Question

Cough Syrup A common brand of cough syrup comes in a 4 -fluid ounce bottle. The active ingredient in the cough syrup is dextromethorphan. For an adult, the standard dose is 2 teaspoons, and a single dose contains 20.0 \(\mathrm{mg}\) of dextromethorphan. Using the relationships, 1 fluid ounce \(=29.6 \mathrm{mL}\) and 1 teaspoon = 5.0 \(\mathrm{mL}\) determine how many grams of dextromethorphan are contained in the bottle.

Short answer

The 4-fluid ounce bottle of cough syrup contains 0.22 grams of dextromethorphan.

Step-by-step solution

Convert fluid ounces to milliliters

We will first convert the 4-fluid ounce bottle into milliliters using the given conversion factor: 1 fluid ounce = 29.6 mL. So, we have: \(4 \text{ fluid ounces} \times 29.6 \frac{\text{mL}}{\text{fluid ounce}} = 118.4 \text{ mL}\)

Find the number of doses in the bottle

The standard dose for an adult is 2 teaspoons, and we are given the conversion factor: 1 teaspoon = 5.0 mL. To find out how many doses are in the bottle, we need to convert the volume of one dose from teaspoons to mL and then divide the total volume of the bottle by the volume of one dose. So, the volume of one dose in mL: \(2 \text{ teaspoons} \times 5.0 \frac{\text{mL}}{\text{teaspoon}} = 10 \text{ mL}\) Now, dividing the total volume of the bottle by the volume of one dose, we get the number of doses: \(\frac{118.4 \text{ mL}}{10 \text{ mL}} = 11.84 \text{ doses}\) Since dosing is usually in whole numbers, we'll round down to the nearest whole number: 11 doses.

Calculate the total amount of dextromethorphan in the bottle

We are given that one dose contains 20.0 mg of dextromethorphan. So, for 11 doses, we need to multiply the dextromethorphan content of one dose by the number of doses: \(11 \text{ doses} \times 20.0 \frac{\text{mg}}{\text{dose}} = 220 \text{ mg}\) Since we need to determine the dextromethorphan content in grams, we'll convert the value from mg to g using the conversion factor: 1 g = 1000 mg. \(\frac{220 \text{ mg}}{1000 \frac{\text{mg}}{\text{g}}} = 0.22 \text{ g}\)

Result

Therefore, the 4-fluid ounce bottle of cough syrup contains 0.22 grams of dextromethorphan.

Key concepts

Medicinal Chemistry

When it comes to medicinal chemistry, understanding the composition and dosage of drug formulations is crucial. Medicinal chemistry involves the design and chemical synthesis of pharmaceuticals, and a deep understanding of dosing, which is critical to both safety and effectiveness. Dextromethorphan, the active ingredient in the cough syrup from our exercise, serves as a cough suppressant. It is an excellent example of how a compound's pharmacological properties are utilized to relieve a specific symptom, in this case, coughing.

Knowing the amount of the active ingredient in a medication—and how it should be administered—is essential for achieving the desired therapeutic effect without causing harm. This exercise shows how to quantitatively determine the total amount of dextromethorphan in a bottle of cough syrup by unit conversions and chemical calculations, which are core skills in the realm of medicinal chemistry.

Unit Conversion

In many scientific calculations, especially in medicinal chemistry, unit conversion is a skill that cannot be overstated. In the context of medication, improper unit conversions can lead to ineffective dosages or, worse, overdoses. The exercise illustrates how to convert from fluid ounces, a common volume unit in the United States, to milliliters, part of the International System of Units (SI).

It's important to memorize or have access to the commonly used conversion factors, like 1 fluid ounce equals 29.6 milliliters, and to maintain precision during these conversions. Mistakes in these conversions could result in incorrect doses being administered. Always double-check calculations and ensure proper units are used throughout the process.

Chemical Calculations

Within the scope of chemical calculations, we are often tasked with determining the quantity of a specific substance within a mixture or solution. This is a fundamental aspect of pharmaceutical dosing. In our exercise, we performed a series of conversions and calculations to determine how much active ingredient is present in a whole bottle of cough syrup.

To be proficient in these calculations, one must be comfortable with mathematical operations, unit conversions, and an understanding of the physical properties and measurements involved. In the calculation, we transition from milliliters to teaspoons, milligrams to grams, and dosage per teaspoon to the total dosage in the bottle, emphasizing the importance of multi-step chemical calculations in medicinal chemistry.