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.

Step by step solution

01

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.

02

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.

03

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 \(96.5 \, \mathrm{mL} = 0.0965 \, \mathrm{Liters}\).

04

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 \(0.857 \, \mathrm{moles} / 0.0965 \, \mathrm{Liters} = 8.88 \, \mathrm{M}\).

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