Question

A sample containing \(33.42 \mathrm{g}\) 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 \(21.6 \mathrm{mL}\). Calculate the density of the metal.

Short answer

The volume of the metal pellets is 8.9 mL, calculated by subtracting the initial water level (12.7 mL) from the final water level (21.6 mL) after adding the metal pellets. The density of the metal is calculated using the formula \(Density = \frac{Mass}{Volume}\), giving us a density of approximately 3.75 g/mL.

Step-by-step solution

Find the volume of the metal pellets

The initial water level in the graduated cylinder was 12.7 mL, and after adding the metal pellets, the water level rose to 21.6 mL. We can find the volume of the metal pellets by this displacement method. Simply subtract the initial water level from the final water level. Volume of metal pellets = Final water level - Initial water level Volume of metal pellets = 21.6 mL - 12.7 mL

Calculate the volume

Subtract the initial water level from the final water level to find the volume of the metal pellets: Volume of metal pellets = 21.6 mL - 12.7 mL Volume of metal pellets = 8.9 mL We have found the volume of the metal pellets to be 8.9 mL.

Use the density formula

Now we have the mass of the metal pellets (33.42 g) and the volume of the metal pellets (8.9 mL). To find the density, use the density formula: Density = Mass / Volume

Calculate the density

Plug in the given mass and calculated volume into the density formula: Density = 33.42 g / 8.9 mL Density ≈ 3.75 g/mL

Write the final answer

We have found that the density of the metal pellets is approximately 3.75 g/mL.

Key concepts

Density Formula

Understanding the density formula is essential when trying to determine how compact an object is. Density is defined as the mass of an object divided by its volume and is expressed in units such as grams per cubic centimeter (g/cm³) or grams per milliliter (g/mL). The density formula is written as:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
This formula is pivotal in determining whether objects will float or sink in a fluid and is used across various scientific disciplines, including chemistry and physics.

Volume Displacement

Volume displacement is a method used to measure the volume of irregularly shaped objects. It is based on the principle of Archimedes, which states that an object submerged in a fluid will displace a volume of fluid equal to its own volume. This concept is particularly useful when dealing with shapes that cannot be easily measured with a ruler or caliper. By submerging the object in water and measuring how much the water level rises in a graduated cylinder, we can accurately determine the volume of the object. This process was exemplified by immersing pellets in water and observing the rise in water level, from 12.7 mL to 21.6 mL, which represents the volume of the pellets.

Graduated Cylinder

A graduated cylinder is a common piece of laboratory equipment used to measure the volume of liquids and irregular solids accurately. It's marked with a scale of milliliters (mL) or sometimes liters (L), allowing for precise volume readings. When using a graduated cylinder, it's important to read the lower meniscus, which is the curve seen at the top of the liquid layer due to surface tension. For the highest accuracy, ensure that the measurement is taken at eye level. In the given problem, after adding the metal pellets, the water level increased from 12.7 mL to 21.6 mL, indicating the volume displaced by the pellets.

Chemistry Problem Solving

Problem solving in chemistry often involves a systematic approach to explore and understand chemical properties and reactions. It usually requires a series of steps, including identifying the problem, collecting relevant data, and applying appropriate formulas or principles. It is crucial to have a firm grasp of the concepts, such as density and volume displacement, to solve problems effectively. In the exercise provided, the problem-solving process involved calculating the density of an irregularly shaped object, a typical challenge in chemistry. Each step, from measuring volume displacement with a graduated cylinder to applying the density formula, was executed methodically to arrive at a solution.