Question

The density of osmium (the densest metal) is \(22.57 \mathrm{~g} / \mathrm{cm}^{3}\) What is the mass of a block of osmium with dimensions \(1.84 \mathrm{~cm} \times 3.61 \mathrm{~cm} \times 2.10 \mathrm{~cm} ?\)

Short answer

The mass of the osmium block with dimensions \(1.84\mathrm{~cm} \times 3.61\mathrm{~cm} \times 2.10\mathrm{~cm}\) is approximately \(314.906\mathrm{~g}\).

Step-by-step solution

Find the volume of the block#line_content#To find the volume of the osmium block, multiply its dimensions: \(Volume = Length \times Width \times Height\)

Step 2: Calculate the volume using given dimensions#line_content#Using the given dimensions, we can calculate the volume of the block: \(Volume = 1.84\mathrm{~cm} \times 3.61\mathrm{~cm} \times 2.10\mathrm{~cm}\)

Compute the volume value#line_content#Perform the multiplication: \(Volume = 13.9448\mathrm{~cm}^{3}\)

Step 4: Find the mass of the block using the density formula#line_content#Now, use the density formula to find the mass of the block: \(Mass = Density \times Volume\)

Calculate the mass using the given density and found volume#line_content#Using the given density and the calculated volume, we can calculate the mass of the osmium block: \(Mass = 22.57 \mathrm{~g}/\mathrm{cm}^{3} \times 13.9448\mathrm{~cm}^{3}\)

Step 6: Compute the mass value#line_content#Perform the multiplication: \(Mass = 314.906\mathrm{~g}\) So, the mass of the block of osmium with dimensions \(1.84\mathrm{~cm} \times 3.61\mathrm{~cm} \times 2.10\mathrm{~cm}\) is approximately \(314.906\mathrm{~g}\).

Key concepts

Volume Calculation

Understanding how to calculate the volume of an object is foundational in density calculations. Volume represents the amount of space an object occupies. For a block-shaped object, like our osmium block, volume is found by multiplying its length, width, and height. This is expressed as:

• \( \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \)

Given the osmium block with dimensions 1.84 cm, 3.61 cm, and 2.10 cm, we compute:

• \( \text{Volume} = 1.84 \text{ cm} \times 3.61 \text{ cm} \times 2.10 \text{ cm} \)
• \( \text{Volume} = 13.9448 \text{ cm}^3 \)

Breaking down the multiplication helps solidify understanding. Multiply two numbers first, then multiply the result by the third. This step-by-step approach ensures accuracy.

Mass Calculation

Mass is a measure of how much matter is in an object. To find the mass of our osmium block, we use the formula:

• \( \text{Mass} = \text{Density} \times \text{Volume} \)

Since density is given as 22.57 g/cm³, and we've calculated the volume to be 13.9448 cm³, we can find the mass:

• \( \text{Mass} = 22.57 \text{ g/cm}^3 \times 13.9448 \text{ cm}^3 \)
• \( \text{Mass} = 314.906 \text{ g} \)

The calculation involves straightforward multiplication. Remember that the unit cm³ cancels out, leaving grams for mass. This clear method ensures the mass is easily understood and accurately calculated.

Measurement Units

Understanding measurement units is key to performing calculations accurately. In this exercise, we used:

• Length, width, and height in centimeters (cm)
• Density in grams per cubic centimeter (g/cm³)
• Mass in grams (g)

We multiply the linear units (cm) to get cubic centimeters (cm³), representing volume. When we calculate mass, the volume unit cancels out with the density unit, resulting in grams.
Being meticulous with units helps prevent errors. It's always a good idea to double-check that all units match up properly before performing calculations.