Explain how you would prepare an aqueous solution of \(0.010 \mathrm{M} \mathrm{KMnO}_{4}(\mathrm{aq})\) starting with (a) solid \(\mathrm{KMnO}_{4} ;\) (b) \(0.050 \mathrm{M} \mathrm{KMnO}_{4}\) (aq).

Understanding molar mass is fundamental in chemistry, particularly when preparing solutions. Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It's calculated by summing the atomic masses of all the atoms in a molecule. For instance, to calculate the molar mass of potassium permanganate (\text{KMnO}_4), add the atomic mass of potassium (K), manganese (Mn), and four oxygen atoms (O).

To find these values, consult the periodic table. Let's break it down: Potassium has an atomic mass of about 39.1 g/mol, manganese is approximately 54.9 g/mol, and oxygen is about 16.0 g/mol. Therefore, the molar mass of \text{KMnO}_4 is calculated as follows:$$ \text{Molar mass of KMnO}_4 = (39.1 + 54.9 + 4 \times 16.0) \text{ g/mol} = 158.0 \text{ g/mol} $$
Knowing the molar mass allows us to converse between the mass of a substance and the number of moles, which is crucial when we're measuring out compounds for a solution in the lab.

Dilution is a process of reducing the concentration of a solute in a solution, typically by adding more solvent. When you have a solution of a higher concentration and need a lower concentration, dilution is the way to go. It's a common laboratory technique that's crucial for various experiments and protocols.

For example, if the task at hand is to dilute a 0.050 M potassium permanganate solution to a 0.010 M solution, you'd apply the dilution formula:$$ C1 \times V1 = C2 \times V2 $$Where:

• \text{C1} is the initial concentration (molarity) of the solution
• \text{V1} is the volume of the solution you're starting with
• \text{C2} is the desired concentration after dilution
• \text{V2} is the final volume of the solution

It's important to remember that while the concentration changes through dilution, the total amount of solute present remains constant. To attain the final concentration, careful measurement of volumes is paramount, using devices such as graduated cylinders or pipettes, and the addition of the solvent (usually distilled water) should be done up to the mark on a calibrated volumetric flask to ensure accuracy in the final solution volume.

Molarity concentration, often simply called molarity, is a measure of the concentration of a solute in a solution. It's defined as the number of moles of solute per liter of solution (\text{M} or mol/L). This concentration measurement is widely used in chemistry because it directly relates to the number of molecules or ions in a given volume of liquid, making stoichiometric calculations and analytical chemistry work straightforward.

To prepare a 0.010 M aqueous solution of \text{KMnO}_4 starting from a solid, we first calculate the number of moles required for a given volume, using the equation:$$ n = M \times V $$Here, \text{M} is molarity, and \text{V} is the volume in liters. Assuming you need 1 liter of the solution:$$ n = 0.010 \text{ moles/L} \times 1 \text{ L} = 0.010 \text{ moles} $$
Next, using the molar mass of \text{KMnO}_4 (158.0 g/mol), calculate the mass needed:$$ m = n \times \text{molar mass}= 0.010 \text{ moles} \times 158.0 \text{ g/mol}= 1.58 \text{ g} $$
Once you have the mass, that amount of solid \text{KMnO}_4 is dissolved in water to achieve the desired volume and molarity. Remember, accurate molarity calculation is important for predicting the behavior of the solution in various chemical reactions and ensuring the right concentration for any biochemical process.