Question

What volume of \(0.25 M \mathrm{HCl}\) solution must be diluted to prepare \(1.00 \mathrm{~L}\) of \(0.040 \mathrm{M} \mathrm{HCl} ?\)

Short answer

To prepare \(1.00 L\) of a \(0.040 M \mathrm{ HCl}\) solution, \(0.16 L\) of the \(0.25 M \mathrm{ HCl}\) solution is needed.

Step-by-step solution

Write down the dilution formula

To find the initial volume of the \(0.25 M \mathrm{ HCl}\) solution needed, we will use the dilution formula: $$C_1 V_1 = C_2 V_2$$

Plug in the known values

We'll now substitute the given values into the equation: $$0.25 \mathrm{M} \cdot V_1 = 0.040 \mathrm{M} \cdot 1.00 \mathrm{L}$$

Solve for V1

To find the initial volume, \(V_1\), we'll rearrange and simplify the equation: $$V_1 = \frac{0.040 \mathrm{M} \cdot 1.00 \mathrm{L}}{0.25 \mathrm{M}}$$

Calculate the volume of the initial solution required

Now, we'll perform the calculation: $$V_1 = \frac{0.040\cancel{\mathrm{M}} \cdot 1.00 \mathrm{L}}{0.25\cancel{\mathrm{M}}} = \frac{0.040 \mathrm{L}}{0.25} = 0.16 \mathrm{L}$$ So, to prepare 1.00 L of a 0.040 M HCl solution, we need 0.16 L of the 0.25 M HCl solution.

Key concepts

Dilution Formula

The dilution formula is a handy tool for chemists. It helps determine the volume needed of a concentrated solution to get a desired concentration after dilution. The general form of this formula is:

• \( C_1 V_1 = C_2 V_2 \)

Here, \( C_1 \) is the initial concentration, \( V_1 \) is the volume of the concentrated solution, \( C_2 \) is the final concentration, and \( V_2 \) is the final volume.
This equation tells us that the product of the concentration and volume before dilution is equal to the product of the concentration and volume after dilution. It is a result of the conservation of the amount of solute.
Knowing how to use this formula is crucial to achieving the desired concentration. Plugging in the given values will help solve for the unknown, making it straightforward to mix solutions accurately.

Molarity

Molarity is a way to express the concentration of a solution. It is defined as the number of moles of solute per liter of solution. The formula for calculating molarity is:

• \( ext{Molarity} (M) = \frac{ ext{moles of solute}}{ ext{liters of solution}} \)

Molarity gives us information about how much solute is present, relative to the volume of the solvent.
In the context of dilution, molarity helps us understand the initial and final concentrations of the solution. It's essential for determining how much of a concentrated solution is needed to obtain a solution with a lower molarity.
For example, in this exercise, we identify the molarity of the more concentrated solution (0.25 M) and the desired molarity after dilution (0.040 M). Knowing these values is vital for performing the dilution calculation accurately.

Volume Calculation

Calculating the volume of a solution needed before or after dilution depends on accurate use of the dilution formula. Solving for volume involves manipulating the dilution formula to find the unknown volume, \( V_1 \).
The equation \( V_1 = \frac{C_2 V_2}{C_1} \) is derived from the dilution formula. By using this version, we easily solve for \( V_1 \), which tells us the exact volume of the concentrated solution required.
It’s important to ensure all units are consistent during calculation, especially keeping concentrations in molarity (M) and volume in liters (L). This consistency ensures straightforward and error-free computation.

• Always re-check the setup of the formula to avoid common mistakes.
• Ensure given values are plugged in correctly.
• Perform calculations with care to get precise results.

Mastering volume calculation is essential for accurate solution preparation in laboratory settings.