Question

Blood alcohol content (BAC) is sometimes reported in weight-volume percent and, when it is, a BAC of \(0.10 \%\) corresponds to \(0.10 \mathrm{g}\) ethyl alcohol per \(100 \mathrm{mL}\) of blood. In many jurisdictions, a person is considered legally intoxicated if his or her BAC is 0.10\%. Suppose that a 68 kg person has a total blood volume of 5.4 L and breaks down ethyl alcohol at a rate of 10.0 grams per hour. \(^{*}\) How many 145 mL glasses of wine, consumed over three hours, will produce a BAC of \(0.10 \%\) in this 68 kg person? Assume the wine has a density of \(1.01 \mathrm{g} / \mathrm{mL}\) and is \(11.5 \%\) ethyl alcohol by mass. (* The rate at which ethyl alcohol is broken down varies dramatically from person to person. The value given here for the rate is a realistic, but not necessarily accurate, value.)

Short answer

The person needs to drink about 2.1 glasses of wine over three hours to produce a BAC of 0.10\%.

Step-by-step solution

Determine the amount of alcohol in one glass

First, determine the amount of alcohol in one 145 mL glass of wine. The wine is 11.5\% ethyl alcohol by mass and has a density of 1.01 g/mL. So, the mass of the wine is \(145 \, \text{mL} \times 1.01 \, \text{g/mL} = 146.45 \, \text{g}\). The amount of alcohol is then \(146.45 \, \text{g} \times 0.115 = 16.84225 \, \text{g}\).

Calculate the total amount of alcohol broken down

Next, determine the total amount of alcohol the person's body breaks down over the three hours, which is \(10.0 \, \text{g/hour} \times 3 \, \text{hours} = 30.0 \, \text{g}\).

Calculate BAC

In order to produce a BAC of 0.10\%, the person needs to have 0.10 g of alcohol per 100 mL of blood in their body after three hours. Since the person has 5.4 L = 5400 mL of blood, the total amount of alcohol to produce a BAC of 0.10\% is \(5400 \, \text{mL} \times 0.10 \, \text{g/100 mL} = 5.4 \, \text{g}\).

Calculate the number of glasses of wine

The total amount of alcohol the person needs to consume, which is not broken down, is the amount that produces a BAC of 0.10\% plus the amount broken down, i.e., \(5.4 \, \text{g} + 30.0 \, \text{g} = 35.4 \, \text{g}\). The number of glasses of wine needed to consume 35.4 g of alcohol is \(35.4 \, \text{g} / 16.84225 \, \text{g/glass} = 2.1 \, \text{glasses}\).

Key concepts

Ethyl Alcohol Metabolism

When you consume alcohol, your body processes it primarily through the liver. Ethyl alcohol metabolism involves a series of enzymatic reactions. Alcohol is broken down into acetaldehyde by alcohol dehydrogenase (ADH) and further into acetic acid by aldehyde dehydrogenase (ALDH). This process allows your body to eventually get rid of alcohol, mainly through urine and breath.

Several factors influence alcohol metabolism:

• Body weight: Larger individuals often metabolize alcohol more slowly.
• Alcohol dehydrogenase levels: Genetic differences can cause varying levels of these enzymes, affecting alcohol metabolism speed.
• Drinking frequency: Regular drinkers may have a slightly enhanced enzyme activity.

In this exercise, it's assumed that the body breaks down 10 grams of ethyl alcohol per hour, which is a rough average. However, you should remember that this rate can vary significantly from one person to another, influencing how quickly one becomes sober after drinking.

Wine Alcohol Content

Wine varies significantly in its alcohol content, which is primarily influenced by its type and fermentation process. For this exercise, the wine is specified to contain 11.5% ethyl alcohol by mass. This means that, out of 100 grams of wine, 11.5 grams are pure alcohol.

To determine how much alcohol is present in a specific wine serving, two key parameters are considered:

• Density of the wine: Given as 1.01 g/mL in this scenario.
• Volume of the serving: Here, a standard glass contains 145 mL.

By calculating the mass of a single wine glass and subsequently applying the alcohol percentage, you can find how much alcohol each glass contains. This calculation is essential to determine how many glasses will lead to a certain blood alcohol content (BAC). It’s important to understand this fundamental concept for responsible drinking and legal considerations.

Density and Mass Calculations

Density is a critical concept in calculating the alcohol content in liquids like wine. It refers to how much mass is contained in a unit volume, expressed as g/mL for this problem. With the wine's density of 1.01 g/mL, you start by finding the mass of one glass:\[\text{Mass of wine} = \text{Volume} \times \text{Density} = 145 \, \text{mL} \times 1.01 \, \text{g/mL} = 146.45 \, \text{g}\]Understanding density aids in translating volume measurements into mass. From there, you can further explore how much actual alcohol is present using the percentage given.Once you have the total mass of the wine, multiplying by the alcohol content percentage gives you the mass of ethyl alcohol:\[\text{Mass of Ethyl Alcohol} = 146.45 \, \text{g} \times 0.115 = 16.84225 \, \text{g}\]This step effectively bridges the gap from the measure in milliliters (volume) to the measure in grams (mass), crucial for all subsequent BAC-related calculations. Hence, density not only grounds these calculations but begins the journey toward solving BAC-related problems.