A paint requires \(42.8 \mathrm{~g}\) of iron(III) oxide to give it the right yellow tint. How many grams of a \(30.0 \%\) by mass solution of iron(III) oxide should be used?

Understanding the concentration of a solution is crucial for precise chemical mixtures. Molarity is among the most widely used units to express such concentrations. It measures the number of moles of solute per liter of solution.

Chemists and other scientists use molarity because it allows them to relate the volume of the solution to the amount of the substance dissolved in it. This is especially useful when reacting substances in a liquid state, where volumes are often measured.

The formula for molarity (\text{M}) is \[ M = \frac{\text{moles of solute}}{\text{liters of solution}} \] Molarity is a central concept in preparing solutions for laboratory experiments, titrations or even making solutions for industrial processes. When dealing with real-world applications like the paint mixture in our exercise, converting this molarity into mass percent further helps in scaling the preparation to larger or smaller batches as needed.

Stoichiometry comes from the Greek words for 'element' and 'measure' and plays a significant role in chemistry. It involves calculating the quantities of reactants and products involved in chemical reactions. Stoichiometry is rooted in the law of conservation of mass where the total mass of reactants equals the total mass of products in a chemical reaction.

The stoichiometric coefficients in a balanced chemical equation tell us in what ratio the substances react or form. For instance, if the equation for the production of water is 2H2 + O2 → 2H2O, it indicates 2 moles of hydrogen react with 1 mole of oxygen to form 2 moles of water.

For the paint problem, stoichiometry helps determine how much of the iron(III) oxide solution is needed to achieve a specific mass of the compound in its dried form within the paint mixture.

Chemical calculations are essential for scientists to evaluate substances and reactions under various conditions. They involve determining molar masses, calculating concentrations in different units, and converting between these units.

Units like percent by mass, molarity, and parts per million (ppm) are often used. For example, percent by mass is the ratio of the mass of the solute to the mass of the whole solution, expressed as a percentage. The formula is: \[ \text{Percent by mass} = \left(\frac{\text{mass of solute}}{\text{total mass of solution}}\right) \times 100\% \]

Executing chemical calculations with precision is critical for the final product's consistency and quality in applications ranging from pharmaceuticals to manufacturing, like in the case of the paint requiring a specific concentration of iron(III) oxide.