Question

Recall that \(1 \mathrm{~mL}=1 \mathrm{~cm}^{3}\). Suppose that the volume of water in a graduated cylinder is \(107.5 \mathrm{~mL}\). A specimen of quartz, tied to a piece of string, is immersed in the water. The new water level reads \(186 \mathrm{~mL}\). What is the volume, in \(\mathrm{cm}^{3}\), of the piece of quartz?

Short answer

The volume of the quartz is 78.5 cm^3.

Step-by-step solution

Understand the Problem

We need to calculate the volume of the quartz by measuring how much the water level increases when the quartz is submerged. Since the volume of water displaced is equal to the volume of the quartz, we can use the change in the water level to find the quartz's volume.

Initial Water Level

The initial volume of water in the graduated cylinder is given as 107.5 mL.

Final Water Level

When the quartz is immersed, the water level rises to 186 mL. This is the new volume of water in the cylinder with the quartz.

Calculate Volume Change

The volume change due to submerging the quartz is the difference between the final and initial water levels. Calculate this by subtracting the initial volume from the final volume: \( 186 \, \text{mL} - 107.5 \, \text{mL}\).

Perform the Calculation

Compute: \[ 186 - 107.5 = 78.5 \, \text{mL} \]The volume of the quartz is 78.5 mL.

Convert Units

Recall that 1 mL is equal to 1 cm^3. Thus, since the volume of the quartz in milliliters is 78.5 mL, it is also 78.5 cm^3.

Key concepts

Graduated Cylinder

A graduated cylinder is a common laboratory tool used to measure the volume of a liquid. It is a tall, narrow container marked with lines - or graduations - that indicate the volume of liquid inside. This allows you to read the exact volume easily.
The key features of a graduated cylinder include:

• A base for stability, reducing the risk of spills.
• Graduations along the side for accurate volume measurement.
• Typically made from glass or plastic for durability and chemical resistance.

A graduated cylinder is especially useful when you need to obtain precise volume measurements for experiments or calculations, such as determining the volume displacement caused by an object submerged in liquid.

Quartz Volume

When calculating the volume of an irregular solid, like a piece of quartz, using a graduated cylinder is ideal. By immersing the object in water, the volume it displaces helps determine its own volume.
Here's how it works:

• Note the initial water level in the cylinder before submersion.
• After submersion, observe the new water level.
• The difference between these two readings gives the volume of the quartz.

Let's say before submersing the quartz, the water level is 107.5 mL, and it rises to 186 mL after putting in the quartz. The volume of the quartz is the rise in water level: 186 mL - 107.5 mL = 78.5 mL. This volume measurement corresponds directly to the physical volume in cubic centimeters (cm³) due to the direct conversion between milliliters and cubic centimeters.

Water Displacement

Water displacement is a method used to determine the volume of an irregularly shaped object. It involves immersing the object in water and measuring how much the water level rises, a concept closely related to Archimedes' principle.
Here’s why water displacement is useful:

• It provides a simple way to gauge an object's volume when its shape doesn't allow for standard measurement techniques.
• By capturing the difference in water levels, we can accurately calculate the object's volume.
• This method is non-destructive, ensuring the specimen remains intact and unchanged.

This technique helps in a variety of scientific contexts, from geology to biology, demonstrating how the fundamentals of physics help us measure and understand the physical properties of materials, like quartz.

Unit Conversion

Unit conversion is the process of converting one unit of measurement to another, ensuring consistency and clarity in calculations. In scientific contexts, precise unit conversions are crucial for accurate data interpretation.
For volume, a key conversion is between milliliters (mL) and cubic centimeters (cm³), as both are commonly used in laboratories:

• 1 mL is exactly equivalent to 1 cm³.
• This straightforward equivalence allows seamless conversion without complex calculations.
• This is particularly useful in contexts such as volume displacement, where results may need to be expressed in different units for different applications.

By understanding these basic conversions, you ensure every component of a scientific measurement is correctly interpreted and applied across different units, improving the accuracy and communication of results.