Question

A lump of pure copper weighs \(25.305 \mathrm{g}\) in air and 22.486 g when submerged in water \((d=0.9982 \mathrm{g} / \mathrm{mL})\) at \(20.0^{\circ} \mathrm{C} .\) Suppose the copper is then rolled into a \(248 \mathrm{cm}^{2}\) foil of uniform thickness. What will this thickness be, in millimeters?

Short answer

The thickness of the copper foil is approximately \(0.0114 cm\) or \(0.114 mm\).

Step-by-step solution

Calculate the Volume of Copper using Archimedes' Principle

First, the difference in weight when the copper is in air and then in water will give us the volume of the water displaced which is also the equal to the volume of the copper. This is based on Archimedes' principle which states that the buoyant force (the force with which a fluid pushes an object upward) exerted on an object equals the weight of the fluid that the object displaces. Therefore, we calculate the volume \(V\) by using the formula: \( V = \frac{25.305g - 22.486g}{0.9982g/mL} \)

Calculate the Volume of Copper

Using the values in the previous formula, we calculate for the volume of the copper which is approximately equal to \( V = 2.823 mL \). Note that mL and cm³ are equivalent units of volume so we can say \( V = 2.823 cm³ \).

Calculate Thickness of the Copper Foil

Now that we have the volume of the copper, we can determine the thickness of the copper foil. A rectangular prism's volume is the product of its length, width, and height. In the context of the foil, these dimensions correspond to the area (\(A\)) and thickness (\(t\)) of the foil. Thus, we solve for \( t \) with the equation: \( t = \frac{V}{A} \)With \( V = 2.823 cm³ \) and \( A = 248 cm² \) in the equation, we can calculate the thickness of the copper foil.

Key concepts

Buoyancy

Buoyancy is the force that allows objects to float or sink in a fluid. It occurs when a fluid exerts an upward force on an object immersed in it. The principle guiding buoyancy was first described by Archimedes, who discovered that the buoyant force on an object is equal to the weight of the fluid displaced by the object.
To see how buoyancy works, think of a piece of copper weighed in air and then submerged in water. In air, the copper's weight is greater than when it is in water. Why? Because the water pushes up against it, counteracting some of its weight.
This upward force is what we call the buoyant force. This is critical for determining volume and eventual measurements of objects, especially when they are submersed.

Copper Density

Density is the measure of how much mass is contained in a given volume. The density of copper is relatively high compared to many other substances. This makes it a dense metal.
Knowing the density of a material can help us understand its properties.

• For example, with a density higher than water, pure copper will sink.
• High density means more mass packed into a smaller space.

When we measure the change in weight of a copper lump in air vs. water, we can use this to calculate its volume, given that density is mass divided by volume. Thus, \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}}\] Understanding copper's density helps determine how much of it we have, given its weight.

Volume Calculation

Calculating the volume of copper involves using the concept of water displacement, rooted in Archimedes' principle. When we consider how much less the copper weighs in water, we are actually calculating the volume of water it displaces.
Here's how it works:

• First, take the weight of copper in air.
• Subtract the weight of copper in water.
• The difference represents the weight of water displaced.

The volume of water displaced equals the volume of the copper. Using the given data in our problem, this calculation is straightforward as \[V = \frac{25.305\, \text{g} - 22.486\, \text{g}}{0.9982\, \text{g/mL}}\] resulting in a copper volume of 2.823 cm³.

Surface Area

Surface area is an important factor when considering the dimensions of the copper foil. In the context of our calculation, it represents the total area covered by one face of the copper foil, measured in square centimeters.
Imagine rolling out the lump of copper into a large, thin plane. This surface would have dimensions equal to its surface area. For this exercise, we know that the foil has a given surface area of 248 cm².
Understanding the surface area helps us translate the volume of copper into a practical measurement. It is the groundwork for subsequently finding out the foil's thickness when the same mass of copper is spread out across this known area.

Thickness Measurement

Once the volume of copper and its surface area are known, measuring thickness becomes straightforward. Thickness is essentially the height dimension in a three-dimensional object, like our copper foil.
Think of the foil as a very flat rectangular prism. To find the thickness (\( t \)) of the foil, we use the formula:
\[t = \frac{\text{Volume}}{\text{Surface Area}} = \frac{2.823\, \text{cm}^3}{248\, \text{cm}^2}\]This calculation offers a direct way to determine how thick the foil is, in this case resulting in a thickness in millimeters after converting from centimeters. Such a measurement is vital for practical tasks like manufacturing or material science, where precise specifications are necessary.