Problem 9 - Chemistry Matter and Change

Question

Calculate Samples \(\mathrm{A}, \mathrm{B}\) , and \(C\) have masses of \(80 \mathrm{g}, 12 \mathrm{g},\) and \(33 \mathrm{g},\) and volumes of \(20 \mathrm{mL}, 4 \mathrm{cm}^{3},\) and 11 \(\mathrm{mL}\) , respectively. Which of the samples have the same density?

Step-by-Step Solution

Answer

Samples B and C have the same density (3 g/mL).

Step 1: Understand the formula for density

Density is calculated using the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). This formula will help us determine the density of each sample.

Step 2: Calculate the density of Sample A

Sample A has a mass of 80 g and a volume of 20 mL. Using the density formula: \( \text{Density of A} = \frac{80\, \text{g}}{20\, \text{mL}} = 4\, \text{g/mL} \).

Step 3: Calculate the density of Sample B

Sample B has a mass of 12 g and a volume of 4 cm\(^3\). Since 1 mL = 1 cm\(^3\), the volume can be considered without conversion. Thus, \( \text{Density of B} = \frac{12\, \text{g}}{4\, \text{cm}^3} = 3\, \text{g/cm}^3 \).

Step 4: Calculate the density of Sample C

Sample C has a mass of 33 g and a volume of 11 mL. Thus, \( \text{Density of C} = \frac{33\, \text{g}}{11\, \text{mL}} = 3\, \text{g/mL} \).

Step 5: Compare the densities of the samples

The densities calculated are: Sample A = 4 g/mL, Sample B = 3 g/cm\(^3\), and Sample C = 3 g/mL. Since the densities of samples B and C are equal, these two have the same density.

Key Concepts

Mass and Volume

When we talk about density, we start with mass and volume. Mass refers to the amount of matter an object contains, while volume is the amount of space that object occupies. In our exercise, Sample A has a mass of 80 grams and a volume of 20 mL, Sample B has a mass of 12 grams and a volume of 4 cm³, and Sample C has a mass of 33 grams and a volume of 11 mL. Understanding mass and volume is crucial because these values are used to calculate density. Remember that mass is typically measured in grams (g) and volume in milliliters (mL) or cubic centimeters (cm³). For liquids, 1 mL is equivalent to 1 cm³.

Unit Conversion

Unit conversion is a fundamental step to ensure consistency in our calculations. It helps in easily comparing different sets of data. In our example, Sample B's volume is given in cm³, but because 1 cm³ equals 1 mL, we don’t need to convert it to find density.

• Always check for different units and make sure they are comparable.
• Use conversion factors, like 1 mL = 1 cm³, when needed to align your units.

This process reduces the chances of mistakes and ensures your calculations are precise.

Density Formula

The density formula is a simple yet powerful tool: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \] This formula tells us how much mass is contained in a given volume.

• For Sample A, density is \( \frac{80 \, \text{g}}{20 \, \text{mL}} = 4 \, \text{g/mL} \).
• For Sample B: \( \frac{12 \, \text{g}}{4 \, \text{cm}^3} = 3 \, \text{g/cm}^3 \).
• For Sample C: \( \frac{33 \, \text{g}}{11 \, \text{mL}} = 3 \, \text{g/mL} \).

Using this formula gives us the ability to directly compare the densities of different materials, regardless of size or form.

Density Comparison

Density comparison allows us to determine which materials are relatively heavier or lighter for their size. In the exercise, we found:

• Sample A with a density of 4 g/mL.
• Sample B with 3 g/cm³.
• Sample C also at 3 g/mL.

Since Sample B and C have the same density, they contain the same amount of mass per unit volume even if their total mass and volume differ. This comparison shows us important information about the substances' properties, like how dense the material is or potentially what substance it might be.