Question

The term "proof," still used to describe the ethanol content of alcoholic beverages, originated in seventeenthcentury England. A sample of whiskey was poured on gunpowder and set afire. If the gunpowder ignited after the whiskey had burned off, this "proved" that the whiskey had not been watered down. The minimum ethanol content for a positive test was about \(50 \%\) by volume. The \(50 \%\) ethanol solution became known as \(^{\prime \prime} 100\) proof." Thus, an 80 -proof whiskey would be \(40 \% \mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\) by volume. Listed in the table below are some data for several aqueous solutions of ethanol. With a minimum amount ofcalculation, determine which of the solutions are more than 100 proof. Assume that the density of pure ethanol is \(0.79 \mathrm{g} / \mathrm{mL}\).$$\begin{array}{cl} \hline \text { Molarity of Ethanol, } M & \text { Density of Solution, g/mL } \\\ \hline 4.00 & 0.970 \\\5.00 & 0.963 \\\6.00 & 0.955 \\\7.00 & 0.947 \\\8.00 & 0.936 \\\9.00 & 0.926 \\\10.00 & 0.913 \\\\\hline\end{array}.$$

Short answer

After calculating the volume percentage for each solution using the given molarity and solution density, solutions with a percentage greater than 50% are more than 100 proof.

Step-by-step solution

Understand proof

The term 'Proof' is used to describe the ethanol content of alcoholic beverages. A 100-proof whiskey has 50% alcohol by volume. Therefore, to determine if a solution is more than 100 proof, we need to calculate if its alcohol content is more than 50% by volume.

Calculate the volume percent of ethanol

The volume percentage of ethanol can be calculated from its molarity and the density of the solution using the formula: \(V \% = \frac{(Molarity \times Molar mass \times 1000)}{(Density \times 10)}\% where Molar mass of ethanol, \(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\), is 46.07 g/mol. Assume the density of pure ethanol is 0.79 g/mL. The 4.00 M solution, for example, would be calculated as: \(V \% = \frac{(4.00 mol/L \times 46.07 g/mol) \times 1000}{0.970 g/mL \times 10}\% \)

Compare with 100 proof

Given that 100 proof equals 50% ethanol by volume, any solution with a percent volume greater than 50% would be more than 100 proof. Apply the proof determination to all solution densities.

Repeat steps 2 and 3

Repeat steps 2 and 3 for the remaining molarities and compare their volume percents to determine if they are more than 100 proof.

Key concepts

Molarity

Understanding molarity is key when determining ethanol content in alcoholic beverages. Molarity measures the concentration of a solute in a solution and is expressed in moles per liter (mol/L). In our exercise, we are given different molarity values for ethanol solutions. To understand this, think of molarity as the number of ethanol molecules present in a given volume of the solution.

When molarity is provided, such as 4.00 M, it means that there are 4.00 moles of ethanol in every liter of the solution. Using molarity, you can calculate various properties of the solution, such as the volume percent of ethanol, which is crucial for determining whether a solution is over 100 proof.

In summary, molarity helps us know how much ethanol is in a given volume of solution, essential for further calculations. It's a foundational concept necessary to compute the volume percent of ethanol, which we will discuss later.

Density

Density is another essential element when analyzing ethanol content in alcoholic solutions. Density is defined as the mass per unit volume of a substance. In the exercise, densities of several ethanol solutions are given. These values are necessary to understand how much ethanol is mixed in water, affecting whether the solution is more than 100 proof.

For instance, the density of a 4.00 M ethanol solution is provided as 0.970 g/mL. This means that each milliliter of this solution weighs 0.970 grams. Density here serves as a bridge to assess the mass and subsequently the volume percent of ethanol in the solution.

• Higher densities usually indicate less space available for ethanol molecules, implying a greater mass of ethanol per volume.
• The densities of solutions were lower in higher molarity, which impacts the calculation of volume percent.

Understanding density helps us interpret how much of the solution is composed of ethanol, guiding us toward calculating if the solution is more than 100 proof.

Alcohol Proof

Alcohol proof is a historical term used to quantify the strength of alcoholic drinks. Specifically, alcohol proof is twice the percentage of alcohol by volume. This means 100-proof alcohol contains 50% alcohol by volume.

To determine if an alcoholic beverage is more than 100 proof, we need to establish if its ethanol content surpasses 50% volume percent. For instance, a whiskey stated as 80 proof contains 40% ethanol by volume. This is below the 100 proof standard.

• Calculating proof allows us to compare the ethanol content across different drinks.
• It is important to remember that proof numbers are double the ethanol volume percentage.

In the context of our exercise, comparing the computed volume percent from molarity and density to the 50% threshold helps us definitively determine if the solution is "more than 100 proof."

Volume Percent

Volume percent is an important concept when determining the ethanol content in beverages and differs slightly from alcohol proof. Volume percent refers to the concentration of a component in a mixture, specifically how much ethanol there is by volume in the solution.

To calculate the volume percent of ethanol from its molarity, we use the formula:\[\text{Volume Percent} = \frac{(\text{Molarity} \times \text{Molar Mass} \times 1000)}{\text{Density} \times 10}\%\]For ethanol, with a molar mass of 46.07 g/mol and known densities of the solution, you can compute the volume percent. For example, if a 4.00 M ethanol solution has a density of 0.970 g/mL, the volume percent calculation is \[\frac{(4.00 \text{ mol/L} \times 46.07 \text{ g/mol}) \times 1000}{0.970 \text{ g/mL} \times 10}\%\]

• Knowing the volume percent informs us how much space ethanol occupies in the mixture.
• It is crucial for verifying whether a solution exceeds the 50% ethanol mark, correlating directly to determining if it's over 100 proof.

Understanding volume percent helps connect how ethanol spreads into the solution, allowing us to better grasp the strength of the beverage.