To what volume should you dilute \(125 \mathrm{~mL}\) of an \(8.00 \mathrm{M}\) \(\mathrm{CuCl}_{2}\) solution so that \(50.0 \mathrm{~mL}\) of the diluted solution contains \(5.9 \mathrm{~g} \mathrm{CuCl}_{2}\) ?

Understanding molarity is essential when studying solutions in chemistry. Molarity, designated as 'M', measures the concentration of a solute in a solution. It is defined as the number of moles of solute divided by the volume of the solution in liters. The formula for calculating molarity is:
\[ M = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \]
To find the molarity of a solution, you should begin by calculating the number of moles of the solute using its mass and molar mass. After finding the moles of solute, divide this value by the total volume of the solution in liters. This measurement is crucial for the comparison of solution concentrations and is employed for various applications such as dilution and reaction stoichiometry.

Molar mass is another fundamental concept in chemistry that links the mass of a substance to its amount in moles. It is expressed in grams per mole (g/mol) and can be thought of as the mass of one mole of a given substance. The molar mass of a compound like copper(II) chloride (\( \text{CuCl}_2 \)) is found by summing the atomic masses of each element, in the proportions they are found in the compound, according to the periodic table. For example:
\[ \text{Molar Mass of } \text{CuCl}_2 = (\text{Mass of Cu} + 2 \times \text{Mass of Cl}) \text{ g/mol} \]
Knowing the molar mass allows you to convert between grams and moles, facilitating calculations involving chemical reactions and solutions.

The dilution equation is a simple yet powerful tool used to determine the volume or concentration changes when diluting a solution. This is particularly important when preparing a chemical solution of a desired molarity. The dilution equation is:
\[ M1 \times V1 = M2 \times V2 \]
where \(M1\) and \(V1\) are the molarity and volume of the concentrated (initial) solution, respectively, and \(M2\) and \(V2\) are the molarity and volume of the diluted (final) solution. The principle behind this equation is the conservation of moles of solute before and after dilution. When you use the dilution equation, remember that volumes need to be in the same units and that the volume of the solvent added for the dilution process can be determined by subtracting \(V1\) from \(V2\).

Using the Dilution Equation

Applying the dilution equation involves solving for the unknown variable, which can be the final volume (\(V2\)) or the final molarity (\(M2\)) of the solution after dilution. It's essential in laboratory settings for creating solutions of precise concentrations.