Question

The initial water level in a 10-mL graduated cylinder is \(5.0 \mathrm{~mL}\). After a sapphire gemstone is added into the cylinder, the water level is \(6.5 \mathrm{~mL}\). What is the volume of the sapphire?

Short answer

The volume of the sapphire is 1.5 mL.

Step-by-step solution

- Initial Condition

Write down the initial water level in the graduated cylinder, which is given as \(5.0\, \mathrm{mL}\).

- Final Water Level

Note the final water level after the sapphire is added, which is \(6.5\, \mathrm{mL}\).

- Calculate Volume of Sapphire

To determine the volume of the sapphire, subtract the initial water level from the final water level: \(6.5\, \mathrm{mL} - 5.0\, \mathrm{mL} = 1.5\, \mathrm{mL}\).

Key concepts

Understanding Graduated Cylinders

A graduated cylinder is a common tool used in laboratories to measure the volume of liquids. These cylinders are tall, narrow containers with markings along their length that indicate specific volume levels.
They come in various sizes, designed to measure different volume ranges, such as 10 mL, 50 mL, or 100 mL.

Key features of graduated cylinders include:

• **Measurements are precise:** The markings allow for accurate volume readings, typically within 0.1 mL graduations depending on the cylinder's size.
• **Easy to use:** The straight design and clear markings make it easy to observe and measure liquids.
• **Versatile:** They can be used to measure not only liquids but also the volume of small, solid objects through the displacement method, as we'll discuss later.

The example in the exercise demonstrated how adding an object like a sapphire gemstone can change the liquid level in the cylinder, which is a vital observation for volume calculations.

Exploring the Displacement Method

The displacement method is an ingenious technique for calculating the volume of irregular-shaped objects. When an object is immersed in a liquid, it displaces an amount of liquid equal to its own volume. This is because matter cannot occupy the same space as liquid.

In the exercise, this technique was utilized to find the volume of a sapphire. By noting the increase in water level in the graduated cylinder, we can determine how much liquid the object displaced:

• The initial water level is noted before the object is introduced.
• The final water level is observed after the object submerged.
• Subtracting the initial volume from the final volume gives the object's volume. For the sapphire, this was calculated as 1.5 mL.

This method is particularly useful because it allows the measurement of objects that may not have a regular geometric shape that can be easily calculated with formulas.

Determining the Volume of Irregular Objects

Finding the volume of irregular objects, like a sapphire, can be tricky without using the displacement method. Irregular objects don’t have easily measurable dimensions like cubes or spheres. This is where the displacement method comes to the rescue.
It's applicable to any object that fits into a graduated cylinder and doesn't dissolve or react with the liquid.

Some benefits of using this method for irregular objects include:

• **Accuracy:** Provides a precise measurement of an object's volume regardless of its shape.
• **Simplicity:** Requires minimal equipment, usually just a graduated cylinder and a liquid such as water.
• **Non-Invasive:** The object does not need to be altered or destroyed to determine its volume.

In practice, as shown in the example, calculating the volume involves measuring the change in liquid level caused by the object's entry into the liquid, thereby giving its volume directly. This method enables us to deal with complex shapes in a straightforward manner, opening up the ability to measure all sorts of irregular items.