Question

Glycerol is a syrupy liquid often used in cosmetics and soaps. A \(2.50-\mathrm{L}\) sample of pure glycerol has a mass of \(3.15 \times 10^{3} \mathrm{~g}\). What is the density of glycerol in grams per cubic centimeter?

Short answer

The density of glycerol is 1.26 g/cm^3.

Step-by-step solution

Identify the Given Information

Identify and list down the given variables. Here, the volume of glycerol is provided as 2.50 L and the mass is given as 3.15 x 10^3 g.

Understand the Formula for Density

Recall the formula for density, which is the mass of a substance divided by its volume. The formula is: density = mass/volume.

Convert Volume from Liters to Cubic Centimeters

Since density is often expressed in grams per cubic centimeter, convert the volume from liters to cubic centimeters using the conversion factor 1 L = 1000 cm^3. Thus, 2.50 L = 2.50 x 1000 cm^3 = 2500 cm^3.

Calculate the Density

Use the density formula with the mass (3.15 x 10^3 g) and the converted volume (2500 cm^3) to find the density. Density = (3.15 x 10^3 g) / (2500 cm^3).

Express the Density in Grams per Cubic Centimeter

Perform the division to find the density in the desired units. Density = (3.15 x 10^3 g) / (2500 cm^3) = 1.26 g/cm^3.

Key concepts

Density Formula

Understanding how to calculate the density of a substance is fundamental in the fields of science and engineering. At its core, density reflects how much mass is packed into a given volume of material. The density formula, \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) is straightforward yet vital for determining characteristics of different substances.

For instance, if we consider glycerol, the substance mentioned in our exercise, we know that its physical properties significantly depend on its density. The formula dictates we need two pieces of information: the mass and the volume of glycerol. Since mass reflects the amount of matter in an object and volume measures the space it occupies, the density ultimately will tell us how concentrated glycerol is.

When students learn how to apply this formula correctly by plugging in the right units and understanding the physical meaning behind it, they can predict and explain the behavior of materials under various conditions—for example, whether glycerol will float or sink when placed in water.

Unit Conversion

Unit conversion is an indispensable skill in science, especially when dealing with measurements. As we see in the glycerol example, we started with liters but needed to convert to cubic centimeters to proceed with density calculations. Why? Because standard units for density are typically grams per cubic centimeter (g/cm³). So, understanding how to convert \( 1 \text{L} \) to \( 1000 \text{cm}^3 \) is crucial.

Common Unit Conversions

It's helpful to memorize some common conversions, such as \( 1 \text{L} = 1000 \text{cm}^3 \) or \( 1 \text{kg} = 1000 \text{g} \) to allow quick and efficient calculations. Conversion factors serve as a multiplier that, when used correctly, will convert a quantity from one unit to another without changing its actual value.

For our glycerol problem, converting 2.50 liters into cubic centimeters was essential to finding the density. When students understand and can correctly use unit conversions, they can tackle problems across various disciplines without hesitation.

Mass-Volume Relationship

The mass-volume relationship is a core principle in understanding material properties and is directly used to calculate density. Mass represents how much matter an object contains, whereas volume measures the space it occupies. The challenge in our glycerol problem isn't just finding the volume or mass, but understanding how these two properties are related through density.

Importance of Consistent Units

Ensuring consistent units across mass and volume is essential for correct density calculation. If mass is measured in grams and volume in liters, converting volume to cubic centimeters is a must to match the preferred density units of g/cm³. Students often miss this detail, leading to incorrect results.

By applying the mass-volume relationship, students gain insights into how materials compare with each other, like why some substances float while others sink, and can use these principles to predict outcomes in practical applications, like whether a certain amount of glycerol can be stored in a container without overflowing.