Question

The concentration of a solution of \(\mathrm{HCl}\) is \(33.1 \%\) by mass, and its density was measured to be \(1.147 \mathrm{~g} / \mathrm{mL}\). How many milliliters of the \(\mathrm{HCl}\) solution are required to obtain \(10.0 \mathrm{~g}\) of \(\mathrm{HCl}\) ?

Short answer

Approximately \(28.6 \mathrm{~mL}\) of the \(\mathrm{HCl}\) solution is required to obtain \(10.0 \mathrm{~g}\) of \(\mathrm{HCl}\).

Step-by-step solution

Calculate the mass of \(\mathrm{HCl}\) per milliliter of the solution

We know that the concentration of a solution of \(\mathrm{HCl}\) is given as \(33.1\%\) by mass. This means that there are \(33.1 \mathrm{~g}\) of \(\mathrm{HCl}\) in \(100 \mathrm{~g}\) of the solution. We also know the density of this solution is given as \(1.147 \mathrm{~g/mL}\). Therefore, the mass of \(1 \mathrm{~mL}\) of the solution is equal to \(1.147 \mathrm{~g}\). Now, we can find the mass of \(\mathrm{HCl}\) in \(1 \mathrm{~mL}\) of the solution by using the percentage concentration: \(\mathrm{HCl}\) mass per \(1 \mathrm{~mL}\) of the solution = \(\frac{33.1 \mathrm{~g} \ of \ \mathrm{HCl}}{100 \mathrm{~g} \ of \ solution} \times 1.147 \mathrm{~g} \ of \ solution\)

Find the required volume to obtain \(10.0 \mathrm{~g}\) of \(\mathrm{HCl}\)

Let \(V\) be the volume in milliliters we want to find. We know that \(10.0 \mathrm{~g}\) of \(\mathrm{HCl}\) is required, so we can set up a proportion: \(\frac{\mathrm{HCl \ mass \ per \ mL} }{1 \mathrm{~mL}} = \frac{10.0 \mathrm{~g} \ of \ \mathrm{HCl}}{V \mathrm{~mL}}\) We can then solve for \(V\): \(V = \frac{10.0 \mathrm{~g} \ of \ \mathrm{HCl} \cdot 1 \mathrm{~mL}}{ \mathrm{HCl \ mass \ per \ mL}}\)

Calculate the volume V

Now we just substitute the value of \(\mathrm{HCl \ mass \ per \ mL}\) from Step 1: \(V = \frac{10.0 \mathrm{~g} \ of \ \mathrm{HCl} \cdot 1 \mathrm{~mL}}{\frac{33.1 \mathrm{~g} \ of \ \mathrm{HCl}}{100 \mathrm{~g} \ of \ solution} \times 1.147 \mathrm{~g} \ of \ solution}\) After solving for \(V\), we get: \(V \approx 28.6 \ \mathrm{mL}\) Hence, approximately \(28.6 \mathrm{~mL}\) of the \(\mathrm{HCl}\) solution is required to obtain \(10.0 \mathrm{~g}\) of \(\mathrm{HCl}\).

Key concepts

Molarity and Molality

Understanding the concentration of solutions is crucial in chemistry, and two key terms often encountered are molarity and molality. Molarity, denoted as M, is the number of moles of solute per liter of solution. It's calculated using the formula:
\[\begin{equation}M = \frac{\text{moles of solute}}{\text{liters of solution}}\end{equation}\]
On the other hand, molality, denoted as m, is the number of moles of solute per kilogram of solvent. Its formula is:
\[\begin{equation}m = \frac{\text{moles of solute}}{\text{kilograms of solvent}}\end{equation}\]
While both terms describe the concentration of a solution, molarity is temperature-dependent since volume changes with temperature, whereas molality is temperature-independent, as mass remains constant regardless of temperature. In the given example, if the problem had asked for molarity or molality, we would first determine the moles of HCl, which requires knowing the molar mass of HCl. Then we'd need the volume in liters to find molarity or the mass of the solvent in kilograms for molality.

Mass Percent

Mass percent is a common way to express the concentration of a component in a solution. It is simply the mass of the solute divided by the total mass of the solution, multiplied by 100 to get a percentage. The formula looks like this:
\[\begin{equation}\text{Mass percent} = \left(\frac{\text{mass of solute}}{\text{total mass of solution}}\right) \times 100\end{equation}\]
This concept is essential in the given exercise, where the concentration of HCl is provided as 33.1% by mass. This means that for every 100 grams of solution, there are 33.1 grams of HCl. To improve understanding of this concept, consider different solution masses and calculate the corresponding mass of HCl to visualize the linear relationship between the total mass of solution and the mass of solute.

Solution Density

Solution density is another fundamental concept, defined as the mass of the solution per unit volume. It is commonly expressed in grams per milliliter (g/mL) or kilograms per liter (kg/L). The formula can be written as:
\[\begin{equation}\text{Density} = \frac{\text{mass of solution}}{\text{volume of solution}}\end{equation}\]
Knowing the density of the solution is useful when you need to convert between mass and volume, as seen in the example. With a density of 1.147 g/mL for the HCl solution, we can determine that each mL of the solution weighs 1.147 grams. This information allows us to calculate how many milliliters of solution are needed to obtain a certain mass of HCl. Remember, the density of a solution can change with temperature and concentration, so it's important to use the density corresponding to the conditions of the problem.