Question

A \(5.00 \mathrm{~L}\) sample of pure glycerol has a mass of \(6.30 \times\) \(10^{3} \mathrm{~g}\). What is the density of glycerol in \(\mathrm{g} / \mathrm{cm}^{3}\) ?

Short answer

The density of glycerol is \(1.26 \, \text{g} / \text{cm}^3\).

Step-by-step solution

Understand the concept of density

Density is defined as the mass per unit volume of a substance. It can be calculated using the formula \(\text{density} = \frac{\text{mass}}{\text{volume}}\).

Convert the volume to the correct unit

The volume is given in liters (\(5.00 \, L\rlap{\text{iters}}\)). To find the density in \(\frac{\text{g}}{\text{cm}^3}\), convert the volume from liters to cubic centimeters (\(1 \, L = 1000 \, cm^3\)). Thus, \(5.00 \, L = 5.00 \times 1000 \, cm^3 = 5000 \, cm^3\).

Calculate the density

With the mass \(6.30 \times 10^3 \, g\) and the converted volume \(5000 \, cm^3\), use the density formula to calculate the density: \(\text{density} = \frac{\text{mass}}{\text{volume}} = \frac{6.30 \times 10^3 \, g}{5000 \, cm^3}\).

Perform the calculation

Perform the division to find the density: \(\text{density} = \frac{6.30 \times 10^3 \, g}{5000 \, cm^3} = 1.26 \, \frac{\text{g}}{\text{cm}^3}\).

Key concepts

Understanding Density

Density is a fundamental concept in science, representing how much matter is packed into a given volume. It indicates how heavy or light a material is for its size. To calculate the density of a substance, we use the density formula:
\[\text{density} = \frac{\text{mass}}{\text{volume}}\]
In this context, mass refers to the amount of matter in the substance, usually measured in grams (g), kilograms (kg), or pounds (lb). Volume measures the space that the substance occupies, which can be in liters (L), cubic centimeters (cm^3), or cubic meters (m^3). The density will be expressed in terms of mass per unit volume, such as grams per cubic centimeter (\(\frac{g}{cm^3}\)) or kilograms per cubic meter (\(\frac{kg}{m^3}\)).
Understanding the concept of density is crucial, especially when you need to compare materials or determine whether a substance will float or sink in a fluid.

Mastering Unit Conversion

Whenever you work with measurements, unit conversion is an indispensable skill. It ensures that your calculations are correct and standardized. In the case of density calculations, we often need to convert volume units to align with the mass units.
For example, converting liters to cubic centimeters is a common task since density is frequently reported in \(\frac{g}{cm^3}\). Here's a vital conversion to remember:
\[1 \, L = 1000 \, cm^3\]
Applying this conversion factor allows you to transform the volume from larger to smaller units (or vice versa) accurately. A systematic approach to unit conversion helps prevent errors and ensures that you compare like for like when evaluating the density of substances.

Applying the Density Formula

After understanding density and mastering unit conversion, you can apply the density formula to real-world problems. This process involves inserting the mass and volume (after conversion) into the formula to calculate density.

Step-by-Step Calculation

Firstly, ensure the units are compatible—mass in grams and volume in cubic centimeters. Then, substitute the values into the density formula:
\[\text{density} = \frac{\text{mass}}{\text{volume}}\]
Next, divide the mass by volume:
\[\text{density} = \frac{6.30 \times 10^3 \, g}{5000 \, cm^3}\]
After performing the division, you'll find the density of the substance, which in our example problem is glycerol. This practical application is vital across various fields including physics, chemistry, and engineering, for tasks like material selection, buoyancy calculations, and quality control.