Question

The concentration of alcohol \(\left(\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{OH}\right)\) in blood, called the "blood alcohol concentration" or \(\mathrm{BAC}\), 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 \(\mathrm{BAC}\) is \(0.08\) or higher. What is the concentration of alcohol, in terms of molarity, in blood if the \(\mathrm{BAC}\) is \(0.08 ?\)

Short answer

The concentration of alcohol in blood when the BAC is 0.08 is approximately 0.0174 mol/L.

Step-by-step solution

Calculate the molar mass of ethanol

First, we need to find the molar mass of ethanol (CH\(_3\)CH\(_2\)OH). The molar mass is the sum of the atomic masses of all the atoms in the molecule. The molar mass of carbon (C) is 12, hydrogen (H) is 1, and oxygen (O) is 16. Therefore, the molar mass of ethanol is: $$\mathrm{Molar~ Mass~ of~ Ethanol}= 2 \times 12 + 6 \times 1 + 1 \times 16 = 24 + 6 + 16 = 46\mathrm{~g/mol}$$

Convert the BAC value to grams per liter

The given BAC value is in grams of alcohol per 100 mL of blood. Since 1 L = 1000 mL, we can convert it to grams per liter by multiplying by 10: $$0.08 \frac{\mathrm{g}}{100 \mathrm{~ml}} \times \frac{1\mathrm{~L}}{1000\mathrm{~mL}} \times \frac{1000}{1}= 0.8 \frac{\mathrm{g}}{\mathrm{L}}$$ The concentration of alcohol in blood is 0.8 g/L.

Calculate the molarity from concentration and molar mass

Now, we can use the definition of molarity to find the molarity of alcohol in blood. Molarity is defined as moles of solute per liter of solution. So we need to divide the concentration in grams per liter (found in Step 2) by the molar mass of ethanol (found in Step 1): $$\mathrm{Molarity} = \frac{\mathrm{Concentration~(g/L)}}{\mathrm{Molar~Mass~(g/mol)}}$$ $$\mathrm{Molarity} = \frac{0.8~\mathrm{g/L}}{46~\mathrm{g/mol}} = 0.0174\mathrm{~mol/L}$$ Therefore, the concentration of alcohol in blood when the BAC is 0.08 is approximately 0.0174 mol/L.

Key concepts

Molarity Calculation

Understanding molarity is essential when dealing with solutions in chemistry. In simple terms, molarity indicates the concentration of a solute in a solution. To calculate the molarity, use the formula:
\[\[\begin{align*}\text{Molarity (M)} = \frac{\text{moles of solute}}{\text{liters of solution}}\end{align*}\]\]
This means dividing the number of moles of the substance by the volume of the solution in liters. When it comes to blood alcohol concentration (BAC), the concept is applied in the same way. The unit of molarity is moles per liter (mol/L), sometimes referred to as 'molar'. By knowing the gram per liter of the alcohol and its molar mass, you can compute its molarity in the blood, helping medical professionals understand the level of intoxication in an individual.

Conversion of Units

Converting units is a fundamental skill in science, ensuring that all measurements are consistent and comparable. When we discuss blood alcohol concentration, it's usually given in grams per 100 milliliters. However, molarity requires the concentration in grams per liter. To convert from g/100mL to g/L, you need to multiply by 10, as there are 1,000 milliliters in a liter.

This conversion is imperative because it aligns the BAC measurement with the molar mass of ethanol in grams per mole, setting the stage to calculate molarity accurately. Consistency in units across the calculation is vital to obtaining a correct and meaningful result.

Molecular Mass of Ethanol

The molecular mass of a compound indicates the sum of the masses of all atoms in a molecule and is vital for converting between moles and grams. For ethanol (\(\mathrm{C}_2\mathrm{H}_5\mathrm{OH}\)), calculating its molecular mass involves summing the atomic masses of carbon (C), hydrogen (H), and oxygen (O). Each carbon atom contributes 12 g/mol, each hydrogen 1 g/mol, and oxygen 16 g/mol. Typically, the molecular mass of ethanol is:

\[\[\begin{align*}\text{Molecular Mass of Ethanol (C}_2\text{H}_5\text{OH)} &= 2(\text{C}) + 6(\text{H}) + 1(\text{O}) \&= 2(12 \text{ g/mol}) + 6(1 \text{ g/mol}) + 16 \text{ g/mol} \&= 46 \text{ g/mol}\end{align*}\]\]
This value is crucial for the final step of calculating molarity since it represents the mass of one mole of ethanol. Understanding the molecular mass helps in converting from the mass of ethanol that one may have consumed to the number of moles of ethanol present in the bloodstream.