Problem 7 - Chemical Principles: The Quest for Insight

Question

When a picce of metal of mass \(3.60 \mathrm{~g}\) is dropped into a graduated cylinder containing \(8.3 \mathrm{~mL}\) of water, the warer level rises to \(9.8 \mathrm{~mL}\). What is the density of the metal (in grams per cubic centimeter)?

Step-by-Step Solution

Answer

The density of the metal is \(2.4\mathrm{~g/cm}^3\).

Step 1: Determine the volume of the metal

To find the volume of the metal, subtract the initial water level from the final water level in the graduated cylinder. This will give the volume of water displaced, which is equal to the volume of the metal.

Step 2: Calculate the volume of the metal

Using the data given, the volume of the metal (V_metal) in milliliters can be calculated as follows: \(V_{metal} = 9.8 \mathrm{~mL} - 8.3 \mathrm{~mL} = 1.5 \mathrm{~mL} = 1.5 \mathrm{~cm}^3\) (since 1 mL = 1 cm^3).

Step 3: Apply the density formula

Density (\(\rho\)) can be calculated using the formula \(\rho = \frac{mass}{volume}\).

Step 4: Calculate the density of the metal

Using the mass of the metal (3.60 g) and the volume of the metal (1.5 cm^3), the density of the metal (\(\rho_metal\)) is calculated as follows: \(\rho_{metal} = \frac{3.60\mathrm{~g}}{1.5\mathrm{~cm}^3} = 2.4\mathrm{~g/cm}^3\).

Key Concepts

Volume Displacement

Understanding volume displacement is vital in the context of density calculation. When an object is placed in a fluid, such as water, it pushes the fluid away to make space for itself. This is known as volume displacement. Imagine filling a tub with water and then getting in; the water level rises because you've displaced an amount of water equivalent to your body volume. Similarly, in our exercise, the piece of metal displaced a certain volume of water when it was submerged in the graduated cylinder.

To calculate the volume displaced by the object, you can simply measure the rise of the fluid's level in the container before and after the object's immersion. The change in fluid level corresponds to the volume of the object. In the exercise, the water level initially was at 8.3 mL and rose to 9.8 mL upon adding the metal, which means the metal displaced 1.5 mL of water, exactly equivalent to the volume of the metal itself.

Graduated Cylinder

A graduated cylinder is a common laboratory equipment used to measure the volume of liquids with a reasonable degree of accuracy. It is a tall, narrow cylinder with volume markings along its side, allowing for precise volume readings. When using a graduated cylinder for volume displacement methods in density calculations, you must carefully observe the initial and final levels of the liquid. As demonstrated in the exercise, precise readings of the water levels before and after submerging the metal are crucial for determining the metal's volume.

For the best results, ensure the graduated cylinder is on a level surface and that you read the volume at eye level to avoid parallax error. The meniscus, or the curve seen at the top of the liquid, should be read from the bottom of the curve to get an accurate measurement. Thorough attention to these details when using the graduated cylinder is key to accurate volume determination.

Mass to Volume Ratio

The mass to volume ratio, commonly known as density, is a measure of how much mass is contained in a given volume. In basic terms, it tells you how 'packed' an object is. The formula to calculate the density \(\rho = \frac{mass}{volume}\) involves dividing the object's mass by its volume.

In our exercise, the mass of the metal was given as 3.60 grams, and the volume was determined to be 1.5 cubic centimeters using volume displacement. Using the density formula, we divide the mass by the volume to obtain the density of the metal. When students grasp the concept of mass to volume ratio, they can understand that density is an intrinsic property of material and is independent of the quantity of material present. The calculated density helps identify or characterize the type of metal based on known density values of various materials.