Question

The alcohol content of hard liquor is normally given in terms of the "proof," which is defined as twice the percentage by volume of ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) present. Calculate the number of grams of alcohol present in 1.00 \(\mathrm{L}\) of 75 -proof gin. The density of ethanol is \(0.798 \mathrm{~g} / \mathrm{mL}\).

Short answer

The gin contains 299.25 grams of ethanol.

Step-by-step solution

Understand Proof Definition

The problem states that proof is twice the percentage by volume of ethanol. Therefore, if gin is 75-proof, then the percentage by volume of ethanol is half of that, which is 37.5%.

Convert Volume Percentage to Volume

Since 37.5% of the volume is ethanol, in 1.00 L of gin, the volume of ethanol is 0.375 L (since 37.5% of 1.00 L = 0.375 L).

Convert Liters to Milliliters

Convert the volume of ethanol in liters to milliliters for easier calculation with density. Thus, 0.375 L is equivalent to 375 mL (since 1 L = 1000 mL).

Calculate Mass Using Density

Use the density of ethanol to find the mass: \[\text{mass} = \text{density} \times \text{volume} = 0.798\, \text{g/mL} \times 375\, \text{mL} = 299.25\, \text{g}.\]

Key concepts

Proof of Alcohol

The term 'proof' is pivotal when discussing alcohol content. It is essentially a measure used to denote the strength of an alcoholic beverage. This term takes its roots from historical methods where liquor was "proved" to be of a certain potency by different means, such as burning. However, in contemporary practice, proof is simply defined as twice the percentage of alcohol by volume (ABV). For instance, if a liquor is labeled as 75-proof, it means the alcoholic content is 37.5% by volume. This conversion from proof to percentage is important for various calculations involving alcohol content, such as finding the exact amount of ethanol in a beverage.

Density of Ethanol

Density is a physical property that represents the mass per unit volume of a substance. For ethanol, a common pure alcohol, the density is given as 0.798 g/mL. This value is crucial when converting between mass and volume, such as when determining how much ethanol is present in a given volume of a beverage. Knowing the density allows us to calculate how many grams of ethanol are in any specified volume by using the formula:

• Density = Mass / Volume

To find the mass from volume, we rearrange the formula:

• Mass = Density × Volume

This conversion is key for various practical purposes, including mixing and selling alcoholic products.

Volume to Mass Conversion

The process of converting volume to mass in calculations involves knowing both the volume and the density of the substance in question. Let's say we have a volume of ethanol measured in milliliters, and we want to find out the mass in grams. Using the density of ethanol (0.798 g/mL), you follow these steps:

• Convert any volume measurements to milliliters if not already.
• Multiply the volume in milliliters by the density (g/mL) to get the mass in grams.

This method ensures accurate results because density provides the relationship between the mass and volume of the substance. Thus, you can reliably calculate how much mass corresponds to any particular volume of ethanol.

Ethanol Percentage by Volume

The ethanol percentage by volume is a standard way to measure how much of a beverage is composed of ethanol. This percentage directly influences the 'proof' of the alcohol. When calculating ethanol content in drinks, it's common to first determine the percentage by volume from the proof value. For example, with a 75-proof beverage, the ethanol percentage is half of that, i.e., 37.5%. From there, this percentage can be used to find out how much of a total volume is ethanol.

• Percentage by Volume = (Proof Value) / 2
• Volume of Ethanol = (Percentage / 100) × Total Volume

Such calculations help in accurately assessing the strength and nature of alcoholic products, providing necessary insights for consumption and product labeling.