative properties, we consider how adding solute to a solvent can alter the physical state of the solution in a way that depends only on the amount of solute particles, not their identity. This includes phenomena such as boiling point elevation, freezing point depression, vapor pressure lowering, and osmotic pressure.

Take boiling point elevation as an example: the more solute particles that are dissolved in a solvent, the higher the boiling point of the solution will be compared to the pure solvent. This effect is essential to understand because it helps us determine unknown properties of the solute like its molecular weight and is used in various applications, from preparing antifreeze solutions to culinary techniques like candy making.

Molality is a critical concept for boiling point elevation and other colligative properties; it measures the concentration of a solution. Defined as the number of moles of solute per kilogram of solvent (not volume!), this unit of concentration is particularly useful when dealing with temperature changes, as it doesn't vary with changes in temperature, unlike its more commonly known counterpart, molarity.

To find molality, we simply divide the moles of solute by the mass of the solvent in kilograms. The formula is expressed as:
\[ m = \frac{n}{M_{solvent}} \]
where \(m\) is the molality, \(n\) is the number of moles of solute, and \(M_{solvent}\) is the mass of the solvent in kilograms. This value is pivotal in calculating the extent to which the boiling point of a solution is elevated.

The van't Hoff factor, symbolized as \(i\), is central to the quantification of the colligative properties of solutions. It indicates the number of particles into which a compound dissociates in a given solvent. For nonelectrolytes, such as sugar in water, the van't Hoff factor is typically 1, meaning the compound remains whole and doesn't split into multiple parts.

However, for electrolytes like table salt (NaCl) in water, the van't Hoff factor is often greater than 1 because salt dissociates into its ions, Na+ and Cl-. This means a single formula unit of salt results in two particles in solution, doubling the effect on the solution's colligative properties for the same molar quantity of solute.

Molar mass calculation is a key step in understanding the composition of a solute in a solution, like determining the molecular formula of a dissolved substance based on its impact on the boiling point. Calculating molar mass involves dividing the mass of the solute by the number of moles of the solute:

\[ Molar \ Mass = \frac{Mass\ of\ Sulfur}{Moles\ of\ Sulfur} \]
This value provides insights into the amount of substance present and is necessary for determining the molecular formula of the compound in question. Knowing the molar mass allows scientists to carry out a host of calculations crucial to research and industry applications.

Determining the molecular formula of a compound is like solving a puzzle. Once the molar mass of a solute in a particular solution is identified through experimentation and calculation, scientists can compare this value to the known molar mass of the pure element or compound.

By doing so, we can infer the number of atoms in a molecule of the solute and thus establish its molecular formula. For example, the molar mass of a sulfur molecule dissolved in carbon disulfide can reveal whether sulfur exists in the solution as an S8 octamer, the typical form of elemental sulfur, or as some other allotrope. This process links the theory of colligative properties with practical identification and characterization of chemical substances.