Calculate the molality of the solute in each of the following solutions: (a) \(10.0 \mathrm{~g}\) of \(\mathrm{NaCl}\) dissolvcd in \(250 \mathrm{~g}\) water; (b) \(0.48 \mathrm{~mol}\) of \(\mathrm{KOH}\) dissolved in \(50.0 \mathrm{~g}\) water; (c) \(1.94 \mathrm{~g}\) of urea, \(\mathrm{CO}\left(\mathrm{NH}_{2}\right)_{2}\), dissolved in \(200 \mathrm{~g}\) water.

Molar mass is a fundamental concept in chemistry, serving as a bridge between the atomic scale and the macroscopic world. It represents the mass of one mole of a substance, typically expressed in grams per mole (\r(g/mol)\r).

One mole contains Avogadro's number of particles, which is approximately \(6.022 \times 10^{23}\) particles—be they atoms, molecules, or ions. To find the molar mass of a compound, such as sodium chloride (NaCl), we sum the molar masses of the individual elements. For instance, sodium has a molar mass of approximately \(22.99 g/mol\), and chlorine's is about \(35.45 g/mol\). Together, for NaCl, the molar mass is \(58.44 g/mol\).

Molar mass is instrumental when converting between mass and moles of a substance, a step essential for solving many chemistry problems, including the calculation of solution concentrations like molality:

Understanding 'moles of solute' is key when dealing with chemical solutions. A mole is a unit that measures the amount of substance and is used to express the quantity of solute present in a solution.

The mole links the microscale of particles, such as atoms and molecules, to the macroscale that we can measure. When we say we have one mole of a substance, we're saying we have \(6.022 \times 10^{23}\) units of that substance. Thus, to calculate the moles of solute, we divide the mass of the solute by its molar mass. For example:
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• If we have \(10.0g\) of NaCl, and the molar mass of NaCl is \(58.44g/mol\), we have approximately \(0.171 moles\) of NaCl.

In the realm of chemistry, solutions are homogeneous mixtures composed of two or more substances. A solution typically has a solvent—which is the substance present in greater amount—and one or more solutes, the substances dissolved in the solvent.

Water is often called the 'universal solvent' because it can dissolve so many different substances. In our exercise examples, water is the solvent in which the solute (NaCl, KOH, or urea) is dissolved. The process of dissolving leads to a uniform distribution of solute particles throughout the solvent, resulting in a clear and uniform solution.

The concentration of a solution is a measure of how much solute is dissolved in a given quantity of solvent. Concentration can be expressed in many ways, including molality, which is detailed in the 'Concentration Units' section.

Concentration units in chemistry provide a measure of how much solute is present in a given amount of solvent or solution. One such unit is molality (\r(m)\r), which is defined as the number of moles of solute per kilogram of solvent.

The formula for molality is:\r
\r\[ m = \frac{{\text{{moles of solute}}}}{{\text{{kilograms of solvent}}}} \]
\rThis unit is especially useful because, unlike volume, mass does not change with temperature. To calculate molality in our examples, we must:
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• Find the mass of the solute in grams.
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• Convert that mass to moles using the molar mass of the solute.
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• Divide the moles of solute by the mass of the solvent in kilograms.

For instance, for a solution with \(0.48 moles\) of KOH in \(50.0 g\) (or \(0.050 kg\)) of water, the molality is \(9.6 m\) (molal). This unit is critical for various applications, including freezing point depression and boiling point elevation calculations.