Problem 76 - General Chemistry

Question

A mineral sample has a mass of \(5.94 \mathrm{~g}\) and a volume of \(0.73 \mathrm{~cm}^{3}\). The mineral is either sphalerite (density = \(4.0 \mathrm{~g} / \mathrm{cm}^{3}\) ), cassiterite (density \(=6.99 \mathrm{~g} / \mathrm{cm}^{3}\) ), or cinnabar (density \(=8.10 \mathrm{~g} / \mathrm{cm}^{3}\) ). Which is it?

Step-by-Step Solution

Answer

The mineral is cinnabar.

Step 1: Understanding the Concept

To identify the mineral, we need to determine its density and compare it to the given densities of sphalerite, cassiterite, and cinnabar. Density is given by the formula: \(\text{Density} = \frac{\text{Mass}}{\text{Volume}}\).

Step 2: Calculating the Density

Substitute the given mass and volume of the mineral sample into the density formula. The mass is \(5.94 \ \text{g}\) and the volume is \(0.73 \ \text{cm}^3\). So, \(\text{Density} = \frac{5.94}{0.73}\approx 8.14 \ \text{g/cm}^3\).

Step 3: Compare with Given Densities

Compare the calculated density \(8.14 \ \text{g/cm}^3\) with the densities of the given minerals: sphalerite (\(4.0 \ \text{g/cm}^3\)), cassiterite (\(6.99 \ \text{g/cm}^3\)), and cinnabar (\(8.10 \ \text{g/cm}^3\)).

Step 4: Identifying the Mineral

The calculated density \(8.14 \ \text{g/cm}^3\) is closest to the density of cinnabar, which is \(8.10 \ \text{g/cm}^3\). Hence, the mineral is cinnabar.

Key Concepts

Understanding Density

Density is a fundamental concept that allows us to understand how much mass is packed into a given volume. It is crucial in various scientific fields, including physics, chemistry, and geology. The formula to calculate density is simple:

• Density = \( \frac{\text{Mass}}{\text{Volume}} \)

Mass is usually measured in grams, and volume in cubic centimeters (cm³), leading to density units of grams per cubic centimeter (g/cm³). It helps in determining how "heavy" an object feels for its size. By calculating density, we can compare whether an object will float or sink in a fluid, identify materials, and understand mineral composition. Breaking down the steps of calculating density can demystify this concept for students: measure mass, measure volume, and divide mass by volume.

Mineral Identification

Mineral identification often involves determining the density of a sample. Since many minerals have unique densities, you can use calculated density to identify unknown minerals. This task is particularly common in geology, where identifying a mineral's composition is necessary for classification and study. In the given problem, we calculated the density of our mineral sample and compared it to the known densities of sphalerite, cassiterite, and cinnabar. Making these comparisons allows us to deduce that the sample is closest to cinnabar. Thus, density serves as a fingerprint, helping to identify minerals and other materials in a practical and scientific context.

Mass and Volume Relationship

The relationship between mass and volume is direct and crucial for calculating density. Mass refers to the amount of matter in an object, while volume indicates the space it occupies. These two properties are interconnected by density, which is often invoked to tell how compact the matter is within a given volume. Measuring Mass and Volume

• Mass: Use a balance to determine the object's mass, usually in grams.
• Volume: For solids with regular shapes, use geometric formulas. For irregular shapes, you can use water displacement methods.

By understanding how mass and volume work with one another, you can determine how materials stack up, literally, in the scientific sense!

Comparing Densities for Mineral Identification

Comparing densities is a practical step in mineral identification. Once you have determined the density of a mineral sample, comparing it to known densities of candidate minerals is the next logical step. By this comparison, we can often identify an unknown mineral. In the exercise, the calculated density of the mineral sample was approximately 8.14 g/cm³. This finding was then compared against the densities of sphalerite (4.0 g/cm³), cassiterite (6.99 g/cm³), and cinnabar (8.10 g/cm³). Since the measured density was close to that of cinnabar, it was reasonable to conclude the mineral in question was cinnabar. Remember, these comparisons are not just about numbers; they provide insights into the type and nature of the materials we are studying. They help us contextualize materials in our daily lives, from identifying gems to understanding planetary compositions.