Question

A chemist has 10 liters of a solution that is 10 percent nitric acid by volume. He wants to dilute the solution to 4 percent strength by adding water. How many liters of water must he add?

Short answer

15 liters of water.

Step-by-step solution

Understand the Original Solution

The chemist has 10 liters of a 10% nitric acid solution. This means that there are 10 liters * 0.10 = 1 liter of pure nitric acid and 9 liters of water in the original solution.

Determine the Desired Concentration

The goal is to dilute the solution to 4% nitric acid by volume. The amount of pure nitric acid will remain 1 liter, but the total volume of the solution will increase after adding water.

Set Up the Equation

Let x be the amount of water to add. The new total volume of the solution will be 10 liters + x liters. The concentration equation can be set up as follows:\[\frac{1 \text{ liter of nitric acid}}{10 + x \text{ liters of total solution}} = 0.04\]

Solve the Equation

Multiply both sides of the equation by \(10 + x\) to clear the denominator:\[1 = 0.04(10 + x)\]Expand and solve for x:\[1 = 0.4 + 0.04x\]Subtract 0.4 from both sides:\[0.6 = 0.04x\]Divide by 0.04:\[x = 15\]

Conclusion

To dilute the solution to 4% nitric acid, 15 liters of water must be added to the original solution.

Key concepts

Percentage Concentration

Percentage concentration tells us how much solute (in this case, nitric acid) is present in a solution relative to the entire volume of the solution. For example, a 10% nitric acid solution means that 10% of the total solution volume is nitric acid, and the remaining 90% is another solvent, usually water.
This concept is vital because it allows chemists to understand and control the strength of solutions. When dealing with dilutions, knowing the percentage concentration helps in adjusting the solution as needed. In the problem given, we are dealing with a 10% nitric acid solution, which means there are 10 parts of nitric acid per 100 parts of solution.

Solution Volume

The volume of a solution is simply the total amount of space that the solution occupies, measured in liters, milliliters, or other volume units. In the problem, we start with 10 liters of a solution that is 10% nitric acid.
Understanding solution volume helps in performing dilutions correctly. When water is added to the solution, the total volume increases, but the amount of nitric acid remains the same. This concept is crucial in calculations because we need to figure out the new volume after dilution. For example, if we add 'x' liters of water to 10 liters of the nitric acid solution, the new volume of the solution becomes 10 + x liters.

Chemical Solution Dilution

Dilution is the process of reducing the concentration of a solute in a solution, typically by adding more solvent. In the problem, we are diluting a 10% nitric acid solution to 4% by adding water.
Using the dilution formula \[ \text{Concentration 1} \times \text{Volume 1} = \text{Concentration 2} \times \text{Volume 2} \], we can set up an equation to find out how much water to add. Here, Concentration 1 is 10% (or 0.10), Volume 1 is 10 liters, Concentration 2 is 4% (or 0.04), and Volume 2 is unknown.
By solving the equation step by step, as shown in the initial solution, we find that 15 liters of water need to be added to achieve the desired 4% concentration. It's essential to follow these calculations closely to ensure accuracy in the dilution process. The concept is easy, but it requires careful arithmetic to avoid mistakes.