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.)

Step by step solution

01

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 \(146.45 \, \text{g} \times 0.115 = 16.84225 \, \text{g}\).

02

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 \(10.0 \, \text{g/hour} \times 3 \, \text{hours} = 30.0 \, \text{g}\).

03

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 \(5400 \, \text{mL} \times 0.10 \, \text{g/100 mL} = 5.4 \, \text{g}\).

04

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 \(35.4 \, \text{g} / 16.84225 \, \text{g/glass} = 2.1 \, \text{glasses}\).

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