(a) What volume of a \(0.778 \mathrm{M} \mathrm{Na}_{2} \mathrm{CO}_{3}\) (aq) solution should be diluted to \(150.0 \mathrm{~mL}\) with water to reduce its concentration to \(0.0234 \mathrm{M} \mathrm{Na}_{2} \mathrm{CO}_{3}(\mathrm{aq})\) ? (b) An experiment requires the use of \(60.0 \mathrm{~mL}\) of \(0.50 \mathrm{M} \mathrm{NaOH}\) (aq). The stockroom assistant can only find a reagent bottle of \(2.5 \mathrm{M} \mathrm{NaOH}\) (aq). How is the \(0.50 \mathrm{M} \mathrm{NaOH}(\) aq \()\) solution to be prepared?

Molarity is a measure of the concentration of a solute in a solution and is expressed in moles per liter (M). It describes how many moles of a substance are present in one liter of a solution. For example, if you have a 1.0 M solution of sodium chloride, that means there is 1 mole of sodium chloride dissolved in every liter of solution. Molarity is a critical concept in chemistry because it allows for the quantification of reaction constituents, which is essential for predicting the outcomes of chemical reactions.

To calculate molarity, use the formula: \(M = \frac{n}{V}\), where \(M\) is molarity, \(n\) is the number of moles of solute, and \(V\) is the volume of the solution in liters. This calculation is pivotal when mixing solutions to achieve a desired concentration for experimental protocols.

The dilution formula is essential for adjusting solution concentrations. It is represented as \(C_1V_1 = C_2V_2\), where \(C_1\) is the initial concentration, \(V_1\) the initial volume, \(C_2\) the final concentration, and \(V_2\) the final volume after dilution. This formula is based on the principle that the amount of solute in the solution remains constant before and after dilution. The formula enables the calculation of how much solvent (usually water) needs to be added to achieve a specific lower concentration or conversely, how much of a concentrated stock solution is required to prepare a more dilute solution. This tool is invaluable in the laboratory when preparing solutions for experiments where precise concentrations are key.

Concentration of solutions describes the amount of solute that is present in a given quantity of solvent or solution. It is a pivotal aspect of many chemical procedures and protocols as it affects reaction rates, yields, and can alter the nature of a reaction altogether. Concentration can be expressed in various ways such as molarity (M), mass percent, volume percent, and parts per million (ppm). Molarity, however, is the most commonly used unit in a laboratory setting for its ease of interconversion and direct use in stoichiometric calculations. It's important to understand how to manipulate concentrations to ensure the correct stoichiometry in reactions and to maintain safety standards when handling chemicals.

Volume calculation in the context of solution preparation is integral to the dilution process. To accurately prepare a desired solution, the volume of solvent to be added or the volume of stock solution to be used must be precisely calculated. The volume of a liquid can be measured in milliliters (mL) or liters (L), with 1,000 mL being equivalent to 1 L. When dealing with solutions, volumes need to be measured accurately using appropriate laboratory equipment such as graduated cylinders, pipettes, or burettes. Accurate volume calculation ensures that the correct molarity and concentration of the solution are maintained, which is critical for the success of chemical experiments.