Question

If \(500 .\) g of water is added to \(75 \mathrm{~g}\) of \(25 \% \mathrm{NaCl}\) solution, what is the percent by mass of \(\mathrm{NaCl}\) in the diluted solution?

Short answer

The percent by mass of NaCl in the diluted solution is approximately \(3.26\%\).

Step-by-step solution

Determine the mass of NaCl in the initial solution

The initial solution consists of 25% NaCl, which means there is 25g of NaCl in every 100g of solution. We are given that we have 75g of this solution. To find the mass of NaCl in the 75g solution, we can use a proportion: \[\frac{25 \text{g NaCl}}{100 \text{g solution}} = \frac{x \text{g NaCl}}{75 \text{g solution}}\]

Solve the proportion to find the mass of NaCl in the initial solution

Cross-multiply and solve for x: \(25 \text{g NaCl} \times 75 \text{g solution} = 100 \text{g solution} \times x \text{g NaCl}\) \[1875 \text{g} = 100x\] Now, divide both sides by 100 to find the mass of NaCl in the 75g solution: \[x = \frac{1875}{100} = 18.75 \text{g NaCl}\]

Determine the total mass of the diluted solution

Now, we need to find the total mass of the diluted solution after adding 500g of water. The initial solution already contains 75g, so the total mass of the diluted solution will be: Total mass = 75g (initial solution) + 500g (water) Total mass = 575g

Calculate the percent by mass of NaCl in the diluted solution

We previously found out that there were 18.75g of NaCl in the initial solution. As no NaCl has been added or removed, there is still 18.75g of NaCl present in the 575g diluted solution. The percent by mass of NaCl in this diluted solution is: \[\% \text{ by mass (NaCl)} = \frac{\text{mass of NaCl}}{\text{total mass of solution}} \times 100\] \[ \% \text{ by mass (NaCl)} = \frac{18.75 \text{g NaCl}}{575 \text{g solution}} \times 100\]

Solve for the final percent by mass

Now, we can plug in our values and solve for the percent by mass of NaCl: \[\% \text{ by mass (NaCl)} = \frac{18.75}{575} \times 100\] \[\% \text{ by mass (NaCl)} = 0.0326 \times 100\] \[\% \text{ by mass (NaCl)} \approx 3.26\%\] The concentration of the NaCl in the diluted solution is approximately 3.26%.

Key concepts

Chemical Solution Concentration

Understanding the concentration of a chemical solution is a fundamental concept in chemistry, crucial for experiments and industrial processes alike. Concentration refers to the amount of a substance (solute) present in a given volume or mass of solution. When working on laboratory or real-life applications, knowing the exact concentration can be critical since chemical reactions depend on the precise amounts of reactants.

Concentration can be expressed in various ways, including molarity, molality, normality, and percent by mass or volume. Percent by mass, the key concept in this example, is a convenient way to describe concentrations, especially in solutions where the volume might change due to temperature or pressure varying but the mass stays consistent.

When water is added to a solution like the NaCl example, it dilutes the original concentration. The process doesn't change the total mass of the solute present; it simply increases the solution's volume or mass. As the mass of the solvent (water in this case) increases, the relative mass of the solute (NaCl) in the total solution decreases, leading to a reduced percent by mass.

Mass Percentage Calculation

Mass percentage is a common method to express the concentration of a component in a mixture or a solution. It is defined as the mass of the solute divided by the total mass of the solution, multiplied by 100 to get a percentage. The calculational formula can be written as: \[\begin{equation}\text{Mass Percentage} = \left(\frac{\text{mass of solute}}{\text{total mass of solution}}\right) \times 100\%\end{equation}\]

This formula allows us to find the concentration of any solute within a solution after certain operations, such as dilution with additional solvent, have taken place. In the NaCl problem, we're observing how the mass percentage of salt changes when water is added. It's a straightforward calculation, yet essential for precision in both academic problems and practical scenarios.

To ensure students fully understand the process for mass percentage calculations, it's crucial to emphasize the steps: identifying the mass of solute, calculating the new total mass post-dilution, and then using those values to find the percentage. Handling the concept mathematically hinges on accurate arithmetic and proportion understanding.

Stoichiometry

Stoichiometry is the section of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It's like the recipe for the reaction, telling you how much of each reactant you need to produce a specific amount of product. The principles of stoichiometry apply to reactions in gas, liquid, or solid phases and involves concepts like the mole ratio, mass-mass calculation, and limiting reactants.

In the context of our NaCl solution problem, stoichiometry comes into play when first determining the initial amount of NaCl in the 25% solution. Using proportionality, we establish the stoichiometric relation between the mass of our solute (NaCl) and the total mass of the solution before and after dilution. A sound understanding of stoichiometry is the backbone that allows students to move seamlessly from theoretical calculations to practical, predictable outcomes in a lab setting or real-world chemical manufacturing.

Although the original problem does not directly involve a chemical reaction, the ability to calculate the amount of reactant (NaCl) based on percentage concentration is a stoichiometric skill. It's these types of calculations that enable chemists to design and control reactions, ensuring the desired products are obtained efficiently and safely.