A 45.8-mL sample of a \(5.8 \mathrm{M} \mathrm{KNO}_{3}\) solution is diluted to \(1.00 \mathrm{~L}\). What volume of the diluted solution contains \(15.0 \mathrm{~g}\) of \(\mathrm{KNO}_{3}\) ? (Hint: Figure out the concentration of the diluted solution first.)

Understanding molarity and dilution is fundamental in mastering chemistry, particularly when dealing with solutions. Molarity, denoted by the symbol M, measures the concentration of a solution. It's defined as the number of moles of solute (the substance being dissolved) per liter of solution. To calculate molarity, you can use the equation:
\[ M = \frac{moles \, of \, solute}{volume \, of \, solution \, in \, liters} \]
Dilution, on the other hand, involves adding more solvent (like water) to a solution, which decreases the concentration of the solute without changing the number of moles of solute present. This process is useful in creating solutions of desired molarities. The relationship between the initial and final molarity and volumes during dilution follows the equation:
\[ M_1V_1 = M_2V_2 \]
where \( M_1 \) and \( M_2 \) are the molarities before and after dilution, respectively, and \( V_1 \) and \( V_2 \) are the corresponding volumes. When you mix a solution to reach a specific molarity, it's crucial to account for the volume increase due to the solvent added.

The molar mass of a compound is an intrinsic property that's critical when dealing with chemical quantities in the lab. It represents the mass of one mole of a substance and is measured in grams per mole (g/mol). The calculation of molar mass involves the summation of the atomic masses of all the atoms in a molecule of the compound, as found on the periodic table. For example, for potassium nitrate (KNO3), the molar mass would be the sum of the atomic masses of potassium (K), nitrogen (N), and three oxygen (O) atoms.
To find the moles of a solid substance like KNO3, you can use the relationship:
\[ moles = \frac{mass \, (g)}{molar \, mass \, (g/mol)} \]
This conversion is essential, as it helps relate the physical mass measured in the laboratory to the theoretical moles, which are used in chemical calculations. Knowing the molar mass allows chemists to calculate how much of a substance is needed to react completely with another substance or to create a solution of a desired concentration.

Concentration refers to the amount of solute present in a given amount of solution or solvent. There are multiple ways to express concentration, with molarity being one of the most common in chemistry. Concentration helps to determine the strength or potency of a solution and plays a key role in stoichiometric calculations for reactions.
When performing dilutions or mixing solutions, it's critical to accurately calculate the concentration to ensure the desired outcome of a reaction. Concentration not only determines how reactive a solution is but also helps in calculating the yield of a reaction. A higher concentration implies a greater number of reactive particles in a given volume, potentially affecting the rate and extent of a reaction.
In the context of the exercise, after diluting a concentrated potassium nitrate solution, the concentration of the resultant solution is needed to find out what volume will contain a certain mass of KNO3. This exercise incorporates the core understanding of concentration by requiring the calculation of molarity after dilution, followed by application of this knowledge to quantify a specific amount of substance based on its concentration.