Question

Iron pyrite \(\left(\mathrm{FeS}_{2}\right)\) is known as Fool's Gold because of its resemblance to gold metal. However, it can easily be distinguished from gold by the difference in their densities. The density of gold is \(19.28 \mathrm{~g}^{\mathrm{cm}} \mathrm{cm}^{3}\) and that of Fool's Gold is \(5.01 \mathrm{~g}^{\prime} \mathrm{cm}^{3}\). What volume of Fool's Gold would have the same mass as a \(4.0 \mathrm{~cm}^{3}\) piece of gold?

Short answer

The volume of Fool's Gold with the same mass as the 4.0 cm^3 piece of gold is approximately 15.4 cm^3.

Step-by-step solution

Calculate the mass of the gold

First, calculate the mass of the gold piece using its volume and density. The formula to calculate mass (m) is mass = density (d) times volume (V). For gold, the density is 19.28 g/cm^3 and the volume is 4.0 cm^3. So the mass of the gold m_gold = d_gold * V_gold = 19.28 g/cm^3 * 4.0 cm^3.

Calculate the volume of Fool's Gold with the same mass

Now, use the mass of the gold to find the volume of the Fool's Gold that would have this same mass. The formula to calculate volume (V) is volume = mass / density. For Fool's Gold, the density is 5.01 g/cm^3. So the volume of the Fool's Gold V_FeS2 = m_gold / d_FeS2.

Plug in the mass of the gold into the volume formula for Fool's Gold

Take the mass of gold calculated in Step 1 and divide it by the density of Fool's Gold. V_FeS2 = (19.28 g/cm^3 * 4.0 cm^3) / 5.01 g/cm^3.

Solve for the volume of the Fool's Gold

Finally, calculate the volume of Fool's Gold by solving the equation from Step 3. V_FeS2 = (77.12 g) / 5.01 g/cm^3.

Key concepts

Mass and Volume Relationship

Understanding the relationship between mass and volume is fundamental in science, particularly in disciplines such as physics, chemistry, and engineering. Mass refers to the amount of matter in an object, while volume is the space that object occupies.

It's crucial to recognize that these two properties are directly proportional under constant conditions for a given substance. This means that if you take a material and double its volume without changing its composition, its mass will also double. Similarly, if you shrink the volume, the mass decreases as well. This relationship is quantified through the concept of density.

Density is defined as the mass of a substance per unit volume and is expressed using the formula: \[ \text{Density} (\rho) = \frac{\text{Mass} (m)}{\text{Volume} (V)} \]. The units often used for density in a laboratory setting are grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³).

Conversion of Mass and Volume

Sometimes in scientific calculations, you need to convert the mass of a substance to its corresponding volume, or vice versa, using the substance's density as a conversion factor. This is particularly useful in stoichiometry calculations in chemistry or when dealing with fluid dynamics in physics.

For conversion, the following formulae are employed depending on what is known and what needs to be found:

• To find mass (m) when density (\( \rho \)) and volume (V) are known, use: \[ m = \rho \times V \].
• To find volume (V) when the mass (m) and density (\( \rho \)) are known, use: \[ V = \frac{m}{\rho} \].

These formulae can also be used in reverse to calculate the density if the mass and volume of a substance are known. It is crucial to ensure that all units are consistent when performing these calculations. For example, if density is given in g/cm³, the mass should be in grams and the volume in cubic centimeters.

Density of Substances

Each substance has a characteristic density that is unique under specific conditions of temperature and pressure. The density of a substance can be used to identify it, as in the case of Fool's Gold (iron pyrite, \( \mathrm{FeS}_{2} \)) and real gold. It also influences whether the substance will float or sink in a fluid and how substances interact with each other.

Densities can vary widely between different materials. For instance, metals generally have higher densities than plastics or cork. Understanding density calculations is crucial for tasks such as mixture separation, alloy creation, and even culinary applications like determining whether an egg is fresh (fresh eggs sink in water due to their higher density compared to spoiled eggs).

Density not only serves as a distinguishing characteristic between substances like iron pyrite and gold in educational exercises, but it's also a practical tool in industries such as material science and geology.