Question

A martini, weighing about 5.0 oz \((142 \mathrm{~g})\), contains \(30.0 \%\) by mass of alcohol. About \(15 \%\) of the alcohol in the martini passes directly into the bloodstream \((7.0 \mathrm{~L}\) for an adult). Estimate the concentration of alcohol in the blood \(\left(\mathrm{g} / \mathrm{cm}^{3}\right)\) of a person who drinks two martinis before dinner. (A concentration of \(0.00080 \mathrm{~g} / \mathrm{cm}^{3}\) or more is frequently considered indicative of intoxication in a "normal" adult.)

Short answer

The blood volume of the adult is 7.0 L. Answer: To find the concentration of alcohol in the blood after drinking two martinis, we need to follow these steps: 1. Calculate the mass of alcohol in one martini: 142 g x 30% = 42.6 g 2. Find the total mass of alcohol consumed: 42.6 g x 2 = 85.2 g 3. Determine the mass of alcohol absorbed: 85.2 g x 15% = 12.78 g 4. Calculate the volume of alcohol in the bloodstream: 12.78 g / 0.789 g/cm³ ≈ 16.2 cm³ 5. Determine the alcohol concentration in the blood: 16.2 cm³ / 7000 cm³ ≈ 0.00232 or 0.232% The concentration of alcohol in the adult's blood after drinking two martinis is approximately 0.232%.

Step-by-step solution

Find the mass of alcohol in one martini

The martini weighs 142 g, and it contains 30.0% of alcohol by mass. So, to find the mass of alcohol in one martini, we will multiply the total mass with the percentage of alcohol. \(Mass_{alcohol} = Mass_{martini} \times \%_{alcohol} = 142 \mathrm{~g} \times 30.0 \% = 142 \mathrm{~g} \times 0.3\)

Calculate the total mass of alcohol consumed

Since the person drinks two martinis before dinner, we need to find the total mass of alcohol consumed by multiplying the mass of alcohol in one martini by 2. \(Total\_Mass_{alcohol} = 2 \times Mass_{alcohol} = 2 \times 142 \mathrm{~g} \times 0.3 \)

Find the mass of alcohol that passes directly into the bloodstream

We are given that 15% of the alcohol in the martini passes directly into the bloodstream. So, we need to find the mass of alcohol absorbed in the bloodstream by multiplying the total mass of alcohol consumed with the percentage of alcohol absorbed. \(Absorbed\_Mass_{alcohol} = Total\_Mass_{alcohol} \times \%_{absorbed} = 2 \times 142 \mathrm{~g} \times 0.3 \times 0.15\)

Calculate the volume of alcohol in the bloodstream

To calculate the volume of alcohol in the bloodstream, we can use the formula \(Volume_{alcohol} = \dfrac{Absorbed\_Mass_{alcohol}}{Density_{alcohol}}\) We know that the density of alcohol \(\rho_{alcohol}=0.789 \mathrm{~g/cm^{ 3}}\) \(Volume_{alcohol}=\dfrac{2 \times 142 \mathrm{~g} \times 0.3 \times 0.15}{0.789 \mathrm{~g/cm^{3}}}\)

Determine the concentration of alcohol in the blood

To find the concentration of alcohol in the blood, we can use the formula \(Concentration_{alcohol} = \dfrac{Absorbed\_Mass_{alcohol}}{Volume_{blood}}\) We are given that the adult's blood volume is 7.0 L, or 7000 cm³. \(Concentration_{alcohol} = \dfrac{2 \times 142 \mathrm{~g} \times 0.3 \times 0.15}{7000 \mathrm{~cm^{3}}}\)

Key concepts

Chemistry of Intoxication

Understanding the chemistry of intoxication involves knowing how alcohol interacts with the body and how its concentration in the blood can be quantified. When a person consumes alcoholic beverages, such as martinis, the alcohol (ethanol) is absorbed into the bloodstream through the stomach and small intestine. It is then distributed throughout the body and can affect the nervous system, resulting in impaired judgment, coordination, and reaction times.

Alcohol is a small molecule that is soluble in water, which means it can easily pass through cell membranes and enter the bloodstream. In the given exercise, we calculate the estimated blood alcohol concentration (BAC) after drinking two martinis. BAC is a measure of the amount of alcohol in the bloodstream and is used legally and medically to indicate levels of intoxication. It's important to note that several factors can influence BAC, such as body weight, gender, metabolism, and the rate of alcohol consumption. However, for the purposes of our calculation, we will focus on the mass of alcohol consumed and the volume of blood in an average adult.

Percent by Mass Calculation

The percent by mass, also known as mass percent, is a fundamental concept in chemistry that measures the concentration of a component in a mixture or solution. It is expressed as the mass of the solute (in this case, alcohol) divided by the total mass of the solution and then multiplied by 100 to obtain a percentage.

For our exercise, a martini contains 30.0% of alcohol by mass. Here's how you'd execute this calculation using a simple formula:\[\begin{equation}\%\text{{ by mass}} = \frac{{\text{{Mass of alcohol}}}}{{\text{{Total mass of martini}}}} \times 100\end{equation}\]Applying this formula to one martini weighing 142 g, we find the actual mass of alcohol present in one drink. By understanding the percent by mass, students can determine the quantity of a substance within a mixture, which is vital for many fields like pharmacology, material science, and, of course, the context of food and beverages.

Solution Concentration

Solution concentration is a quantitative measure of the amount of solute in a given volume of solvent. Blood alcohol concentration (BAC) is a specific example of solution concentration where the solute is alcohol and the solvent is blood. It is typically expressed in grams per cubic centimeter (g/cm³) or as a percentage.

To calculate the concentration of alcohol in the bloodstream from our exercise, we need to first find the mass of alcohol absorbed into the bloodstream and then relate it to the volume of blood in the body. The BAC can be calculated using the following formula:\[\begin{equation}\text{{BAC}} = \frac{{\text{{Mass of absorbed alcohol}}}}{{\text{{Volume of blood}}}}\end{equation}\]For a typical adult with 7.0 L of blood, after consuming two martinis, we divide the absorbed mass of alcohol by this volume, taking into account the density of alcohol to obtain the concentration in g/cm³. Such calculations are critical for toxicology, medical diagnosis, and forensic science, as they quantify the level of intoxication and can have legal implications as well.