Question

A sample containing 33.42 g of metal pellets is poured into a graduated cylinder initially containing 12.7 mL of water, causing the water level in the cylinder to rise to 21.6 mL. Calculate the density of the metal.

Short answer

The density of the metal can be calculated using the formula: Density = Mass / Volume. After adding the metal pellets into the graduated cylinder, the water level rises from 12.7 mL to 21.6 mL, which means the volume of the metal pellets is 21.6 mL - 12.7 mL = 8.9 mL. Given the mass of the metal pellets is 33.42 g, we can now find the density: Density = 33.42 g / 8.9 mL ≈ 3.76 g/mL.

Step-by-step solution

Determine the volume of the metal pellets

To find the volume of the metal pellets, we need to subtract the initial volume of the water in the graduated cylinder from the final volume of the water after the metal was added. This is because the change in water volume will be equal to the volume of the metal pellets. Initial water volume: 12.7 mL Final water volume: 21.6 mL Volume of metal pellets = Final water volume - Initial water volume

Calculate the volume of the metal pellets

Now we'll subtract the initial water volume from the final water volume to find the volume of the metal pellets. Volume of metal pellets = 21.6 mL - 12.7 mL = 8.9 mL

Calculate the density of the metal

Now that we have the mass and volume of the metal pellets, we can calculate the density using the following formula: Density = Mass / Volume We are given the mass of the metal pellets (33.42 g) and have calculated the volume (8.9 mL). Now, we can plug these values into the formula: Density = 33.42 g / 8.9 mL = 3.76 g/mL The density of the metal is approximately 3.76 g/mL.

Key concepts

Volume Measurement

Volume measurement is an essential concept in determining the space occupied by an object or substance. In our exercise, we measured the volume of metal pellets by noting the change in water level in a graduated cylinder. The initial volume of water was 12.7 mL, and after adding the metal pellets, this volume increased to 21.6 mL.

To find the volume of the metal, we subtracted the initial volume from the final volume:

• Initial water volume: 12.7 mL
• Final water volume: 21.6 mL
• Volume of metal = Final volume - Initial volume

Thus, the volume of the metal pellets was 8.9 mL, representing the space they occupy when submerged in water.

Mass Measurement

The measurement of mass helps us understand how much matter is contained within an object or substance. Mass is typically measured using a balance and is often expressed in grams. In our example, the mass of the metal pellets is given as 33.42 grams.

Mass measurement is crucial for calculating density, as it allows us to determine the amount of material present. In laboratory settings, accurate mass measurements ensure that calculations, like those for density, are correct.

Graduated Cylinder

A graduated cylinder is a common laboratory tool used for measuring the volume of a liquid. Graduated cylinders are marked with a scale, usually in milliliters (mL), that allows us to measure the volume of liquids precisely.

When using a graduated cylinder for measuring, it's important to read the volume at the bottom of the meniscus (the curved surface of the liquid) at eye level to ensure accurate results. Graduated cylinders are also useful for measuring the volume of an irregular solid, like metal pellets, through the displacement method. In our situation, the metal pellets caused the water in the cylinder to rise, thus allowing us to determine their volume.

Density Formula

The density formula is a fundamental equation in the study of matter, allowing us to relate an object's mass to its volume. Density (\( \rho \)) is calculated using the formula:

\[ \rho = \frac{\text{Mass}}{\text{Volume}} \]
By dividing the mass of an object by its volume, we obtain information about how much mass is concentrated within a given space. In our exercise, the mass was 33.42 grams, and the volume was 8.9 mL, allowing us to calculate the density:

• Density = 33.42 g / 8.9 mL
• Density = 3.76 g/mL

The density tells us how tightly packed the particles within a substance are, sometimes indicating its material composition.

Displacement Method

The displacement method is a practical technique used to determine the volume of an irregularly shaped object. When an object is submerged in a liquid, it displaces an amount of liquid equal to its own volume. This method is particularly useful in cases where it is difficult to measure dimensions directly.

In the exercise, by adding metal pellets to a graduated cylinder filled with water, the water level rose. The change in volume equals the volume of the pellets. This method is reliable and easy, making it a favorite in classroom and laboratory settings. To ensure accuracy, it is crucial that the liquid completely covers the object and that there are no air bubbles affecting measurements.