Question

What is the density \((\mathrm{g} / \mathrm{mL})\) of each of the following samples? a. A medication, if \(3.00 \mathrm{~mL}\) has a mass of \(3.85 \mathrm{~g}\). b. The fluid in a car battery, if it has a volume of \(125 \mathrm{~mL}\) and a mass of \(155 \mathrm{~g}\). c. A \(5.00-\mathrm{mL}\) urine sample from a patient suffering from symptoms resembling those of diabetes mellitus. The mass of the urine sample is \(5.025 \mathrm{~g}\). d. A syrup is added to an empty container with a mass of \(115.25 \mathrm{~g}\). When \(0.100\) pint of syrup is added, the total mass of the container and syrup is \(182.48 \mathrm{~g}\). \((1 \mathrm{qt}=2 \mathrm{pt})\)

Short answer

a. 1.28 g/mL, b. 1.24 g/mL, c. 1.005 g/mL, d. 1.42 g/mL

Step-by-step solution

- Understand the formula for density

Density is calculated using the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

- Calculate the density for the medication (a)

Given:Volume: 3.00 mL Mass: 3.85 g \[ \text{Density} = \frac{3.85 \, g}{3.00 \, mL} = 1.28 \, g/mL \]

- Calculate the density for the battery fluid (b)

Given:Volume: 125 mL Mass: 155 g \[ \text{Density} = \frac{155 \, g}{125 \, mL} = 1.24 \, g/mL \]

- Calculate the density for the urine sample (c)

Given:Volume: 5.00 mL Mass: 5.025 g \[ \text{Density} = \frac{5.025 \, g}{5.00 \, mL} = 1.005 \, g/mL \]

- Calculate the density for the syrup (d)

First, convert the volume from pints to mL: \[ 0.100 \, \text{pint} \times \frac{473.176 \, \text{mL}}{1 \, \text{pint}} = 47.3176 \, \text{mL} \] Next, calculate the mass of the syrup: \[ 182.48 \, g - 115.25 \, g = 67.23 \, g \] Finally, calculate the density: \[ \text{Density} = \frac{67.23 \, g}{47.3176 \, mL} = 1.42 \, g/mL \]

Key concepts

density formula

Density tells us how much mass is packed into a certain volume. The formula to calculate density is straightforward: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] This formula highlights a simple relationship between mass and volume, where density (\text{g/mL}) is found by dividing mass (\text{g}) by volume (\text{mL}). It helps us understand how heavy a substance is for a fixed volume.
Let's see this formula in action in the solutions provided.

mass

Mass is a measure of how much matter is in an object. The unit commonly used for mass in these calculations is grams (g). Understanding the mass of an object is essential because it lets you compute its density when combined with volume.
For instance, in step 2, the medication has a mass of 3.85 grams, while in step 3, the battery fluid has a mass of 155 grams. By knowing these masses, we can proceed to find their densities.

volume

Volume measures the space an object occupies. In our calculations, volume is measured in milliliters (mL).
For example, the medication's volume in step 2 is 3.00 mL, while the battery fluid in step 3 has a volume of 125 mL. Similar to mass, knowing the volume is essential for calculating density using the density formula.

unit conversions

Sometimes, we need to convert units to make our calculations easier. Let’s look at step 5 for an example. Initially, we have the volume in pints (0.100 pint). To use the density formula, we need to turn that into mL using the conversion factor: \[ 0.100 \, \text{pint} \times \frac{473.176 \, \text{mL}}{1 \, \text{pint}} = 47.3176 \, \text{mL} \] After converting the volume, we can use it directly in the density calculation. Always make sure your units match the ones you use in the formula to ensure accuracy in your results.