Problem 83 - Chemistry: The Central Science in SI Units, Global Edition

Question

A thief plans to steal a cylindrical platinum medal with a radius of \(2.3 \mathrm{~cm}\) and a thickness of \(0.8 \mathrm{~cm}\) from a jewellery store. If the platinum has a density of \(21.45 \mathrm{~g} / \mathrm{cm}^{3},\) what is the mass of the medal in \(\mathrm{kg} ?\) [The volume of a cylinder is \(\left.V=\pi r^{2} h .\right]\)

Step-by-Step Solution

Answer

The mass of the platinum medal is approximately \(0.28615 \, \mathrm{kg}\).

Step 1: Identify the given measurements

We are provided with the following measurements: - The radius of the cylindrical medal, \(r = 2.3 \, \text{cm}\) - The thickness (or height) of the medal, \(h = 0.8 \, \text{cm}\) - The density of platinum, \(d = 21.45 \, \frac{\text{g}}{\text{cm}^3}\)

Step 2: Calculate the volume of the medal

We can find the volume of the cylindrical medal using the formula: \[V = \pi r^2 h\] Substitute the given values of radius and height into the formula: \[V = \pi (2.3)^2 (0.8)\] Now, compute the volume of the medal: \[V \approx 13.34 \, \text{cm}^3\]

Step 3: Find the mass of the medal using the density formula

We know that the density, \(d\), is defined as the mass, \(m\), divided by the volume, \(V\). So we can find the mass using the formula: \[m = d V\] Substitute the given density and calculated volume into the formula: \[m = 21.45 (13.34)\] Now, compute the mass of the medal: \[m \approx 286.15 \, \text{g}\]

Step 4: Convert the mass from grams to kilograms

To convert the mass of the medal from grams to kilograms, divide the mass in grams by 1,000: \[m_{\text{kg}} = \frac{286.15}{1,000}\] \[m_{\text{kg}} = 0.28615 \, \text{kg}\] So, the mass of the platinum medal is approximately \(0.28615 \, \mathrm{kg}\).

Key Concepts

Volume of a Cylinder

A cylinder is a 3D shape with two parallel circular bases and a curved surface in between. To find the volume of a cylinder, the formula is used: \[ V = \pi r^2 h \] where:

• \( V \) is the volume of the cylinder
• \( r \) is the radius of the base circle
• \( h \) is the height (or thickness) of the cylinder

The calculation involves squaring the radius, multiplying it by the height, and then by \( \pi \) (approximately 3.14159).
In the case of the cylindrical medal:

• \( r = 2.3 \, \text{cm} \)
• \( h = 0.8 \, \text{cm} \)

Plugging these values into the formula gives:\[ V = \pi (2.3)^2 (0.8) \]You multiply \( 2.3^2 \) to get 5.29, then multiply this by 0.8, and finally by \( \pi \) to calculate approximately 13.34 \( \text{cm}^3 \).
This volume tells us the amount of space inside the cylinder.

Mass Conversion

Converting mass from one unit to another is a straightforward process.
Typically, mass is expressed in grams (g) or kilograms (kg), and they are related by the factor of 1,000. That means:

• 1 kilogram (kg) = 1,000 grams (g)
• To convert from grams to kilograms, divide the number of grams by 1,000.

For example, if a platinum medal weighs 286.15 grams, its weight in kilograms is calculated as:\[ m_{\text{kg}} = \frac{286.15}{1,000} \]This results in approximately 0.28615 kg.
Such conversions are common in scientific calculations to ensure consistency with SI units.

Density Formula

Density is a measure of how much mass is contained in a given volume. The formula to calculate density is:\[ d = \frac{m}{V} \]where:

• \( d \) is the density
• \( m \) is the mass
• \( V \) is the volume

In this exercise, you are given the density (\( 21.45 \, \text{g/cm}^3 \)) and need the mass based on the calculated volume of \( 13.34 \, \text{cm}^3 \). Re-arranging the density formula to solve for mass gives:\[ m = d \times V \]Substituting the known values:\[ m = 21.45 \times 13.34 \]The multiplication yields approximately 286.15 grams.
Understanding this relationship allows you to calculate the mass of a specific volume of material if its density is known.