Question

A solution is prepared by dissolving \(22.0 \mathrm{~g}\) of \(\mathrm{NaOH}\) in \(118.0 \mathrm{~g}\) of water. The \(\mathrm{NaOH}\) solution has a density of \(1.15 \mathrm{~g} / \mathrm{mL}\). a. What is the mass percent \((\mathrm{m} / \mathrm{m})\) of the \(\mathrm{NaOH}\) solution? b. What is the total volume (mL) of the solution? c. What is the molarity (M) of the solution?

Short answer

a. 15.71% b. 121.74 mL c. 4.52 M

Step-by-step solution

- Calculate Mass Percent

The mass percent (m/m) is calculated using the formula: \[\text{Mass percent} = \left(\frac{\text{mass of solute}}{\text{mass of solution}}\right) \times 100\%\]In this exercise, the solute is NaOH and the solvent is water.\[\text{Mass of NaOH} = 22.0 \text{ g}\]\[\text{Mass of water} = 118.0 \text{ g}\]\[\text{Mass of solution} = 22.0 \text{ g} + 118.0 \text{ g} = 140.0 \text{ g}\]Now plug in the values:\[\text{Mass percent} = \left(\frac{22.0}{140.0}\right) \times 100\% \approx 15.71\%\]

- Calculate Total Volume

The total volume of the solution can be found by using the density formula:\[\text{Density} = \frac{\text{mass}}{\text{volume}}\]Rearranged to find volume, we get:\[\text{Volume} = \frac{\text{mass}}{\text{density}}\]\[\text{Total mass of solution} = 140.0 \text{ g}\]\[\text{Density of solution} = 1.15 \text{ g/mL}\]\[\text{Volume} = \frac{140.0 \text{ g}}{1.15 \text{ g/mL}} \approx 121.74 \text{ mL}\]

- Calculate Molarity

Molarity (M) is calculated using the formula: \[\text{Molarity} = \frac{\text{moles of solute}}{\text{volume of solution in liters}}\]First, calculate the moles of NaOH:\[\text{Moles of NaOH} = \frac{\text{mass}}{\text{molar mass}}\]\[\text{Molar mass of NaOH} = 22.99 + 15.999 + 1.008 = 40.0 \text{ g/mol}\]\[\text{Moles of NaOH} = \frac{22.0 \text{ g}}{40.0 \text{ g/mol}} = 0.55 \text{ moles}\]Convert the volume from milliliters to liters:\[\text{Volume in liters} = \frac{121.74 \text{ mL}}{1000} \approx 0.1217 \text{ L}\]Finally, calculate the molarity:\[\text{Molarity} = \frac{0.55 \text{ moles}}{0.1217 \text{ L}} \approx 4.52 \text{ M}\]

Key concepts

Mass Percent

Mass percent is a way to express the concentration of a solution. It tells you what percentage of the total mass of the solution is made up of the solute. To find the mass percent, you use the formula: \[ \text{Mass percent} = \frac{\text{mass of solute}}{\text{mass of solution}} \times 100\text{\text{%}} \] In this exercise, the solute is NaOH and the solvent is water. First, you calculate the total mass of the solution by adding the mass of NaOH (22.0 g) to the mass of water (118.0 g), which gives you 140.0 g. Then you use the formula to find the mass percent: \[ \text{Mass percent} = \frac{22.0 \text{ g}}{140.0 \text{ g}} \times 100\text{\text{%}} \] This yields a mass percent of approximately 15.71\%. Simple, isn’t it? Understanding mass percent can help you compare the concentrations of different solutions easily.

Density

Density is a measure of how much mass is contained in a given volume. It is calculated using the formula: \[ \text{Density} = \frac{\text{mass}}{\text{volume}} \] For the NaOH solution in this exercise, we already know its density is 1.15 g/mL. To find the volume of the solution, you rearrange the formula to solve for volume: \[ \text{Volume} = \frac{\text{mass}}{\text{density}} \] Given that the total mass of the solution is 140.0 g, you can find the volume: \[ \text{Volume} = \frac{140.0 \text{ g}}{1.15 \text{ g/mL}} \] This calculation shows that the volume is approximately 121.74 mL. Knowing the density and volume of a substance is crucial in a variety of scientific calculations.

Molarity

Molarity is a measure of concentration that tells you how many moles of solute are present in one liter of solution. The formula for molarity is: \[ \text{Molarity} = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \] To find the molarity of the NaOH solution, first calculate the moles of NaOH using its mass (22.0 g) and molar mass (40.0 g/mol): \[ \text{Moles of NaOH} = \frac{22.0 \text{ g}}{40.0 \text{ g/mol}} = 0.55 \text{ moles} \] Next, convert the volume of the solution from milliliters to liters: \[ \frac{121.74 \text{ mL}}{1000} \] This gives you approximately 0.1217 L. Finally, use these values to find the molarity: \[ \text{Molarity} = \frac{0.55 \text{ moles}}{0.1217 \text{ L}} \] This results in a molarity of approximately 4.52 M. Molarity is extremely useful for comparing how concentrated different solutions are.

Moles of Solute

Understanding moles of solute is key to many concentration calculations. A mole is a measure of the amount of substance, based on Avogadro’s number, which is \[ 6.022 \times 10^{23} \] particles. To calculate the moles of a solute, you use its mass and molar mass. The formula is: \[ \text{Moles of solute} = \frac{\text{mass}}{\text{molar mass}} \] For the NaOH solution, the mass is 22.0 g, and the molar mass of NaOH is 40.0 g/mol. Plugging in these values gives you: \[ \text{Moles of solute} = \frac{22.0 \text{ g}}{40.0 \text{ g/mol}} = 0.55 \text{ moles} \] Knowing the number of moles helps you to understand the amount of substance you're working with, which is crucial for chemical reactions and solution preparation.

Volume of Solution

Volume is the amount of space that a substance or object occupies. In the context of solutions, knowing the volume is important for calculating concentrations like molarity. The volume of a solution can be calculated if you know its density and total mass. The formula for volume is: \[ \text{Volume} = \frac{\text{mass}}{\text{density}} \] From the exercise, the mass of the NaOH solution is 140.0 g and its density is 1.15 g/mL. Using the formula, you find the volume to be: \[ \text{Volume} = \frac{140.0 \text{ g}}{1.15 \text{ g/mL}} \] which is approximately 121.74 mL. Converting this to liters by dividing by 1000, you get roughly 0.1217 L. Understanding the volume is important for many laboratory procedures and theoretical calculations.