Question

In the United States, alcohol is rated according to "proof", which is usually defined as twice the percent by volume of pure ethanol in solution, measured at \(60^{\circ} \mathrm{F}\). What is the molarity of ethanol in a "92 proof' solution of ethanol in water. The density of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right.\), molecular weight \(=\) \(46 \mathrm{~g} /\) mole \()\) is \(0.80 \mathrm{~g} / \mathrm{cm}^{3}\) at \(60^{\circ} \mathrm{F} ;\) the density of water \(\left(\mathrm{H}_{2} \mathrm{O}\right.\), molecular weight \(=18 \mathrm{~g} /\) mole \()\) is \(1.0 \mathrm{~g} / \mathrm{cm}^{3}\) at \(60^{\circ} \mathrm{F}\).

Short answer

The molarity of ethanol in the "92 proof" solution is approximately 8 M.

Step-by-step solution

Find the percentage of ethanol in the solution

Since "proof" is twice the percent by volume of pure ethanol in the solution, we can find the ethanol percentage by dividing the "proof" by 2. In this case, the solution is "92 proof," so we have: Ethanol percentage = 92 / 2 = 46%

Find the mass of ethanol in 1 L of the solution

We know that the density of ethanol is 0.80 g/cm³ and that the density of water is 1.0 g/cm³. First, we need to find the volume of ethanol and water in 1 L solution. Since 1 L = 1000 cm³, we can express the ethanol and water volumes as follows: Ethanol volume = (46/100) * 1000 = 460 cm³ Water volume = 1000 - 460 = 540 cm³ Now let's find the mass of ethanol: Ethanol mass = Density of ethanol * Ethanol volume Ethanol mass = 0.80 g/cm³ * 460 cm³ ≈ 368 g

Find the moles of ethanol in the solution

We can now find the moles of ethanol by dividing the mass of ethanol by its molecular weight: Moles of ethanol = Ethanol mass / Molecular weight of ethanol Moles of ethanol = 368 g / 46 g/mol ≈ 8 mol

Determine the molarity of ethanol in the solution

Finally, we'll find the molarity of ethanol by dividing the moles of ethanol by the total volume of the solution in liters: Molarity of ethanol = Moles of ethanol / Total volume of solution in liters Molarity of ethanol = 8 mol / 1 L = 8 M So, the molarity of ethanol in the "92 proof" solution is 8 M.

Key concepts

Ethanol Proof

When discussing alcoholic beverages, you often hear the term "proof." This was originally a way to verify the alcohol strength by mixing the spirit with gunpowder and seeing if it would ignite. In modern times, especially in the United States, proof is streamlined to be exactly twice the percentage of ethanol, or pure alcohol, by volume.

For example, a "92 proof" spirit implies that it consists of 46% ethanol by volume. This relationship is simply expressed as:

• Proof = 2 x Ethanol (%) by volume

Understanding proof can be essential not only in determining the strength of an alcoholic drink but also in calculating other properties of the solution, such as molarity.

Density of Ethanol

Density is a measure that relates the mass of a substance to its volume. For ethanol, this value is crucial in many calculations due to its fluid properties. At a specific temperature (in this case, 60°F), the density of ethanol is about 0.80 g/cm³.

This means that in every 1 cm³ of pure ethanol, there is a mass of 0.80 grams. Density has real applications when figuring out substances in a mixture since it aids in converting volume-based measures (like a percentage volume) to mass-based measures, which can then be used in further calculations such as determining the quantity of a component in moles.

• Density = Mass/Volume

Molecular Weight

Molecular weight is a fundamental concept in chemistry, guiding us on how much one mole of any substance weighs. For ethanol, whose chemical formula is C₂H₅OH, the molecular weight is determined by summing the atomic masses of its atoms.

• Carbon (C) = 12 g/mol
• Hydrogen (H) = 1 g/mol
• Oxygen (O) = 16 g/mol

Adding these up for ethanol:

• 2 Carbon = 24 g/mol
• 6 Hydrogen = 6 g/mol
• 1 Oxygen = 16 g/mol

The total molecular weight of ethanol (C₂H₅OH) is 46 g/mol. Knowing the molecular weight is crucial for calculating the number of moles from a given mass, providing a stepping stone to find concentrations like molarity.

Solution Concentration

In chemistry, understanding the concentration of solutions is pivotal in many processes and reactions. Concentration can be expressed in various ways, but molarity (M) is among the most common. Molarity represents the number of moles of solute (the substance being dissolved) per liter of solution.

For example, a solution with a molarity of 8 M contains 8 moles of the solute per liter of total solution. Molarity allows for consistent and easy calculations across chemical equations because it standardizes the amount of reactive substance within a given volume.

The formula to calculate molarity is:

• Molarity = Moles of solute / Liters of solution

Calculating molarity requires comprehending the total number of moles, easily derived from mass and molecular weight, and knowing the total volume of the solution in liters.