Problem 65 - Chemistry: The Central Science in SI Units, Global Edition

Question

The concentration of alcohol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right)\) in blood, called the "blood alcohol concentration" or BAC, is given in units of grams of alcohol per \(100 \mathrm{~mL}\) of blood. The legal definition of intoxication, in many states of the United States, is that the BAC is 0.08 or higher. What is the concentration of alcohol, in terms of molarity, in blood if the BAC is \(0.08 ?\)

Step-by-Step Solution

Answer

The concentration of alcohol in the blood when the BAC is 0.08 is approximately \(1.74 \times 10^{-5} \text{ M}\).

Step 1: Calculate the Molecular Weight of Alcohol

The molecular weight of alcohol can be calculated by adding up the weights of all the atoms in the molecule, which is: Molecular weight of alcohol = Molecular weight of C + Molecular weight of C + Molecular weight of H × 3 + Molecular weight of H × 2 + Molecular weight of O + Molecular weight of H - Molecular weight of C = 12.01 g/mol - Molecular weight of H = 1.01 g/mol - Molecular weight of O = 16.00 g/mol So, the molecular weight of alcohol is: \(12.01 + 12.01 + 3×1.01 + 2×1.01 + 16.00 + 1.01 = 46.07 \text{ g/mol}\)

Step 2: Calculate the Mass of Alcohol in Blood

The given BAC value is 0.08, which means there are 0.08 grams of alcohol per 100 mL of blood. We need to convert the volume of blood to liters: \(0.08 \frac{\text{g}}{100 \text{ mL}} × \frac{1 \text{ L}}{1000 \text{ mL}} = 0.0008 \frac{\text{g}}{\text{L}}\)

Step 3: Convert the Mass to Moles

Now, we can convert the mass of alcohol to moles using the molecular weight calculated in Step 1: \(\text{Moles of alcohol} = \frac{\text{Mass of alcohol}}{\text{Molecular weight of alcohol}}\) \(\text{Moles of alcohol} = \frac{0.0008 \text{ g}}{46.07 \text{ g/mol}} = 1.736×10^{-5} \text{ mol}\)

Step 4: Calculate the Molarity of Alcohol in Blood

Finally, the molarity is the number of moles of solute (alcohol) per liter of solvent (blood). Since we already have calculated the moles of alcohol (1.736 × 10^{-5} mol) and the blood volume in liters (1 L), we can now calculate the molarity: Molarity = \(\frac{\text{Moles of alcohol}}{\text{Volume of blood in liters}}\) Molarity = \(\frac{1.736 \times 10^{-5} \text{ mol}}{1 \text{ L}} = 1.736 \times 10^{-5}\text{ M}\) Therefore, the concentration of alcohol in the blood when the BAC is 0.08 is approximately \(1.74 \times 10^{-5} \text{ M}\).

Key Concepts

Blood Alcohol Concentration (BAC)

Blood Alcohol Concentration, commonly abbreviated as BAC, is a metric used to measure the concentration of alcohol in a person's bloodstream. It is typically expressed in grams of alcohol per 100 milliliters ( ext{mL}) of blood. For example, a BAC of 0.08 means there are 0.08 grams of alcohol in every 100 ext{mL} of blood.

BAC is crucial for legal and medical purposes because it helps determine alcohol impairment levels. In many jurisdictions, a BAC of 0.08 or higher is considered legally intoxicated.

• This means that a person with a BAC at or above this threshold is considered impaired and may be subjected to legal consequences, such as fines or imprisonment, if caught driving.

Understanding your BAC can also help you manage your drinking habits better, ensuring that you stay within safe and legal limits. Factors affecting BAC include the amount of alcohol consumed, body weight, and the rate at which a person metabolizes alcohol.

Molecular Weight Calculation

Molecular weight is an essential concept in chemistry that refers to the sum of the atomic weights of all atoms in a molecule. It is usually expressed in grams per mole ( ext{g/mol}). Calculating the molecular weight allows us to convert mass into moles, which is a crucial step in various chemical calculations.

For ethanol ( ext{CH}_3 ext{CH}_2 ext{OH}), the molecular weight is calculated as follows:

• Atomic weight of Carbon ( ext{C}): 12.01 ext{g/mol}
• Atomic weight of Hydrogen ( ext{H}): 1.01 ext{g/mol}
• Atomic weight of Oxygen ( ext{O}): 16.00 ext{g/mol}

To find the molecular weight of ethanol, we add up the atomic weights:

12.01 + 12.01 + 3×1.01 + 2×1.01 + 16.00 + 1.01 = 46.07 ext{ g/mol}

This calculation tells us that one mole of ethanol weighs 46.07 grams, which is crucial for further calculations involving moles and concentrations.

Conversion to Moles

Converting mass to moles is a fundamental process in chemistry that allows chemists to understand and quantify chemical reactions. Moles, being a basic unit in chemistry, reflect the number of molecules or atoms present in a given sample.

To convert mass to moles, we use the formula:

\[ ext{Moles} = \frac{ ext{Mass}}{ ext{Molecular Weight}}\]

In our exercise, with 0.0008 grams of ethanol and a molecular weight of 46.07 ext{g/mol}, the number of moles is calculated as:

\[ ext{Moles of ethanol} = \frac{0.0008 ext{ g}}{46.07 ext{ g/mol}} = 1.736 imes 10^{-5} ext{ mol}\]

Finding the moles is a crucial step to then calculate molarity. Molarity is simply the moles of solute (ethanol, in this case) per liter of solution. Understanding these conversions ensures accurate and meaningful application in lab settings and practical scenarios.