Question

The term proof is defined as twice the percent by volume of pure ethanol in solution. Thus, a solution that is 95$\mathrm{\%}$ (by volume) ethanol is 190 proof. What is the molarity of ethanol in a 92 proof ethanol-water solution? Assume the density of ethanol, ${\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}$ , is 0.79 $\mathrm{g}/{\mathrm{cm}}^3$ and the density of water is 1.0 $\mathrm{g}/{\mathrm{cm}}^3$ .

Short answer

The molarity of ethanol in a 92 proof ethanol-water solution is approximately 7.888 mol/L.

Step-by-step solution

Calculate Percent by Volume of Ethanol in the Solution

First, we need to find the percent by volume of ethanol in the solution, which we can do by dividing the proof by two. Given: Proof = 92 Percent by Volume of Ethanol = (Proof) / 2 = $92/2$ = 46% Ethanol is 46% (by volume) in the solution.

Establish Ratios of Ethanol and Water

Since we now know that the solution is 46% (by volume) ethanol, we can also determine that it is 54% water. Let the volume of the solution = 100 cm³. Given: Volume of ethanol = 46 cm³ Volume of water = 54 cm³

Calculate Masses of Ethanol and Water

Using the given densities, we will now calculate the masses of ethanol and water in the solution. Given: Density of Ethanol = 0.79 g/cm³ Density of Water = 1.0 g/cm³ Mass of ethanol = (Volume of Ethanol) × (Density of Ethanol) = (46 cm³) × (0.79 g/cm³) = 36.34 g Mass of water = (Volume of Water) × (Density of Water) = (54 cm³) × (1.0 g/cm³) = 54 g

Calculate the moles of ethanol and water

We need to calculate the moles of ethanol and water based on their molecular weights. Given: Molecular Weight of Ethanol = 46.07 g/mol Molecular Weight of Water = 18.02 g/mol Moles of Ethanol = (Mass of Ethanol) / (Molecular Weight of Ethanol) = $36.34\text{g}/46.07\text{g/mol}$ = 0.7888 mol Moles of Water = (Mass of Water) / (Molecular Weight of Water) = $54\text{g}/18.02\text{g/mol}$ = 2.9978 mol

Calculate the Molarity of Ethanol in the Solution

Finally, we can calculate the molarity of the ethanol solution by dividing the moles of ethanol by the volume of the solution in liters: Given: Volume of Solution = 100cm³ = 0.1 L Molarity of Ethanol = (Moles of Ethanol) / (Volume of Solution in Liters) = $0.7888\text{mol}/0.1\text{L}$ = 7.888 mol/L The molarity of ethanol in a 92 proof ethanol-water solution is approximately 7.888 mol/L.

Key concepts

Ethanol Concentration

When discussing the concentration of ethanol in a solution, we're referring to how much ethanol is present compared to the total volume of the solution. In alcoholic beverages, this is often expressed as "proof," which is a measure used primarily in the United States to describe the strength of the alcohol. A 92 proof ethanol solution means it contains 46% ethanol by volume. This is calculated by dividing the proof number by two, as proof is twice the volume percent of ethanol in the solution. Understanding ethanol concentration is essential in chemistry because it affects the physical and chemical behaviors of the solution. The amount of ethanol can influence the solution's flammability, boiling point, and even how the solution interacts with other substances.

Density of Liquids

Density is a property that describes how much mass a substance has in a given volume and is usually expressed in grams per cubic centimeter ($ext{g/cm}^3$). For ethanol, the density is 0.79 $ext{g/cm}^3$, while for water, it's 1.0 $ext{g/cm}^3$. When you know the density of a liquid and its volume, you can calculate its mass by multiplying these two values. For example, if we know a solution is 46% ethanol by volume and the total volume is 100 $ext{cm}^3$, we can find the mass of the ethanol by multiplying 46 $ext{cm}^3$ by 0.79 $ext{g/cm}^3$, which results in a mass of 36.34 grams.This concept is crucial in a variety of real-world applications, from chemistry lab experiments to industrial processes, where precise measurements are necessary.

Volume Percent

Volume percent is a way of expressing the concentration of a component in a mixture or solution. It is defined as the volume of the component divided by the total volume of the mixture, multiplied by 100.In the case of alcoholic solutions like ethanol-water solutions, knowing the volume percent helps us understand how much alcohol is present. For example, a 46% volume percent of ethanol means that out of every 100 $ext{cm}^3$ of the solution, 46 $ext{cm}^3$ are pure ethanol.This measurement is widely used in both the beverage industry and chemical manufacturing because it's a straightforward way to discuss the composition of a solution. It also directly connects to practical applications, such as the proof measurement in alcoholic beverages.

Molecular Weight Calculations

To find the number of moles of a substance, you need to know its mass and its molecular weight (also known as molar mass). The molecular weight of a compound is the sum of the masses of all the atoms in a molecule.In our case, the molecular weight of ethanol $ext{C}_{2}ext{H}_{5}extOH$ is approximately 46.07 $extg/mol$. By dividing the mass of ethanol in the solution by its molecular weight, we can calculate the number of moles. For instance, with 36.34 grams of ethanol, we divide by 46.07 $extg/mol$, resulting in approximately 0.789 moles of ethanol.Understanding molecular weight calculations is fundamental in chemistry for preparing solutions with precise concentrations and for predicting the outcomes of chemical reactions, where it's important to know the exact amounts of reactants involved.