Question

Osmium (Os) is the densest element known (density = \(\left.22.57 \mathrm{~g} / \mathrm{cm}^{3}\right) .\) Calculate the mass in pounds and in kilograms of an Os sphere \(15 \mathrm{~cm}\) in diameter (about the size of a grapefruit) (volume of a sphere of radius \(r\) is \(\frac{4}{3} \pi r^{3}\) ).

Short answer

The mass of the Os sphere is approximately 39.91 kg or 87.98 lbs.

Step-by-step solution

Convert Diameter to Radius

The diameter of the Os sphere is given as 15 cm. To find the radius, divide the diameter by 2:\[ r = \frac{d}{2} = \frac{15}{2} = 7.5 \text{ cm} \]

Calculate the Volume of the Sphere

Using the sphere volume formula \( V = \frac{4}{3} \pi r^3 \), substitute the radius we calculated:\[ V = \frac{4}{3} \pi (7.5)^3 \approx 1767.15 \text{ cm}^3 \]

Calculate Mass in Grams

Multiply the volume by the density to find the mass in grams:\[ \text{Mass}_{\text{g}} = \text{Density} \times \text{Volume} = 22.57 \, \text{g/cm}^3 \times 1767.15 \, \text{cm}^3 \approx 39905.58 \, \text{g} \]

Convert Grams to Kilograms

To convert grams to kilograms, divide by 1000:\[ \text{Mass}_{\text{kg}} = \frac{39905.58}{1000} \approx 39.91 \, \text{kg} \]

Convert Kilograms to Pounds

1 kilogram is approximately 2.20462 pounds. Multiply the mass in kilograms by this conversion factor:\[ \text{Mass}_{\text{lbs}} = 39.91 \, \text{kg} \times 2.20462 \, \text{lbs/kg} \approx 87.98 \, \text{lbs} \]

Key concepts

Volume of a Sphere

Understanding the volume of a sphere is essential for many calculations, especially when dealing with geometric shapes. A sphere is a perfectly round 3D object, and its volume tells us how much space it occupies. To find the volume, you need to know the formula: \[ V = \frac{4}{3} \pi r^3 \]Here:

• \( V \) represents the volume.
• \( r \) is the radius of the sphere.

For example, if the sphere's diameter is 15 cm, you first need to find the radius by dividing the diameter by 2. This gives a radius of 7.5 cm. Plugging this into the formula, we get:\[ V = \frac{4}{3} \pi (7.5)^3 \approx 1767.15 \, \text{cm}^3 \]This calculation shows the sphere's volume in cubic centimeters, which is a measure of how much space it occupies inside.

Conversion of Units

Conversions are crucial when working with different measurement systems. In our example, the calculations require converting grams to kilograms, and then to pounds. Here's how you approach unit conversion:

• First, to convert grams to kilograms, remember that 1 kilogram is 1000 grams. You simply divide the mass in grams by 1000.
• Next, to convert kilograms to pounds, use the conversion factor: 1 kilogram is approximately 2.20462 pounds.

For instance, if you have a mass of 39905.58 grams, converting to kilograms gives:\[ \text{Mass}_{\text{kg}} = \frac{39905.58}{1000} \approx 39.91 \, \text{kg} \]Converting this to pounds involves multiplying by 2.20462:\[ \text{Mass}_{\text{lbs}} = 39.91 \, \text{kg} \times 2.20462 \, \text{lbs/kg} \approx 87.98 \, \text{lbs} \]These conversions ensure that you're working with the correct units for any context or requirement.

Mass Calculation

Calculating the mass of an object involves understanding its density and volume. Density represents how much mass is contained in a given volume, commonly expressed in grams per cubic centimeter (g/cm³).To find the mass (\( \text{Mass}_{\text{g}} \)), use this formula:\[ \text{Mass}_{\text{g}} = \text{Density} \times \text{Volume} \]For a sphere made of Osmium, which has a density of \( 22.57 \, \text{g/cm}^3 \), and a volume calculated earlier as \( 1767.15 \, \text{cm}^3 \), the mass can be calculated by:\[ \text{Mass}_{\text{g}} = 22.57 \, \text{g/cm}^3 \times 1767.15 \, \text{cm}^3 \approx 39905.58 \, \text{g} \]Understanding mass calculation provides insights into the characteristics of materials and their applications.

Chemical Elements

Chemical elements are fundamental substances that constitute all matter. Each element, like Osmium (Os), has unique properties. Osmium is known for being the densest naturally occurring element, with a density of \( 22.57 \, \text{g/cm}^3 \).

• In the periodic table, Osmium is represented by the symbol \( \text{Os} \) and has an atomic number of 76.
• Its high density is why it makes an ideal candidate for calculations involving density.
• The properties of chemical elements like Osmium help in a variety of applications, from scientific research to industrial uses.

Understanding these characteristics of chemical elements allows for better comprehension of their practical uses and scientific significance.