Question

The density of osmium is reported by one source to be \(22,610 \mathrm{kg} / \mathrm{m}^{3} .\) What is this density in \(\mathrm{g} / \mathrm{cm}^{3} ?\) What is the mass of a block of osmium measuring 10.0 \(\mathrm{cm} \times 8.0 \mathrm{cm} \times 9.0 \mathrm{cm} ?\)

Short answer

The density of osmium is 22.61 g/cm³, and the mass of a block of osmium measuring 10.0 cm × 8.0 cm × 9.0 cm is 16,279.2 g.

Step-by-step solution

Conversion of units for density

First, let's convert the density of osmium from kg/m³ to g/cm³. We need to multiply the given density by the conversion factors: \(1 kg = 1000 g\) and \(1 m = 100 cm\). Density in kg/m³ = 22,610 kg/m³ To convert it to g/cm³, we have to multiply by \((1000 g/1 kg)\) and \((1 m / 100 cm)^3\): \(22,610 \frac{kg}{m^3} \times \frac{1000 g}{1 kg} \times (\frac{1 m }{100 cm})^3\)

Calculate the new density of Osmium

Now, let's simplify the expression we've got: \(22,610 \times \frac{1000}{1} \times \frac{1}{100^3}\) \(22,610 \times 1000 \times \frac{1}{1,000,000}\) \(22,610 \times \frac{1}{1,000}\) = 22.61 g/cm³ The density of osmium is 22.61 g/cm³.

Calculate the volume of the block

Now let's find the volume of the block with dimensions 10.0 cm x 8.0 cm x 9.0 cm. We can calculate the volume using the formula: Volume = Length × Width × Height Volume = 10.0 cm × 8.0 cm × 9.0 cm = 720 cm³ The volume of the block is 720 cm³.

Calculate the mass of the block

Finally, let's find the mass of the block using the formula: Mass = Density × Volume We know the density of osmium is 22.61 g/cm³ and the volume of the block is 720 cm³. Mass = 22.61 g/cm³ × 720 cm³ = 16,279.2 g The mass of the block of osmium is 16,279.2 g.

Key concepts

Osmium Properties

Osmium is a fascinating element and one of the densest naturally occurring substances found on Earth. Its striking property of high density makes it valuable in various industrial applications. It has a density reported to be around 22,610 kg/m³, which translates to 22.61 g/cm³ after conversion. This extreme density is due to osmium's tightly packed crystal structure. Osmium is often used in applications requiring durability and hardness, such as fountain pen nibs, electrical contacts, and even in some medical implants due to its resilience.

Unit Conversion

Converting units is a vital skill in science and everyday life. Let's see how we convert the density of osmium from kg/m³ to g/cm³. First, understand that 1 kg equals 1000 g, and 1 meter is 100 centimeters. When converting the volume, remember that \(1 \, \text{m}^3 = (100 \text{cm})^3 = 1,000,000 \, \text{cm}^3\).

\[22,610 \, \frac{\text{kg}}{\text{m}^3} \times \frac{1000 \, \text{g}}{1 \, \text{kg}} \times (\frac{1 \, \text{m}}{100 \, \text{cm}})^3 = 22.61 \, \text{g/cm}^3\]
By breaking this down step by step, you ensure accuracy. Always double-check conversions to prevent errors in calculations.

Volume Calculation

Calculating the volume of a geometric shape is straightforward if you know the dimensions. For a block, use the formula:

\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
Given dimensions of 10.0 cm, 8.0 cm, and 9.0 cm, the calculation is:

\[ \text{Volume} = 10.0 \, \text{cm} \times 8.0 \, \text{cm} \times 9.0 \, \text{cm} = 720 \, \text{cm}^3 \]
Making sure to multiply accurately ensures you determine the correct volume. With this volume and the known density, one can then calculate the mass using the formula:

\[ \text{Mass} = \text{Density} \times \text{Volume} \]
This results in a mass of 16,279.2 g for the osmium block. Accurate calculations lead to precise and meaningful results.