Problem 112 - Chemistry Matter and Change

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.

Step-by-Step Solution

Answer

The bottle contains 0.2368 grams of dextromethorphan.

Step 1: Convert Fluid Ounces to Milliliters

First, convert the total volume of the cough syrup in the bottle from fluid ounces to milliliters. Since the bottle contains 4 fluid ounces and 1 fluid ounce is equivalent to 29.6 mL, multiply these values: \(4 \text{ fluid ounces} \times 29.6 \frac{\text{mL}}{\text{fluid ounce}} = 118.4 \text{ mL}\). So, the entire bottle contains 118.4 mL of syrup.

Step 2: Convert Teaspoons to Milliliters

Next, convert the standard dose for an adult from teaspoons to milliliters. Each dose is 2 teaspoons, where 1 teaspoon equals 5.0 mL. Therefore, \(2 \text{ teaspoons} \times 5.0 \frac{\text{mL}}{\text{teaspoon}} = 10.0 \text{ mL}\). The standard adult dose is 10.0 mL.

Step 3: Calculate Milligrams of Dextromethorphan per mL

Determine how much dextromethorphan is present per milliliter of syrup. A single dose of 10.0 mL contains 20.0 mg of dextromethorphan. Thus, divide the milligrams by the dose volume to find the concentration: \(\frac{20.0 \text{ mg}}{10.0 \text{ mL}} = 2.0 \text{ mg/mL}\). Each milliliter of syrup contains 2.0 mg of dextromethorphan.

Step 4: Calculate Total Milligrams of Dextromethorphan in Bottle

Now, calculate the total amount of dextromethorphan in the entire bottle by multiplying the total milliliters of the syrup by the concentration: \(118.4 \text{ mL} \times 2.0 \frac{\text{mg}}{\text{mL}} = 236.8 \text{ mg}\). The bottle contains 236.8 mg of dextromethorphan.

Step 5: Convert Milligrams to Grams

Finally, convert the total dextromethorphan in milligrams to grams. Since 1 gram is equal to 1000 milligrams, divide the total milligrams by 1000: \(\frac{236.8 \text{ mg}}{1000} = 0.2368 \text{ g}\). Therefore, the bottle contains 0.2368 grams of dextromethorphan.

Key Concepts

Unit Conversion

When dealing with different units of measure, it's crucial to convert them into a common framework for easy understanding.
Conversions serve a pivotal role in calculating medication dosages correctly. In this exercise, you need to translate the volume from everyday measurement units—like fluid ounces and teaspoons—into milliliters, which is the metric standard.

• To convert fluid ounces to milliliters, multiply the number of fluid ounces by the conversion factor 29.6 mL/fluid ounce.
• Likewise, convert teaspoons to milliliters using the factor 5.0 mL/teaspoon.

By standardizing your measurements to milliliters, the calculations become more straightforward, reducing the risk of error.

Concentration Determination

Concentration refers to the amount of a substance within a set volume. It's a critical concept in pharmacy and chemistry as it determines how much of the active ingredient is available in each dose.
For cough syrup, concentration is determined by dividing the amount of dextromethorphan (in milligrams) by the volume of syrup (in milliliters) that contains it.

• For example, 20 mg of dextromethorphan distributed over 10 mL of syrup results in a concentration of 2 mg/mL.

Understanding concentration helps in accurately computing how much medication each milliliter possesses, ensuring precise dosing.

Cough Syrup Composition

Cough syrup is a liquid medication formulated to suppress coughs. Its composition consists of an active ingredient, which in this case is dextromethorphan, dissolved within a syrup base. The syrup not only mediates the dose but also enhances palatability, making it easier to swallow.
In this exercise, you're tasked with discerning how much active ingredient—dextromethorphan—is present in the entire bottle.

• This involves calculating the total volume of the syrup and understanding the concentration of dextromethorphan per milliliter.
• Careful measurement and conversion ensure each dosage administered is both safe and effective.

Comprehension of cough syrup composition is fundamental for effective medication administration.

Metric Measurements in Chemistry

The metric system is widely used in scientific fields like chemistry because of its ease of conversion and global standardization.
In developing a better understanding of the task at hand, we used metric units like milliliters and milligrams to calculate the quantity of dextromethorphan in the syrup.

• Milliliters (mL) are used to measure liquid volume, making them perfect for dosing liquid medications.
• Milligrams (mg) are used for weighing substances, ideal for measuring active ingredients like dextromethorphan.
• It's simple to convert between these measurements using proportional factors: 1000 mg makes up a gram, and 1000 mL makes a liter.

Using metric units allows us to arrive at precise, reliable results, which is crucial in medical and scientific contexts.