Question

An aluminum engine block has a volume of \(4.77 \mathrm{~L}\) and a mass of \(12.88 \mathrm{~kg}\). What is the density of the aluminum in grams per cubic centimeter?

Short answer

The density of the aluminum is 2.70 grams per cubic centimeter.

Step-by-step solution

Convert Volume to Cubic Centimeters

First, convert the volume from liters to cubic centimeters. There are 1000 cubic centimeters in 1 liter, so multiply the volume in liters by 1000.

Convert Mass to Grams

Next, convert the mass from kilograms to grams. There are 1000 grams in 1 kilogram, so multiply the mass in kilograms by 1000.

Calculate Density

Finally, to find the density, divide the mass in grams by the volume in cubic centimeters. The formula for density is \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \].

Key concepts

Unit Conversion

Understanding unit conversion is fundamental in accurately performing mathematical operations, especially in science and engineering.
Conversion of units is essentially changing the measure of a quantity from one unit to another without changing the actual amount. In our exercise, unit conversion is key to find the density of aluminum.

For example, to convert kilometers to meters, knowing that 1 kilometer equals 1000 meters is essential. Similarly, when converting larger units to smaller ones, like liters to cubic centimeters or kilograms to grams, the conversion involves multiplication by an appropriate factor. For liters to cubic centimeters, every 1 liter equals 1000 cubic centimeters, while for kilograms to grams, 1 kilogram equals 1000 grams.

Grasping unit conversions allows students to tackle a wide array of problems accurately and confidently. Always refer to unit conversion charts or conversion factors as a reliable reference point during calculations.

Mass to Grams Conversion

Converting mass to grams is a common task in scientific measurements as grams are a more convenient unit for lab work.
Mass tells us how much matter is in an object and is usually measured in kilograms. When solving for density or working in smaller scales, we prefer grams. To convert kilograms to grams, simply multiply the mass value by 1000 (since 1 kilogram is equivalent to 1000 grams).

For instance, a mass of 12.88 kilograms, as mentioned in the aluminum block example, converted to grams would be:
\( 12.88 \text{ kg} \times 1000 = 12880 \text{ grams} \).

This conversion is indispensable for calculating density because it standardizes the mass unit, making it compatible with volume in cubic centimeters, as density is often expressed in grams per cubic centimeter.

Volume to Cubic Centimeters

Volume describes the amount of space an object occupies, and it's often measured in liters for larger volumes and cubic centimeters for smaller ones.
Converting volume to cubic centimeters is akin to translating a measure from a larger container into the scale of a sugar cube. To do this, we utilize the fact that 1 liter is equivalent to 1000 cubic centimeters.

By multiplying the volume in liters by 1000, you get the volume in cubic centimeters. If we take the example from the exercise, a volume of 4.77 liters is converted as follows:
\( 4.77 \text{ L} \times 1000 = 4770 \text{ cm}^3 \).

This step is crucial when you need to calculate density, which requires volume in cubic centimeters, especially when the mass is already converted to grams.

Density Formula

Density is a measure of how much mass is contained in a given unit volume of a substance. It's a fundamental concept in understanding materials' properties and their behavior in different conditions.
The density formula is quite straightforward:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]

In this formula, mass must be in grams and volume in cubic centimeters to determine the density in grams per cubic centimeter (g/cm³). From the exercise with the aluminum block, after converting mass to grams and volume to cubic centimeters, you divide the mass by the volume to find the block's density:
\[ \text{Density of aluminum} = \frac{12880 \text{ grams}}{4770 \text{ cm}^3} \approx 2.70 \text{ g/cm}^3 \].

The density calculation enables us to understand the relative heaviness of different materials and plays a pivotal role in many scientific and industrial applications.