Question

The molecular weight of ethanol (CH₃CH₂OH) is 46 and its density is 0.789 g/cm^3.

^{a. What is the molarity of ethanol in beer that is 5% ethanol by volume? [Alcohol content of beer varies from about 4% (lite beer) to 8% (stout)].}

^{b. The legal limit for a driver's blood alcohol content varies, but 80 mg of ethanol per 100 ml of blood (usually referred to as a blood alcohol level of 0.08) is typical. What is the molarity of ethanol in a person at the legal limit?}

^{c. How many 12-oz (355 ml) bottles of 5% alcohol beer could a 70 kg person drink and remain under the legal limit? A 70 kg person contains about 40 liters of water. Ignore the metabolism of ethanol, and assume that the water content of the person remains constant.}

^{d. Ethanol is metabolized at a constant rate of about 120 mg per hour per kg body weight regardless of its concentration. If a 70 kg person were at twice the legal limit (160 mg/100 ml), how long would it take for their blood alcohol level to fall below the legal limit?}

Answer 1

Ethanol:
MW = 46
Density = 0.789 g/ml

a.
5% = 5 mL in 100 mL
5 mL = 5 X 0.789 = 3.945 g

M = (3.945/46)(1000/100)
= 0.8576

b.
80 mg (= 0.08 g) of ethanol in 100 mL

M = (0.08/46)(1000/100)
= 1.73 X 10^-2

c.
Maximum number of moles of ethanol a person can have,
= 0.0173 X 40
= 0.692 M

Molarity = Moles per litre
Number of litres of beer required = 0.692/0.8576
= 0.806 L
= 806 ml
= 806/355 = 2.27 bottles

d.
Twice the legal limit = 160 mg/dL
Volume of blood = 40000 mL
Amount of alcohol = (160 mg/100 mL)(40000 mL)
= 64000 mg

Legal limit = (80 mg/100 mL)(40000 mL)
= 32000 mg

So,
Amount of alcohol to be metabolized = 64000 - 32000 = 32000 mg
Metabolic rate = 120 mg/hr/kg
So,
A 70 kg person metabolizes 70 X 120 = 7200 mg/hr

= 32000/7200
= 4.44 hrs