Problem 86 - Chemistry The Central Science In Si Units

Question

A \(10.0 \mathrm{~g}\) block of gold is hammered into a thin gold sheet which has an area of \(150 \mathrm{~cm}^{2}\). Given the density of gold is \(19.3 \mathrm{~g} / \mathrm{cm}^{3}\), what is the approximate thickness of the gold sheet in millimeters?

Step-by-Step Solution

Answer

The thickness of the gold sheet is approximately 0.0345 mm.

Step 1: Find the Volume of the Gold

Since we know the mass of the gold and its density, we can find the volume of the gold block using the formula for density: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\). Rearranging the formula gives us \(\text{Volume} = \frac{\text{Mass}}{\text{Density}}\). Substituting the given values, we have: \(\text{Volume} = \frac{10.0 \, \text{g}}{19.3 \, \text{g/cm}^3} \approx 0.518 \, \text{cm}^3\).

Step 2: Relate Volume, Area, and Thickness

The volume of a sheet can also be expressed as the product of its area and thickness using the formula: \(\text{Volume} = \text{Area} \times \text{Thickness}\). Since we have the area and the volume, we can rearrange this formula to find the thickness: \(\text{Thickness} = \frac{\text{Volume}}{\text{Area}}\).

Step 3: Calculate the Thickness in Centimeters

Using the formula from the previous step, plug in the values for volume and area to find the thickness in centimeters: \(\text{Thickness} = \frac{0.518 \, \text{cm}^3}{150 \, \text{cm}^2} \approx 0.00345 \, \text{cm}\).

Step 4: Convert Thickness to Millimeters

Remember that 1 centimeter is equivalent to 10 millimeters. Therefore, multiply the thickness in centimeters by 10 to convert it to millimeters: \(0.00345 \, \text{cm} \times 10 = 0.0345 \, \text{mm}\).

Key Concepts

Volume Calculation

To determine the volume of an object, especially when its mass and density are known, we can use the handy density formula: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\). Rearranging this equation gives us the formula for volume: \(\text{Volume} = \frac{\text{Mass}}{\text{Density}}\). This is particularly useful when dealing with solids like gold or other metals.

For example, if you have a block of gold weighing 10 grams and the density of gold is known to be \(19.3 \, \text{g/cm}^3\), you can easily substitute these values into the formula to get the volume of the gold block:

• Mass = 10.0 g
• Density = 19.3 g/cm\(^3\)
• Volume = \(\frac{10.0 \, \text{g}}{19.3 \, \text{g/cm}^3} \approx 0.518 \, \text{cm}^3\)

This calculation reveals that this particular block of gold takes up approximately 0.518 cubic centimeters of space.

Area and Thickness Relationship

Once you have the volume of an object, like a thin sheet, understanding how area and thickness relate is vital for further calculations. For objects like a metal sheet, the volume can also be expressed as the multiplication of its area and its thickness:

\[\text{Volume} = \text{Area} \times \text{Thickness}\]

In this formula, if you know the volume and the area, you can rearrange the equation to solve for thickness:

\[\text{Thickness} = \frac{\text{Volume}}{\text{Area}}\]

Suppose you have a gold sheet with an area of 150 cm\(^2\) and a volume of 0.518 cm\(^3\). Using the formula for thickness:

• Area = 150 cm\(^2\)
• Volume = 0.518 cm\(^3\)
• Thickness = \(\frac{0.518 \, \text{cm}^3}{150 \, \text{cm}^2} \approx 0.00345 \, \text{cm}\)

This means the gold sheet would be approximately 0.00345 cm thick.

Unit Conversion

When we perform calculations in science and engineering, unit conversion is often necessary to express results in the desired units. In this exercise, converting the thickness from centimeters to millimeters provides a more intuitive measurement for thin materials.

Remember, the relationship between centimeters and millimeters is straightforward: 1 cm equals 10 mm.

For the calculated thickness of 0.00345 cm, we convert it to millimeters by multiplying:

• Thickness in cm: 0.00345 cm
• Conversion factor: 10 mm/cm
• Thickness in mm: \(0.00345 \, \text{cm} \times 10 = 0.0345 \, \text{mm}\)

This converts it to 0.0345 mm, offering a clearer perspective on just how thin the gold sheet truly is.