Problem 7 - Chemistry for Allied Health

Question

For a solution that contains \(15.0 \mathrm{~mL}\) of methanol \(125 \mathrm{~mL}\) of ethanol, find each of the following. The density of methanol is \(0.792 \mathrm{~g} / \mathrm{mL}\) and the density of ethanol is \(0.789 \mathrm{~g} / \mathrm{mL}\). a. mass percent b. mass/volume percent c. volume percent

Step-by-Step Solution

Answer

a. Mass percent: ≈ 10.75% b. Mass/volume percent: ≈ 8.49% c. Volume percent: ≈ 10.71%

Step 1: Calculate Mass of Methanol

First, calculate the mass of methanol using its volume and density. The formula to use is:\[\text{Mass} = \text{Volume} \times \text{Density}\]Substitute the given values:\[\text{Mass of Methanol} = 15.0 \, \text{mL} \times 0.792 \, \text{g/mL} = 11.88 \, \text{g}\]

Step 2: Calculate Mass of Ethanol

Next, calculate the mass of ethanol using its volume and density with the same formula:\[\text{Mass of Ethanol} = 125 \, \text{mL} \times 0.789 \, \text{g/mL} = 98.625 \, \text{g}\]

Step 3: Calculate Total Mass of the Solution

Add the masses of methanol and ethanol to find the total mass of the solution:\[\text{Total Mass} = 11.88 \, \text{g} + 98.625 \, \text{g} = 110.505 \, \text{g}\]

Step 4: Calculate Mass Percent of Methanol

The mass percent of methanol is calculated using the formula:\[\text{Mass Percent} = \left(\frac{\text{Mass of Methanol}}{\text{Total Mass}}\right) \times 100\%\]Substituting the values:\[\text{Mass Percent of Methanol} = \left(\frac{11.88}{110.505}\right) \times 100\% \approx 10.75\%\]

Step 5: Calculate Mass/Volume Percent of Methanol

The mass/volume percent is given by:\[\text{Mass/Volume Percent} = \left(\frac{\text{Mass of Methanol}}{\text{Volume of Solution}}\right) \times 100\%\]The total volume is the sum of methanol and ethanol volumes:\[\text{Total Volume} = 15.0 \, \text{mL} + 125 \, \text{mL} = 140.0 \, \text{mL}\]Substituting the values:\[\text{Mass/Volume Percent} = \left(\frac{11.88}{140.0}\right) \times 100\% \approx 8.49\%\]

Step 6: Calculate Volume Percent of Methanol

The volume percent is calculated using:\[\text{Volume Percent} = \left(\frac{\text{Volume of Methanol}}{\text{Total Volume}}\right) \times 100\%\]Substitute the values:\[\text{Volume Percent of Methanol} = \left(\frac{15.0}{140.0}\right) \times 100\% \approx 10.71\%\]

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume. It tells us how tightly packed the particles of a substance are within a specific space. The formula for density is written as:\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]To use this formula, simply divide the mass of the substance by its volume. For example, with the density of methanol being \(0.792 \, \text{g/mL}\), it means every milliliter of methanol contains \(0.792\) grams of mass. Similarly, ethanol has a density of \(0.789 \, \text{g/mL}\). Knowing the densities allows for the calculation of each substance's mass when its volume is known.
By multiplying the volume of a liquid by its density, you can find its mass, which is crucial for further calculations like determining percentages in a solution.

Mass/Volume Percent

The mass/volume percent is a way to express the concentration of a component in a solution. It combines the mass of one part of the solution with the total volume of the solution. The formula to calculate the mass/volume percent is:\[\text{Mass/Volume Percent} = \left(\frac{\text{Mass of Component}}{\text{Volume of Solution}}\right) \times 100\%\]For instance, in the case of methanol and ethanol, you need to first calculate the mass of methanol, which is done using its density and volume. Once you have this, divide it by the total volume of the solution (which is the sum of volumes of methanol and ethanol) and multiply by 100 to convert it into a percentage.
This gives an easy-to-understand representation of how much of the solution's volume is comprised of methanol based on its mass.

Volume Percent

Volume percent is a measure used to express how much of a solution is made up of a particular liquid based purely on volume. This is especially useful when dealing with liquid mixtures. The formula for calculating volume percent is:\[\text{Volume Percent} = \left(\frac{\text{Volume of Component}}{\text{Total Volume of Solution}}\right) \times 100\%\]To find the volume percent, simply divide the volume of the specific component (like methanol in the given exercise) by the total volume of the entire solution, which here is methanol plus ethanol. Then multiply the result by 100 to convert it into a percentage form.
This percentage provides a simple and clear way of understanding the proportion that a particular liquid occupies in a mixture, which can be important for various applications requiring specific liquid ratios.