Question

You have \(0.2\) liter of an acid solution whose acid concentration is 16 moles per liter. You want to dilute the solution with water so that its acid concentration is only 12 moles per liter. Use the given model to determine how many liters of water you should add to the solution.

Short answer

0.067 liters of water should be added to the solution.

Step-by-step solution

Identify Given Variables

The initial concentration of the acid solution is 16 moles/liter and its volume is 0.2 liters. The desired final concentration is 12 moles/liter.

Interpret the Problem Using a Formula

To alter the concentration of a solution by adding water, we will use the formula \( C_1V_1 = C_2V_2 \). Here, \(C_1 = 16 moles/liter\) (initial concentration), \(V_1 = 0.2 liters\) (initial volume), and \(C_2 = 12 moles/liter\) (final concentration). We want to find \(V_2\) (final volume).

Substitute the Values into the Formula

To obtain \(V_2\), the values of \(C_1\), \(V_1\) and \(C_2\) are substituted into the formula, which gives us \(16 moles/liter \times 0.2 liters = 12 moles/liter \times V_2\).

Calculate \(V_2\)

Solving for \(V_2\) in the equation yields \(V_2 = (16 moles/liter \times 0.2 liters) / 12 moles/liter = 0.267 liters\). This is the final volume of the solution.

Determine the Volume of Water to be Added

The volume of water added is the difference between the final volume and the initial volume: \(0.267 liters - 0.2 liters = 0.067 liters\)

Key concepts

Acid Concentration

Acid concentration tells us how much acid is present in a given volume of liquid. It is typically expressed in moles per liter. Essentially, it indicates the number of moles of acid that are dissolved in one liter of the solution.
For example, in our problem, the initial acid solution has a concentration of 16 moles per liter. This means that each liter of solution contains 16 moles of acid. Concentration is a crucial factor because it helps us understand the strength of the solution. Higher concentrations mean more acid is present, making the solution stronger.
Understanding acid concentration is key when diluting solutions. By adding water, the number of moles of acid stays the same, but the total volume of the solution increases, which lowers the concentration.

Moles per Liter

Moles per liter is a way to express concentration, also known as molarity. It refers to how many moles of a substance are present in one liter of the solution. The mole is a fundamental unit in chemistry, representing \(6.022 \times 10^{23}\) entities, usually atoms or molecules.
This unit is very useful when measuring the substances involved in chemical reactions because it allows chemists to discuss concentrations in practical terms rather than very large numbers.

• The initial acid solution in the problem has a concentration of 16 moles per liter, meaning 16 moles of acid are dissolved in every liter of the solution, making it a fairly concentrated solution.
• The final desired concentration is 12 moles per liter, which is a little less concentrated, indicating that we need to add water to dilute the original solution.

Understanding moles per liter helps us to precisely adjust the concentration of a solution by adding or removing a solvent, such as water.

Dilution Formula

The dilution formula, \C_1V_1 = C_2V_2\, is a pivotal tool in chemistry. It relates the initial and final concentrations and volumes of a solution. This equation is based on the principle that the number of moles of solute remains constant during the dilution process.

• \C_1\ is the initial concentration. In our case, it's 16 moles per liter.
• \V_1\ is the initial volume, 0.2 liters in this example.
• \C_2\ is the final desired concentration, 12 moles per liter.
• \V_2\ represents the final volume after dilution, which we must find.

By rearranging the formula, we solve for \(V_2\), integrating the initial conditions to find the final volume necessary for the desired concentration. This solution will guide us to determining the precise amount of water added to achieve the correct dilution.

Solution Volume

The solution volume refers to the total volume of the liquid that contains the dissolved solute, which in this case is acid. It is a significant factor in calculating concentrations. When we speak of solution volume in dilution problems, we are usually interested in how the volume changes as we add more solvent.
Initially, the solution volume was 0.2 liters. To decrease the concentration from 16 moles per liter to 12 moles per liter, we modified the solution volume.

• First, we calculated the new solution volume using the dilution formula, arriving at 0.267 liters, the total volume after adding water.
• Finally, the difference between the new volume (0.267 liters) and the original volume (0.2 liters) is the amount of water added, which was 0.067 liters.

Understanding and controlling the volume of a solution is essential to achieve the desired concentration, particularly when dealing with solutions in a laboratory setting.