Question

The density of osmium is reported by one source to be $22610\mathrm{kg}/$ ${\mathrm{m}}^3$. 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 the block of osmium is 16,278.8 g.

Step-by-step solution

Convert the density from kg/m³ to g/cm³

To convert the density from kg/m³ to g/cm³, we will need to perform a series of unit conversions: 1. Convert kg to g (1 kg = 1000 g) 2. Convert m³ to cm³ (1 m³ = 1,000,000 cm³)

Apply conversion factors

We can now apply the conversion factors to the given density value: $Densit{y}_{g/cm³}=Densit{y}_{kg/m³}\ast \frac{1000g}{1kg}\ast \frac{1m³}{1,000,000cm³}$

Calculate the new density

Substitute the given density value and perform the calculation: $Densit{y}_{g/cm³}=22,610kg/m³\ast \frac{1000g}{1kg}\ast \frac{1m³}{1,000,000cm³}=22.61g/cm³$ So, the density of osmium in g/cm³ is 22.61 g/cm³.

Calculate the volume of the osmium block

To find the mass of the block, we need to first calculate its volume. We can do this using the given dimensions: Volume = Length × Width × Height Using the given values: Volume = 10.0 cm × 8.0 cm × 9.0 cm = 720 cm³

Calculate the mass of the osmium block

Now that we have the volume of the block and the density of osmium in g/cm³, we can calculate the mass using the following formula: Mass = Density × Volume Substitute the values: Mass = 22.61 g/cm³ × 720 cm³ = 16,278.8 g So, the mass of the block of osmium is 16,278.8 g.

Key concepts

Unit Conversion

When you come across a problem that involves different measurement units, unit conversion is an essential tool to use. In sciences like physics and chemistry, you'll often need to convert units to make sense of your measurements in relation to different scales or to perform calculations accurately.

For example, converting from kilograms to grams involves multiplying by 1000, since 1 kilogram equals 1000 grams. Similarly, converting cubic meters to cubic centimeters requires a multiplication by 1,000,000 because 1 cubic meter contains 1,000,000 cubic centimeters. Always remember to use precise conversion factors and be careful with decimal places and zeroes; one small mistake can lead to a significant error in your results. This type of conversion allows you to scale up or down between units, ensuring that you can work with figures that are either more manageable or that align with the standard units required for a particular context or formula.

Mass Calculation

The mass calculation is a fundamental concept in physics and engineering. It refers to the process of determining the amount of matter in an object. Mass is typically measured in kilograms or grams. To calculate the mass when you know the density and volume of an object, you use the formula: \
$$Mass=Density\times Volume$$

In our textbook exercise involving a block of osmium, we first convert the density to compatible units (g/cm³). Then, we use the block's dimensions to calculate its volume in cubic centimeters. Finally, we multiply this volume by the converted density to find the mass in grams. Calculating mass is not only crucial in academic problems but also in real-world applications such as engineering, where it's necessary for designing structures or in commerce for shipping and packaging.

Volume Calculation

The concept of volume calculation is vital when dealing with three-dimensional spaces. Volume refers to the amount of space that an object occupies and is usually expressed in cubic units.

The volume of a rectangular block, such as the osmium block in the exercise, can be found using the formula: \
$$Volume=Length\times Width\times Height$$

When the block's dimensions are provided, as in our example (10.0 cm by 8.0 cm by 9.0 cm), we simply multiply these dimensions to obtain the volume in cubic centimeters (cm³). It's crucial to make sure that all dimensions are in the same units before performing the multiplication, to avoid any confusion and errors in your final calculations. Understanding how to compute volume is essential not just in academic settings but also in everyday situations, such as determining the space needed for storage or the quantity of materials needed for construction.