Question

If \(12.00 \mathrm{~L}\) of a \(6.00 \mathrm{M} \mathrm{HNO}_{3}\) solution needs to be diluted to \(0.750 \mathrm{M}\), what will be its final volume?

Short answer

The final volume is 96.00 L.

Step-by-step solution

Understanding the Problem

We need to dilute a 12.00 L solution of 6.00 M HNO₃ to a concentration of 0.750 M. We aim to find the final volume of the solution after dilution.

Writing the Dilution Formula

Dilution problems can be solved using the formula: \( C_1V_1 = C_2V_2 \), where \( C_1 \) and \( V_1 \) are the initial concentration and volume, and \( C_2 \) and \( V_2 \) are the final concentration and volume, respectively.

Plugging in Known Values

We have \( C_1 = 6.00 \) M, \( V_1 = 12.00 \) L, and \( C_2 = 0.750 \) M. Plug these values into the formula: \( 6.00 \times 12.00 = 0.750 \times V_2 \).

Solving for the Unknown Volume

Solve for \( V_2 \): \[V_2 = \frac{6.00 \times 12.00}{0.750}\]Calculate this to find: \[ V_2 = 96.00 \text{ L} \].

Key concepts

Molarity

Molarity is a measure of the concentration of a solute in a solution. It is defined as the number of moles of solute per liter of solution. The formula used to calculate molarity is:

• \( M = \frac{n}{V} \)

where:

• \( M \) is the molarity,
• \( n \) is the number of moles of solute,
• \( V \) is the volume of solution in liters.

Understanding molarity is crucial when it comes to dilution as it helps determine how much solvent needs to be added to achieve the desired concentration. For example, in an initial solution of 6.00 M HNO₃, this means there are 6.00 moles of HNO₃ per liter of solution. The process of dilution involves adding more solvent to decrease this concentration, ensuring we achieve a more diluted solution.
Knowing the molarity of solutions allows us to precisely control and use them in chemical reactions or experiments.

Dilution Formula

The dilution formula is a simplified equation used to determine the concentration of a solution after it has been diluted. The formula is:

• \( C_1V_1 = C_2V_2 \)

where:

• \( C_1 \) is the initial concentration,
• \( V_1 \) is the initial volume,
• \( C_2 \) is the final concentration,
• \( V_2 \) is the final volume.

This formula essentially describes the conservation of moles of solute in the solution. The amount of solute before dilution (\( C_1V_1 \)) equals the amount of solute after dilution (\( C_2V_2 \)). By rearranging the formula, you can solve for any one of the variables if the other three are known. In our exercise, we knew the initial molarity and volume, as well as the desired reduced concentration (\( C_2 = 0.750 \) M), allowing us to solve for the final volume. Utilizing the formula correctly ensures that the solution is prepared to the needed specification for its intended use.

Volume Calculation

Volume calculation in dilution is a straightforward process once you understand the dilution formula. With the variables established:

• \( C_1 = 6.00 \) M,
• \( V_1 = 12.00 \) L,
• \( C_2 = 0.750 \) M,

you insert the known values into the equation \( C_1V_1 = C_2V_2 \):

• \( 6.00 \times 12.00 = 0.750 \times V_2 \)

From here, isolate \( V_2 \) to solve for the unknown volume of the final diluted solution:

• \( V_2 = \frac{6.00 \times 12.00}{0.750} \)

Upon calculation, \( V_2 = 96.00 \) liters. This means the final volume of the solution needed to reach the desired concentration of 0.750 M is 96.00 L. The calculation of volume is essential as it tells us exactly how much more solvent needs to be added to achieve the required dilution for specific applications or experiments.