Problem 125 - General Chemistry

Question

The density of quartz mineral was determined by adding a weighed piece to a graduated cylinder containing \(52.2 \mathrm{~mL}\) water. After the quartz was submerged, the water level was \(67.1 \mathrm{~mL}\). The quartz piece weighed \(39.8 \mathrm{~g}\). What was the density of the quartz?

Step-by-Step Solution

Answer

The density of the quartz is approximately 2.67 g/mL.

Step 1: Determine the Volume of Quartz

To find the volume of the quartz, subtract the initial water volume from the final water volume. The formula is: \[ \text{Volume of quartz} = \text{Final water level} - \text{Initial water level} = 67.1 \, \text{mL} - 52.2 \, \text{mL} \] By calculating, we find that the volume of quartz is equal to 14.9 mL.

Step 2: Convert Volume to the Correct Unit (if necessary)

Since the density is typically expressed in g/mL, and the volume we calculated is already in mL, no conversion is needed in this case.

Step 3: Calculate the Density of Quartz

Density is defined as the mass divided by the volume. Use the given mass of the quartz and the volume calculated:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{39.8 \, \text{g}}{14.9 \, \text{mL}} \]Calculate this expression to find the density of the quartz.

Step 4: Perform the Calculation

Carry out the division to find the density:\[ \frac{39.8 \, \text{g}}{14.9 \, \text{mL}} \approx 2.67 \, \text{g/mL} \] Thus, the density of the quartz is approximately 2.67 g/mL.

Key Concepts

Quartz Density

When discussing the density of quartz, we are referring to a key physical property of this mineral. Density is a measure of how much mass an object contains in a given volume. For quartz, this property is crucial because it helps in identifying and categorizing the mineral. The density is typically expressed in grams per milliliter (g/mL) or in kilograms per cubic meter (kg/m³).
In the example provided, we calculated the density of a quartz sample by submerging it in water and measuring how much the water level rose, which provided us with the volume of the quartz. Once the volume was determined, we used the mass of the quartz to calculate its density using the formula:

• Density = Mass ÷ Volume

This method is an easy and practical way to determine the density of solid objects like minerals. For the quartz sample, the calculated density was approximately 2.67 g/mL, which is typical for quartz and reflects its compact and structured crystalline form.

Mass and Volume Relationship

Understanding the relationship between mass and volume is key for calculating density. Mass is the amount of matter in an object and is typically measured in grams (g). Volume, on the other hand, is the space that the object occupies, which is measured in milliliters (mL) or cubic centimeters (cm³).
To determine the density of an object, such as our quartz sample, you need both its mass and volume. In our case, the quartz had a mass of 39.8 g and, from the displacement of water method, a volume of 14.9 mL. The mass and volume relationship can be visualized as:

• More mass in the same volume results in a higher density.
• The same mass spread over a larger volume results in a lower density.

This relationship is crucial for understanding why certain materials float or sink in water, as materials with a density greater than water will sink, whereas those with a lower density will float.

Physical Properties of Minerals

Minerals, like quartz, have distinctive physical properties that help in their identification. These properties include hardness, color, luster, and clearly, density. Each mineral's density is a unique characteristic that depends on its internal structure and composition.
Quartz is well-known for its clarity and hardness, ranking 7 on the Mohs scale. Its density, around 2.67 g/mL as in our example, is typical and aids in confirming the identity of the mineral. Such density suggests a close-packed structure typical of crystalline minerals, meaning atoms are tightly bound in a repeating pattern, contributing to its stability and resistance to weathering.
Understanding these physical properties, such as density, helps geologists and gemologists to classify minerals and determine their potential uses in various applications, from jewelry to industrial components.