Question

A graduated cylinder was filled with water to the \(15.0 \mathrm{~mL}\) mark and weighed on a balance. Its mass was \(27.35 \mathrm{~g}\). An object made of silver was placed in the cylinder and completely submerged in the water. The water level rose to \(18.3 \mathrm{~mL}\). When reweighed, the cylinder, water, and silver object had a total mass of \(62.00 \mathrm{~g}\). Calculate the density of silver in \(\mathrm{g} \mathrm{cm}^{-3}\).

Short answer

Density of Silver = (62.00 g - 27.35 g) / (18.3 mL - 15.0 mL) = 34.65 g / 3.3 mL = 10.50 g/cm^3.

Step-by-step solution

Calculate the Mass of the Silver Object

To find the mass of the silver object, subtract the mass of the water and the cylinder before the object was added from the total mass after the object was added. That is, Mass of Silver = Total Mass (cylinder, water, and silver) - Mass (cylinder and water alone).

Calculate the Volume of the Silver Object

The volume displaced by the silver object is equal to the increase in the water level in the graduated cylinder. Calculate the volume by subtracting the initial water level from the final water level. So, Volume of Silver = Final water level - Initial water level.

Calculate the Density of the Silver Object

Density is equal to mass divided by volume. Use the mass and volume of the silver object obtained from previous steps to calculate its density. Thus, Density of Silver = Mass of Silver / Volume of Silver.

Key concepts

Mass and Volume

Understanding the relationship between mass and volume is critical to grasping the concept of density. Mass refers to the amount of matter in an object, typically measured in grams (g) or kilograms (kg). Volume, on the other hand, is the measure of the space that an object occupies and is commonly expressed in cubic centimeters (cm³) or milliliters (mL), knowing that 1 cm³ is equivalent to 1 mL.

To find an object's density, you must first accurately measure its mass and volume. In the given exercise, the mass was obtained using a balance, which provides the gravitational mass of the object based on the force it exerts under gravity. The volume was determined by the amount of water displaced by the silver object in the graduated cylinder, which is a reliable method for measuring volume, especially when the object does not have a regular shape.

Exercise Improvement Advice:

It is important to ensure that the balance is calibrated properly before measuring the mass to avoid any inaccuracies. Likewise, reading the volume from the graduated cylinder should be done at eye level, at the bottom of the meniscus (the curve seen at the top of the liquid), to ensure precision in the volume measurement.

Graduated Cylinder Usage

A graduated cylinder is a common laboratory apparatus used to measure the volume of liquids accurately. Its usage is paramount in conducting experiments requiring precise volume measurements, such as the displacement method for determining the volume of an irregularly shaped object. When using a graduated cylinder, it is vital to place it on a flat surface and view the measurement at eye level to avoid parallax error.

To measure the volume accurately, take note of the liquid’s meniscus and ensure your line of sight is perpendicular to the measurement scale. Graduated cylinders come in various sizes, and selecting the appropriate size can increase the accuracy of the measurement. Smaller graduated cylinders tend to have more precise measurement increments.

Exercise Improvement Advice:

When an object is submerged in the liquid within the graduated cylinder, it is important to ensure that it does not touch the sides, as this can affect the accuracy of the volume reading. Additionally, the object must be fully submerged without causing the liquid to overflow.

Displacement Method for Density

The displacement method is a widely used technique to determine the volume of an object. When an object is submerged in water, it displaces a volume of water equal to the volume of the object. By measuring the difference in water level before and after submerging the object, you can find the displaced volume, which is the same as the volume of the object.

This method is particularly useful for irregularly shaped objects that cannot have their volume calculated by simple formulas. After determining the volume using the displacement method, and having the mass from a balance, the density can be calculated using the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
It's essential for the accuracy of the density calculation that both mass and volume are measured precisely.

Exercise Improvement Advice:

Make sure to account for any water that may have been added or removed during the process, other than the displacement caused by the object. Additionally, temperature and purity of the water can affect the result, so these should be controlled as much as possible to ensure an accurate determination of the object's volume and, thus, its density.