Question

(a) If \(25.0 \mathrm{~cm}^{3}\) of an unknown substance has a mass of \(195 \mathrm{~g}\), what is the density of the substance in grams per cubic centimeter? (b) How many cubic centimeters does \(500.0 \mathrm{~g}\) of the substance occupy? (c) Does this substance sink or float in mercury, which has a density of \(13.6 \mathrm{~g} / \mathrm{mL} ?\)

Short answer

(a) The density of the unknown substance is 7.8 g/cm³. (b) 500.0 g of the substance will occupy approximately 64.1 cm³. (c) The substance will float in mercury since its density is lower than mercury's density (7.8 g/cm³ < 13.6 g/cm³).

Step-by-step solution

Identify the given values

We are given the volume and mass of the unknown substance: Volume = 25.0 cm³ Mass = 195 g

Find the density

We will now use the density formula to find the density of the substance: Density = Mass / Volume Density = 195 g / 25.0 cm³

Calculate the density

The density of the unknown substance is: Density = \( \frac{195}{25.0} \) g/cm³ Density = 7.8 g/cm³ Exercise (b):

Identify the given value and the density found in exercise (a)

We are given the mass of the unknown substance: Mass = 500.0 g Density from exercise (a) = 7.8 g/cm³

Calculate the volume occupied by 500 g of the substance

We will rearrange the density formula to find the volume: Volume = Mass / Density Volume = 500.0 g / 7.8 g/cm³

Calculate the volume occupied

The volume occupied by 500.0 g of the unknown substance is: Volume = \( \frac{500.0}{7.8} \) cm³ Volume ≈ 64.1 cm³ Exercise (c):

Compare the density of the unknown substance with the density of mercury

Density of unknown substance = 7.8 g/cm³ Density of mercury = 13.6 g/cm³

Determine if the substance will sink or float

Since the density of the unknown substance is lower than the density of mercury, the substance will float in mercury.

Key concepts

Density Formula

Understanding the concept of density is fundamental in physics and chemistry. Density is essentially a measurement that compares the amount of matter an object has to its volume. The denser an object, the more mass it has in a given volume. The density formula is a simple but powerful tool given by the equation:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Here, the mass is typically measured in grams (g) and the volume in cubic centimeters (cm³) or milliliters (mL), which are equivalent. The resulting density is therefore expressed in grams per cubic centimeter (g/cm³) or grams per milliliter (g/mL).
For instance, to calculate the density of a substance, you'll need to know the mass and the volume of the substance, as in exercise (a). By simply dividing the mass by the volume, you'll obtain the density, a characteristic property that can aid in substance identification.

Mass-Volume Relationship

The relationship between mass and volume is at the heart of the density calculation. Since the density is mass per unit volume, any change in the mass or volume will affect the density. This mass-volume relationship helps us solve various problems, such as exercise (b).
To find the volume that a known mass of a substance occupies, you can rearrange the density formula:
\[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \]
This equation reminds us that if you have a greater mass of a substance, you'd expect it to occupy a larger volume if the density remains constant. Conversely, for a given mass, a substance with higher density will occupy less volume than a substance with lower density. Understanding this relationship can provide insights into the behavior of materials under different conditions.

Buoyancy Principle

The buoyancy principle, also known as Archimedes' Principle, states that an object submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object. This principle is critical when considering whether objects will sink or float in a given fluid.
According to this principle, whether a substance sinks or floats depends on its density compared to the fluid's density. In exercise (c), we compared the density of an unknown substance to that of mercury. Since the substance's density (7.8 g/cm³) is less than mercury's density (13.6 g/cm³), the substance will float. Objects with a density lower than the fluid's density will float on the surface, whereas those with a greater density will sink. This is why ships made of steel can float on water; they are designed to displace enough water that their overall density is less than that of water.