Problem 26 - General Chemistry: Principles and Modern Applications

Question

All of the following minerals are semiprecious or precious stones. Determine the mass percent of the indicated element. (a) \(\mathrm{Zr}\) in zircon, \(\mathrm{ZrSiO}_{4}\) (b) Be in beryl (emerald), \(\mathrm{Be}_{3} \mathrm{Al}_{2} \mathrm{Si}_{6} \mathrm{O}_{18}\) (c) Fe in almandine (garnet), \(\mathrm{Fe}_{3} \mathrm{Al}_{2} \mathrm{Si}_{3} \mathrm{O}_{12}\) (d) S in lazurite (lapis lazuli), \(\mathrm{Na}_{4} \mathrm{SSi}_{3} \mathrm{Al}_{3} \mathrm{O}_{12}\)

Step-by-Step Solution

Answer

The mass percent of each element in the mineral is calculated by the relation of its molar mass in the compound over the total molar mass of the compound. The solutions are obtained by applying this formula to each case.

Step 1: Molar mass calculation

For every mineral, calculate the molar mass which is the sum of the atomic masses of every atom in a single molecule of that substance. The atomic mass of each element can be found on a standard periodic table. Also calculate the molar mass for the specific element in question.

Step 2: Calculation for Zircon

For zircon, \(\mathrm{ZrSiO}_{4}\), use the atomic masses to calculate the molar mass: \( \text{Molar Mass of ZrSiO}_{4} = 1* \text{Molar mass of Zr} + 1 * \text{Molar mass of Si} + 4 * \text{Molar mass of O}\). The percent of Zr in Zircon is then the molar mass of Zr divided by the molar mass of ZrSiO4, times 100.

Step 3: Calculation for Beryl (Emerald)

The process for calculating the percent of Be in beryl (\(\mathrm{Be}_{3} \mathrm{Al}_{2} \mathrm{Si}_{6}\mathrm{O}_{18}\)) is the same as for zircon, but with different elements and quantities involved. Similarly, calculate the molar mass and then use it to find the percentage.

Step 4: Calculation for Almandine (Garnet)

Follow similar steps to calculate for almandine (\(\mathrm{Fe}_{3} \mathrm{Al}_{2} \mathrm{Si}_{3}\mathrm{O}_{12}\)), evaluating the percent of Fe (iron).

Step 5: Calculation for Lazurite (Lapis Lazuli)

Finally, for lazurite (\(\mathrm{Na}_{4} \mathrm{SSi}_{3} \mathrm{Al}_{3} \mathrm{O}_{12}\)), calculate the molar mass for the whole mineral and then isolate for S (sulfur) to find the mass percent.

Key Concepts

Molar Mass

When discussing chemistry and minerals, molar mass is an essential concept to understand. Molar mass refers to the mass of one mole of a given substance. It is numerically equal to the sum of the atomic masses of the atoms in a molecule. These atomic masses are easily found on the periodic table and are measured in atomic mass units (amu). For example, in order to calculate the molar mass of zircon, with the formula \( \mathrm{ZrSiO}_4 \), you must add together the atomic masses of one zirconium (Zr), one silicon (Si), and four oxygen (O) atoms.

• Zirconium (Zr): Atomic mass ≈ 91.22 amu
• Silicon (Si): Atomic mass ≈ 28.09 amu
• Oxygen (O): Atomic mass ≈ 16.00 amu

To find the molar mass of \( \mathrm{ZrSiO}_4 \), calculate it as follows:\[\text{Molar Mass} = (91.22) + (28.09) + (4 \times 16.00) = 183.30 \text{ grams per mole}\]Understanding molar mass is crucial because it allows you to convert between grams and moles, making it a fundamental part of chemical reactions and calculations.

Atomic Mass

To understand mass percent calculations in semiprecious stones, it's important to grasp the concept of atomic mass. Atomic mass is the mass of a single atom, usually measured in atomic mass units (amu). This value considers both the protons and neutrons in an atom's nucleus. Let's take the example of beryl, which is a mineral with the formula \( \mathrm{Be}_3 \mathrm{Al}_2 \mathrm{Si}_6 \mathrm{O}_{18} \). In this compound, the atomic masses of the elements are:

• Beryllium (Be): Atomic mass ≈ 9.01 amu
• Aluminum (Al): Atomic mass ≈ 26.98 amu
• Silicon (Si): Atomic mass ≈ 28.09 amu
• Oxygen (O): Atomic mass ≈ 16.00 amu

The total molar mass of beryl is calculated using these atomic masses, giving the sum for the entire structure of the mineral.For finding mass percent, focus on the specific element. For example, to calculate the mass percent of Beryllium in beryl, you would isolate its contribution compared to the entire molar mass of \( \mathrm{Be}_3 \mathrm{Al}_2 \mathrm{Si}_6 \mathrm{O}_{18} \). Understanding atomic mass enables these kind of specific calculations and comparisons.

Semiprecious Stones

Semiprecious stones, like those in our exercise, are mineral formations valued for their beauty, rarity, and physical properties. Unlike precious stones such as diamonds and rubies, semiprecious stones are generally more abundant and accessible. However, they still play a prominent role in jewelry and ornamentation. Examples from the exercise include:

• Zircon: A mineral associated with vibrant colors and brilliance, often used as a diamond substitute.
• Beryl (Emerald): Known for its green hues, emerald is one of the more esteemed varieties of beryl.
• Almandine (Garnet): Almandine is a type of garnet often characterized by its deep red color.
• Lazurite (Lapis Lazuli): Famous for its intense bluish-purple shade and is commonly used for decorative purposes.

In scientific terms, understanding the composition of these stones through chemical formulas allows us to explore their properties further. These stones are composed of various elements whose weights and proportions can be calculated using the concepts of molar mass and atomic mass. By analyzing their chemical makeup, we can better understand the unique traits that each semiprecious stone offers.