Question

Lipitor, a pharmaceutical drug that has been shown to lower "bad" cholesterol levels while raising "good" cholesterol levels in patients taking the drug, had over \(\$ 11\) billion in sales in 2006 . Assuming one \(2.5\) -g pill contains \(4.0 \%\) of the active ingredient by mass, what mass in \(\mathrm{kg}\) of active ingredient is present in one bottle of 100 . pills?

Short answer

First, find the mass of the active ingredient in one pill: \(2.5 \times 0.040 = 0.1 \text{g}\). Next, calculate the mass of the active ingredient in one bottle: \(0.1 \times 100 = 10 \text{g}\). Finally, convert the mass to kg: \(\frac{10}{1000} = 0.01 \text{kg}\). So, the mass of the active ingredient in one bottle of 100 pills is \(0.01 \mathrm{kg}\).

Step-by-step solution

Calculate the mass of the active ingredient in one pill.

To calculate the mass of the active ingredient in one pill, we need to multiply the mass of one pill by the percentage of the active ingredient. We are given that one pill has a mass of 2.5g and that the active ingredient is 4.0% of the mass of one pill. So, the mass of the active ingredient in one pill can be calculated as: \[ \text{mass of active ingredient in one pill} = (\text{mass of one pill}) \times (\text{percentage of active ingredient}) \]

Calculate the mass of the active ingredient in one bottle.

Now that we have the mass of the active ingredient in one pill, we can find the mass of the active ingredient in one bottle containing 100 pills by multiplying the mass of the active ingredient in one pill by 100: \[ \text{mass of active ingredient in one bottle} = (\text{mass of active ingredient in one pill}) \times 100 \]

Convert the mass of the active ingredient in one bottle to kg.

We need to convert the mass of the active ingredient in one bottle, which is in grams, to kilograms. To do this, we will divide the mass in grams by 1000: \[ \text{mass of active ingredient in one bottle (kg)} = \frac{\text{mass of active ingredient in one bottle (g)}}{1000}\] Now, let's put everything together and calculate the mass of the active ingredient in one bottle in kg.

Key concepts

Chemical Mass Calculation

Understanding chemical mass calculation is crucial in fields like pharmacy where precision is essential. It involves determining the amount of a substance by using its mass and the properties of its components. In the context of pharmaceuticals, calculating the mass of an active ingredient is vital for dosage accuracy and patient safety.

Let's consider a scenario where we need to calculate the mass of the active ingredient in a medication. The mass of one pill is given, and we know the percentage that the active ingredient contributes to this mass. The formula for calculating the mass of the active ingredient in one pill would be: \[\begin{equation}\text{mass of active ingredient in one pill} = (\text{mass of one pill}) \times (\text{percentage of active ingredient})\end{equation}\]
If a pill weighs 2.5g and the active ingredient contributes 4.0% of this mass, we use the formula above to determine the mass of the active ingredient per pill. This basic step in chemical mass calculation serves as the foundation for determining the total mass of the ingredient in any quantity of pills—a fundamental practice in preparing and dispensing medications.

Percentage Composition

In pharmaceutical calculations, understanding percentage composition is essential as it defines the proportion of a specific substance within a mixture. When it comes to medication, this could refer to the ratio of an active ingredient to the overall mass of the pill.

In our example involving Lipitor pills, the percentage composition is provided for us; the active ingredient makes up 4.0% of the mass of one pill. This means that for every 100 grams of the pill, 4 grams will be the active ingredient. The calculation is represented as follows: \[\begin{equation}4.0 \% = \frac{4.0}{100} = 0.04\end{equation}\]

Why is this important?

Knowing the percentage composition allows pharmacists to accurately determine the dosage for patients and ensure that the amount of active ingredient is appropriate for the treatment of various conditions. The precision in measuring and understanding this percentage is what ensures the efficacy and safety of the medication being consumed.

Unit Conversion

Unit conversion is an omnipresent requirement in science, particularly in pharmacy where dosages may be provided in different units than those used for mass production or chemical synthesis. Converting mass units from grams to kilograms—or vice versa—is a common task.

In our example, after calculating the total mass of the active ingredient in a bottle of 100 pills in grams, the final step is to convert this mass into kilograms, as most pharmaceutical drug mass is standardly measured in larger scales. This conversion is necessary because it aligns with the standard units used in various aspects of healthcare and pharmaceutical logistics. The unit conversion formula from grams to kilograms is summarized as: \[\begin{equation}\text{mass in kilograms} = \frac{\text{mass in grams}}{1000}\end{equation}\]
Through unit conversion, pharmacists can ensure that the dosage instructions provided to patients are aligned with the mass units they are familiar with, ultimately leading to better understanding and adherence to medication regimens.