Question

A sample containing \(33.42 \mathrm{~g}\) of metal pellets is poured into a graduated cylinder initially containing \(12.7 \mathrm{~mL}\) 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 density of the metal is approximately \(3.75 \mathrm{~g/mL}\).

Step-by-step solution

Mass of the metal

We are given that the mass of the metal is \(33.42 \mathrm{~g}\). Step 2: Find the volume of the metal.

Volume of the metal

The initial water level in the graduated cylinder is \(12.7 \mathrm{~mL}\) and the final water level is \(21.6 \mathrm{~mL}\). The change in volume is equal to the volume of the metal pellets, which can be calculated as: Volume of the metal = Final water level - Initial water level Volume of the metal = \(21.6 \mathrm{~mL}\) - \(12.7 \mathrm{~mL}\) Volume of the metal = \(8.9 \mathrm{~mL}\) Step 3: Calculate the density of the metal.

Density of the metal

Now that we have the mass and volume of the metal, we can calculate the density using the formula: Density = Mass / Volume Density = \(\dfrac{33.42 \mathrm{~g}}{8.9 \mathrm{~mL}}\) Density = \(3.75 \mathrm{~g/mL}\) The density of the metal is approximately \(3.75 \mathrm{~g/mL}\).

Key concepts

Mass and Volume

Understanding the concepts of mass and volume is crucial when it comes to determining the density of an object. Mass refers to the amount of matter in an object and is typically measured in grams (g) or kilograms (kg). In contrast, volume measures the space that matter occupies and is usually expressed in milliliters (mL) or liters (L).

To calculate the density of a substance, we need to know both its mass and its volume. The relationship between these two properties is directly used to compute density, which essentially shows us how much mass is contained in a given volume of the substance. In the provided exercise, we are given the mass of the metal pellets as 33.42 grams and we calculate their volume based on the rise in water level inside a graduated cylinder from 12.7 mL to 21.6 mL, resulting in a volume of 8.9 mL.

Graduated Cylinder

A graduated cylinder is a common laboratory equipment used to measure the volume of liquids accurately. Students should be familiar with how to read the graduations marked on the cylinder to determine the volume of liquid present. The method used in the exercise involves a simple yet elegant way to find the volume of irregularly-shaped objects, such as metal pellets, using water displacement.

When the object is immersed in water in a graduated cylinder, the water level rises by an amount equal to the volume of the object. This is due to the fact that the object displaces a volume of water equivalent to its own volume. By reading the new water level and subtracting the initial water level, we can determine the object's volume. In our exercise, the displaced water volume was calculated by subtracting the initial water level (12.7 mL) from the final water level (21.6 mL). It's important for students to ensure that the measurements are at eye level to the graduated cylinder to avoid parallax error and obtain the most accurate volume measurement.

Unit Conversion

In scientific calculations, it is often necessary to convert units to ensure that all the quantities are in the correct units for a given formula to work properly. In the context of density calculations, we typically want the mass in grams and the volume in milliliters to calculate density in grams per milliliter (g/mL).

Unit conversion is a fundamental skill in science as it allows for consistency and comparability of measurements. For example, if we are provided mass in kilograms, we must convert it to grams (1 kg = 1000 g) before using it in the density formula. In the case of the exercise we are considering, no unit conversion is needed since the mass is already in grams and the volume is in milliliters. However, students should always check and ensure that units are consistent to avoid any calculation errors.