Question

If \(500 .\) g of water is added to \(75 \mathrm{g}\) of \(25 \% \mathrm{NaCl} \mathrm{so}-\) lution, 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

Find the mass of NaCl in the original solution

We have 75g of 25% NaCl solution. To find the mass of NaCl in this solution, we will calculate 25% of 75g: Mass of NaCl = (25/100) × 75g = (1/4) × 75g = 18.75g

Calculate the mass of the diluted solution

Now that we have added 500g of water to the original solution, the mass of the diluted solution will be the sum of the mass of the original solution and the added water: Total mass of diluted solution = mass of original solution + mass of added water = 75g + 500g = 575g

Calculate the percent by mass of NaCl in the diluted solution

Now that we have the mass of NaCl in the solution and the total mass of the diluted solution, we can find the percent by mass of NaCl in the diluted solution using the formula: Percent by mass of NaCl = (Mass of NaCl in the solution/Total mass of diluted solution) × 100% Percent by mass of NaCl = (18.75g/575g) × 100% = \( \frac{18.75}{575} \) × 100% = 0.0326 × 100% ≈ 3.26% Therefore, the percent by mass of NaCl in the diluted solution is approximately 3.26%.

Key concepts

Solution Concentration

When discussing solution concentration, we refer to the amount of a substance, known as the solute, that is dissolved in a particular volume of solvent to create a solution. It's essential to understand that a solution is a homogeneous mixture consisting of these two parts: the solute and the solvent.

Concentration can be expressed in various ways, including molarity, molality, and as in this case, percent by mass. In everyday terms, the concentration tells us how 'strong' or 'weak' a solution is based on how much solute is present compared to the total amount of the solution.

For instance, if you add table salt to water, the resulting saline solution's concentration would be higher if more salt is dissolved. Understanding this concept is paramount when it comes to making solutions in a laboratory or adjusting them for various applications.

Mass Percentage

The mass percentage is a way of expressing the concentration of a component in a mixture or a solution. It's calculated by taking the mass of the component of interest (usually the solute) and dividing it by the total mass of the solution, then multiplying by 100%.

Mathematically, it is represented as: \[\text{Mass percentage} = \left( \frac{\text{Mass of the component}}{\text{Total mass of the solution}} \right) \times 100%\]
It's a valuable tool in chemistry for describing the composition of a solution. For students grappling with this concept, visualizing or actually mixing different concentrations can be very helpful. For example, mixing more sugar in a fixed amount of water will result in a sweet syrup with a high mass percentage of sugar.

Dilution of Solutions

The process of dilution involves adding more solvent to a solution without adding more of the solute. This simple action decreases the solution's concentration because the solute is now spread over a greater volume of solvent.

In practical applications, dilution is used to achieve a desired concentration. It's crucial in various fields such as medicine, where specific drug concentrations are required for treatments, or in culinary arts, where flavors are tuned to the desired strength.

Mathematically, the concept of dilution can be captured by the formula: \[V_1C_1 = V_2C_2\] where \(V_1\) and \(C_1\) are the volume and concentration of the initial solution, and \(V_2\) and \(C_2\) are the volume and concentration after dilution, respectively. This formula assumes that the amount of solute remains constant during the dilution process.