Question

Iron has density \(7.87 \mathrm{g} / \mathrm{cm}^{3}\). If \(52.4 \mathrm{g}\) of iron is added to \(75.0 \mathrm{mL}\) of water in a graduated cylinder, to what volume reading will the water level in the cylinder rise?

Short answer

The water level in the cylinder will rise to approximately \(81.66\,\mathrm{mL}\).

Step-by-step solution

Write the density formula for iron

We will use the density formula, which is: \[ \text{density} = \frac{\text{mass}}{\text{volume}} \] For iron, the given density is \(7.87 \mathrm{g/cm^3}\). We need to find the volume it occupies when the mass is \(52.4 \mathrm{g}\).

Rearrange the formula to find the volume of iron

Rearrange the density formula to solve for the volume of iron: \[ \text{volume} = \frac{\text{mass}}{\text{density}} \]

Substitute the given values and calculate the iron's volume

Now, substitute the mass and density of iron into the formula: \[ \text{volume} = \frac{52.4 \mathrm{g}}{7.87 \mathrm{g/cm^3}} \] Calculate the volume: \[ \text{volume} \approx 6.66 \mathrm{cm^3} \]

Add the iron volume to the initial water volume

Given that the initial volume of water is \(75.0\,\mathrm{mL}\), or \(75.0\,\mathrm{cm^3}\), in the graduated cylinder, we will now add the volume of the iron to the water's initial volume: \[ \text{final volume} = \text{initial water volume} + \text{iron volume} \] \[ \text{final volume} = 75.0\,\mathrm{cm^3} + 6.66\,\mathrm{cm^3} \]

Determine the final volume reading

Calculate the final volume reading in the graduated cylinder: \[ \text{final volume} \approx 81.66\,\mathrm{cm^3}\, \mathrm{(mL)} \] The water level in the cylinder will rise to approximately \(81.66\,\mathrm{mL}\).

Key concepts

Understanding Volume Calculation

Volume calculation is a fundamental concept in understanding how much space an object occupies. In the context of this exercise, we need to determine how much space the added iron takes up in the graduated cylinder. To find the volume of a substance when you know its mass and density, you use the formula:
\[\text{volume} = \frac{\text{mass}}{\text{density}}\]This formula is derived from the basic principles of density, which is the mass of an object divided by its volume. Calculating volume is essential for determining how much a material will affect a liquid's level in a container, as it directly adds to the total space taken by substances within the cylinder.
In our exercise:

• The mass is given as 52.4 grams.
• The density of iron is 7.87 g/cm³.

Using these values in our formula gives us the volume that the iron occupies.

Using a Graduated Cylinder

A graduated cylinder is a crucial tool in many scientific experiments for measuring liquid volumes accurately. It is a tall, cylindrical piece of glassware marked with units of volume along its side. It provides precise readings, making it highly useful for determining the exact volume of a liquid or mixture. In our scenario, we can use it to measure both the initial water volume and the final volume after iron is added.
How to use it effectively:

• Always place the cylinder on a flat surface to get correct readings.
• Read the volume at the bottom of the meniscus, the curve formed by the liquid.
• Ensure your line of sight is level with the liquid surface to avoid errors.

When the iron is added to the water in the cylinder, the water level rises by the volume of the iron due to the displacement of water.

Understanding the Density Formula

The density formula is essential for connecting the physical properties of mass and volume. Density is a measure of how much mass is contained within a defined volume. The formula for density is:
\[\text{density} = \frac{\text{mass}}{\text{volume}}\]This formula allows you to solve for any of the three variables—density, mass, or volume—if you have the other two.
Key points about the density formula:

• It helps identify how concentrated a material's mass is in a given space.
• A higher density means more mass in less space.
• A lower density indicates less mass in more space.

In material science and physics, this formula is used to identify substances and calculate their properties, including when substances like iron displace water.

Exploring the Mass and Density Relationship

The relationship between mass and density is straightforward: density is mass divided by volume. This relationship is vital for understanding how different substances will behave under similar conditions. It tells us a lot about the nature of materials and their interactions.
When you look at an object:

• Mass is the amount of matter it contains, measured in grams or kilograms.
• Density shows how tightly packed the matter within that space is.

If you know two of these values, the density equation can help you find the third. In our example, knowing the mass of iron and its density allows us to calculate its volume, giving insight into how it interacts with the water in the graduated cylinder. Understanding these concepts helps in predicting and measuring the rise in water level accurately when an object is submerged.