Question

Eighty proof whiskey is \(40 \%\) ethanol \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}\right)\) by volume. A man has \(7.0 \mathrm{~L}\) of blood and drinks \(28 \mathrm{~mL}\) of the whiskey, of which \(22 \%\) of the ethanol goes into his blood. (a) What concentration (in \(\mathrm{g} / \mathrm{mL}\) ) of ethanol is in his blood \((d\) of ethanol \(=0.789 \mathrm{~g} / \mathrm{mL}\) )? (b) What volume (in \(\mathrm{mL}\) ) of whiskey would raise his blood alcohol level to \(8.0 \times 10^{-4} \mathrm{~g} / \mathrm{mL},\) the level at which a person is considered intoxicated?

Short answer

The ethanol concentration in blood is approximately 2.78 × 10^{-4} g/mL. To reach 8.0 × 10^{-4} g/mL, 80.7 mL of whiskey is required.

Step-by-step solution

Calculate the Volume of Ethanol in Whiskey

The whiskey is 40% ethanol by volume. Hence, the volume of ethanol in 28 mL of whiskey is:\[ V_{ethanol} = 0.40 \times 28 \text{ mL} = 11.2 \text{ mL} \]

Determine Ethanol Absorbed into Blood

Only 22% of the ethanol consumed goes into the blood. Therefore, the volume of ethanol absorbed is:\[ V_{absorbed} = 0.22 \times 11.2 \text{ mL} = 2.464 \text{ mL} \]

Convert Volume to Mass of Ethanol

Use the density of ethanol to convert the volume absorbed into mass:\[ m_{ethanol} = V_{absorbed} \times d_{ethanol} = 2.464 \text{ mL} \times 0.789 \text{ g/mL} = 1.944 \text{ g} \]

Calculate Concentration of Ethanol in Blood

The blood volume is 7.0 liters, which is 7000 mL. The concentration of ethanol in the blood is then:\[ C_{ethanol} = \frac{m_{ethanol}}{V_{blood}} = \frac{1.944 \text{ g}}{7000 \text{ mL}} \approx 2.78 \times 10^{-4} \text{ g/mL} \]

Calculate Required Mass of Ethanol for Intoxication

To reach the intoxication level of 8.0 × 10^{-4} g/mL, the total mass of ethanol needed in the blood is:\[ m_{required} = C_{intoxication} \times V_{blood} = 8.0 \times 10^{-4} \text{ g/mL} \times 7000 \text{ mL} = 5.6 \text{ g} \]

Determine Total Volume of Ethanol Needed

Convert the required mass of ethanol back to volume:\[ V_{required} = \frac{m_{required}}{d_{ethanol}} = \frac{5.6 \text{ g}}{0.789 \text{ g/mL}} \approx 7.1 \text{ mL} \]

Calculate Total Volume of Whiskey Needed

Considering that 22% of ethanol is absorbed and whiskey is 40% ethanol, the required volume of whiskey is:\[ V_{whiskey} = \frac{7.1 \text{ mL}}{0.22 \times 0.40} \approx 80.7 \text{ mL} \]

Key concepts

Ethanol Absorption

Ethanol absorption is the process by which ethanol (alcohol) passes from the consumed liquid into the bloodstream. When you drink whiskey, not all of the ethanol gets absorbed into your blood immediately. In this exercise, only 22% of the ethanol is absorbed. To determine the volume of ethanol absorbed from a 28 mL drink of 40% ethanol whiskey, we multiply the volume of ethanol in the whiskey by the absorption rate:

\[ V_{absorbed} = 0.22 \times V_{ethanol} \] Remember, volume absorbed is a fraction of the total volume of ethanol present in the drink.

Concentration Calculation

Determining the concentration of ethanol in the blood involves calculating how much ethanol is in a given volume of blood. Concentration is usually expressed in grams per milliliter (g/mL). To find it, you first need the total mass of ethanol absorbed into the blood and then divide that by the blood volume. For example:

\[ C_{ethanol} = \frac{m_{ethanol}}{V_{blood}} \] This formula helps you understand how the ethanol in your bloodstream increases with more consumption.

Density and Mass Conversion

Density is a measure of how much mass is present in a given volume of a substance. For ethanol, it is often given as 0.789 g/mL. To convert the volume of ethanol into mass, use the following formula:

\[ m_{ethanol} = V_{absorbed} \times d_{ethanol} \] This relationship is crucial when you want to understand how much actual ethanol you have taken into your body from the liquid consumed.

Volume Conversion

Volume conversion is essential when dealing with different units, such as liters to milliliters. Understanding that 1 liter equals 1000 milliliters is fundamental in accurately deriving values needed in concentration and density calculations. For instance, converting blood volume from liters to milliliters helps in more straightforward calculations:

\[ 7.0 \text{ L} = 7000 \text{ mL} \] Such conversions ensure precision and avoid errors in the calculations of mass, volume and concentration.

Proof Whiskey

Proof is a measure of the strength of alcoholic beverages such as whiskey. Eighty proof means the whiskey contains 40% ethanol by volume. This is a standard measure used to calculate the amount of alcohol in a drink. When given the proof, you can easily determine the ethanol content:

\[ V_{ethanol} = 0.40 \times V_{whiskey} \] Thus, understanding proof helps in accurately estimating the alcohol intake from a beverage.

Intoxication Level

The intoxication level is the ethanol concentration in the blood at which a person is considered impaired or drunk. A common benchmark is 0.08% (or 8.0 × 10^-4 g/mL). To determine how much more whiskey a person can drink before reaching this level, calculate the additional amount of ethanol needed and then convert it back to the volume of whiskey:

\[ V_{required} = \frac{m_{required}}{0.22 \times 0.40} \] This tells you the quantity of whiskey needed to attain the legal intoxication threshold.