Question

Water and ethanol, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}(\mathrm{l}),\) are miscible, that is, they can be mixed in all proportions. However, when these liquids are mixed, the total volume of the resulting solution is not equal to the sum of the pure liquid volumes, and we say that the volumes are not additive. For example, when \(50.0 \mathrm{mL}\) of water and \(50.0 \mathrm{mL}\) of \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}(\mathrm{l}),\) are mixed at \(20^{\circ} \mathrm{C},\) the total volume of the solution is \(96.5 \mathrm{mL}\), not \(100.0 \mathrm{mL}\). (The volumes are not additive because the interactions and packing of water molecules are slightly different from the interactions and packing of \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) molecules.) Calculate the molarity of \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) in a solution prepared by mixing \(50.0 \mathrm{mL}\) of water and \(50.0 \mathrm{mL}\) of \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}(\mathrm{l})\) at \(20^{\circ} \mathrm{C} .\) At this temperature, the densities of water and ethanol are 0.99821 \(\mathrm{g} / \mathrm{mL}\) and \(0.7893 \mathrm{g} / \mathrm{mL},\) respectively.

Short answer

The molarity of ethanol in the solution at 20°C is approximately 8.88 M.

Step-by-step solution

Calculate the mass of ethanol

Firstly, calculate the mass of ethanol which is the mass per volume (density) times the volume. For ethanol, this would be \(0.7893 \, \mathrm{g/mL} \times 50.0 \, \mathrm{mL} = 39.465 \, \mathrm{g}\) of ethanol.

Determine the number of moles of ethanol

The molar mass of ethanol is about 46.07 g/mol. Therefore, determine the number of moles of ethanol by dividing total mass of ethanol by the molar mass of ethanol. This will be \(39.465 \, \mathrm{g} / 46.07 \, \mathrm{g/mol} = 0.857 \, \mathrm{moles}\) of ethanol.

Calculate the volume of the solution

Add the volumes of water and ethanol that are mixed. But, the volumes are not additive as some space is lost when mixing. Therefore, the total volume is not 100.0 mL but 96.5 mL. Convert this volume to liters. This becomes \(96.5 \, \mathrm{mL} = 0.0965 \, \mathrm{Liters}\).

Calculate the molarity of ethanol

Finally, to determine the molarity, divide the number of moles of ethanol by the volume of the solution in liters. This gives the molarity of ethanol as \(0.857 \, \mathrm{moles} / 0.0965 \, \mathrm{Liters} = 8.88 \, \mathrm{M}\).

Key concepts

Density of Liquids

Density is a fundamental property of liquids that relates mass to volume. Specifically, it measures how much mass of a substance exists in a unit volume, typically expressed in grams per milliliter (g/mL) for liquids. To compute the density of a liquid, the formula is: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\] This means that when you know the density of a liquid and the volume, you can calculate its mass by rearranging the formula to: \[\text{Mass} = \text{Density} \times \text{Volume}\] In the example of ethanol given in the original problem, with a density of 0.7893 g/mL and a volume of 50.0 mL, the mass is computed as follows: \[0.7893 \, \text{g/mL} \times 50.0 \, \text{mL} = 39.465 \, \text{g}\] Understanding density is crucial as it affects how substances interact when mixed, which leads us to explore concepts where volumes are not additive.

Volumes Not Additive

The property that certain liquid combinations, like water and ethanol, exhibit is known as non-additive volumes. Typically, when you sum up volumes of two separate pure liquids, you would expect the total volume to equal the individual volumes added together. However, this isn’t always the case. When two liquids mix, their particles rearrange into a new, preferably more compact structure due to molecular interactions. This rearrangement can lead to a contraction in the volume. In our case, mixing 50.0 mL of water and 50.0 mL of ethanol results in only 96.5 mL of a solution, not the expected 100.0 mL. This discrepancy occurs because:

• Molecules of different liquids may fill the intermolecular spaces more efficiently, causing a reduction in total volume.
• The interactions between different molecules may differ in strength compared to those within the same kind of molecules, influencing how closely the molecules pack.

Understanding that volumes are not strictly additive is important in calculating accurate concentrations, such as molarity, which depends on knowing the precise volume of the solution.

Miscibility of Ethanol and Water

Miscibility refers to the ability of two substances to mix and form a uniform solution. Ethanol and water are fully miscible, meaning they can mix in any ratio to form a single phase without separating. This property is due to their molecular structure and the types of bonding they can form. Ethanol (\(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\)), like water, forms hydrogen bonds due to the presence of an OH group. When mixed, these molecules are attracted to one another, allowing for a thorough and complete mixing. The miscibility of these two substances is crucial when you want to create solutions with specific concentrations on a molecular level:

• Hydrogen bonding is the primary reason behind this miscibility. The polar nature of both molecules allows them to dissolve into one another readily.
• This is unlike oil and water, which are not miscible due to oil being non-polar and therefore not capable of forming hydrogen bonds with water.

Understanding miscibility is key when preparing chemical solutions, as it affects homogeneity and the accurate calculation of solution properties, including the molarity of solutes such as ethanol.