Question

Aspirin has a density of \(1.40 \mathrm{~g} / \mathrm{cm}^{3} .\) What is the volume in cubic centimeters of an aspirin tablet weighing \(250 \mathrm{mg}\) ? Of a tablet weighing \(500 \mathrm{lb}\) ?

Short answer

The volume of a 250 mg tablet is approximately 0.179 cm³, while a 500 lb tablet is about 162,000 cm³.

Step-by-step solution

Understanding Density

Density is defined as the mass per unit volume, given by the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). In this problem, we need to find the volume in cubic centimeters (cm³) for two different masses of aspirin: 250 milligrams (mg) and 500 pounds (lb).

Converting Units for 250 mg

Since the density is given in grams per cubic centimeter and the mass is in milligrams, we must first convert milligrams to grams. \(250 \text{ mg} = 0.250 \text{ g}\).

Calculating Volume for 250 mg

Use the formula \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). For a mass of 0.250 g, the volume \( V = \frac{0.250 \text{ g}}{1.40 \text{ g/cm}^3} \approx 0.179 \text{ cm}^3 \).

Converting Units for 500 lb

Convert the weight from pounds to grams. There are approximately 453.592 grams in a pound. Thus, \(500 \text{ lb} = 500 \times 453.592 \text{ g} = 226796 \text{ g}\).

Calculating Volume for 500 lb

Use the formula \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). For a mass of 226796 g, the volume \( V = \frac{226796 \text{ g}}{1.40 \text{ g/cm}^3} \approx 162,000 \text{ cm}^3 \).

Key concepts

Unit Conversion

Unit conversion is a crucial skill in solving problems related to density and other scientific measurements. It allows you to switch between different units of measurement, making calculations more consistent and understandable. For example, in our problem, we need to convert units of mass to match the units given in the density.
When dealing with density problems, here are a few common conversions you might encounter:

• From milligrams (mg) to grams (g): To convert, divide the mass in milligrams by 1000, because there are 1000 milligrams in a gram. For example, 250 mg is converted to grams as follows: \( 250 \text{ mg} = \frac{250}{1000} \text{ g} = 0.250 \text{ g} \).
• From pounds (lb) to grams (g): Knowing that 1 pound is approximately 453.592 grams, multiply the mass in pounds by this conversion factor. Like converting 500 lb to grams results in: \( 500 \text{ lb} \times 453.592 = 226796 \text{ g} \).

These conversions help prepare the mass to be used accurately with the given density units in the calculations.

Volume Calculation

Volume calculation is the key step to determine how much space a given mass occupies, based on its density. The formula to find the volume, when you know the mass and density, is:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]In the given problem, once the mass unit is converted to grams, you can easily calculate the volume:

• For the 250 mg mass (converted to 0.250 g), use the density formula to find the volume of the aspirin: \( V = \frac{0.250 \text{ g}}{1.40 \text{ g/cm}^3} \approx 0.179 \text{ cm}^3 \).
• For the 500 lb mass (converted to 226796 g), the volume becomes: \( V = \frac{226796 \text{ g}}{1.40 \text{ g/cm}^3} \approx 162,000 \text{ cm}^3 \).

Remember, the volume is typically expressed in cubic centimeters (cm³), and these calculations tell you precisely how much space each mass takes up.

Density Formula

The density formula is fundamental in science and engineering, helping you understand how matter is packed within a given space. The formula is expressed as:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This formula shows that density represents the amount of mass per unit of volume. In problems where you are provided with density and mass, you can rearrange the formula to solve for volume instead:\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]Common applications include determining whether an object will float or sink in a fluid, or in assessing the purity of a material. In this problem, using the density of aspirin \(1.40 \text{ g/cm}^3\), allowed us to find out how much space 250 mg and 500 lb of aspirin will occupy as 0.179 cm³ and 162,000 cm³, respectively.