Problem 86 - General Chemistry: Principles and Modern Applications

Question

The important natural sources of boron compounds are the minerals kernite, \(\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7} \cdot 4 \mathrm{H}_{2} \mathrm{O}\) and borax, \(\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7} \cdot 10 \mathrm{H}_{2} \mathrm{O} .\) How much additional mass of mineral must be processed per kilogram of boron obtained if the mineral is borax rather than kernite?

Step-by-Step Solution

Answer

The additional mass of mineral that must be processed per kilogram of boron obtained if the mineral is borax rather than kernite is approximately 0.12 kg.

Step 1: Determine the Molar Mass of Boron

Firstly, use the periodic table to determine the molar mass of each element in the compounds. Boron (B) is 10.81 g/mol.

Step 2: Calculate the molar mass of Kernite and Borax

Next, calculate the molar mass of kernite \(\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7} \cdot 4 \mathrm{H}_{2} \mathrm{O}\) and borax \(\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7} \cdot 10 \mathrm{H}_{2} \mathrm{O}\). For Kernite, it is (2*22.99) + (4*10.81) + (7*16.00) + (8*1.008) = 381.376 g/mol. For Borax, it is (2*22.99) + (4*10.81) + (7*16.00) + (20*1.008) = 411.376 g/mol.

Step 3: Find the proportion of Boron

Now determine the mass fraction of boron in kernite and borax. The mass of Boron in kernite is \(4*10.81\) = 43.24 g. The mass fraction of boron in kernite is \(43.24 / 381.376\) and for borax is same, but divided by the molar mass of borax: \(43.24 / 411.376\)

Step 4: Compare the mass needed

Finally, we will determine how much more borax is needed to yield the same amount of boron as kernite. If 1 kg of boron is obtained from kernite, the mass of borax needed will be \(1 / (43.24 / 411.376)\) kg. Then subtract the mass of kernite (1kg) to determine the additional mass needed.

Key Concepts

Kernite

Kernite is a significant mineral source of boron, with the chemical formula \(\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7} \cdot 4 \mathrm{H}_{2} \mathrm{O}\).
It belongs to a class of minerals known as borates, which are salts containing boron.
Kernite specifically is a hydrated sodium borate. Kernite is a valuable natural resource because it concentrates boron, an element used in many industrial applications and products, such as glass and ceramics.
The boron atoms in kernite are bonded in a way that stabilizes the compound while storing water molecules as seen in its chemical formula.
This hydration level (four water molecules for each formula unit) affects its molar mass, impacting how we calculate the proportion of boron in the mineral.Knowing the molar mass of kernite is crucial for determining its boron content. This molar mass is calculated with the individual atomic masses of the components: two sodium atoms, four boron atoms, seven oxygen atoms from the \(\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7}\) core, and an additional four water molecules, making for a total molar mass of \(381.376\ g/mol\).
This calculation provides insight into how efficiently we can extract boron from kernite compared to other boron-containing minerals.

Borax

Borax, with the chemical formula \(\mathrm{Na}_{2} \mathrm{B}_{4} \mathrm{O}_{7} \cdot 10 \mathrm{H}_{2} \mathrm{O}\), is another primary source of boron like kernite.
It is classified as a tetraborate mineral and is known for its white crystalline appearance. Unlike kernite, borax contains more water molecules, evidenced by the ten water molecules in its structure.
This increased hydration level affects not only its appearance and physical properties but also its molar mass. The molar mass for borax is heavier: \(411.376\ g/mol\). The additional water content in borax leads to a lower percentage of boron by mass compared to kernite. To extract an equivalent amount of boron, a larger quantity of borax would be needed.
This is an important factor to consider in industries relying on boron extraction, as processing larger quantities implies additional costs and resources.

Molar Mass Calculation

Calculating the molar mass of compounds like kernite and borax is a foundational skill in chemistry that helps understand the composition of minerals.
Molar mass is the mass of one mole of a substance, expressed in grams per mole \(g/mol\) and is calculated by summing up the atomic masses of all the atoms in a chemical formula.

Steps to Calculate Molar Mass:

• Find the atomic masses of each element in the compound from the periodic table. For instance, Sodium (Na) is approximately \(22.99\ g/mol\), Boron (B) is \(10.81\ g/mol\), and Oxygen (O) is \(16.00\ g/mol\). Hydrogen (H), found in the water of hydration, is \(1.008\ g/mol\).
• Multiply the atomic mass of each element by the number of times it appears in the compound. For kernite: \((2 \times 22.99) + (4 \times 10.81) + (7 \times 16.00) + (8 \times 1.008)\).
• Add up these results to get the total molar mass. Kernite totals \(381.376\ g/mol\) while borax gives \(411.376\ g/mol\).

The molar mass is vital to determine how much of each element, like boron, is present in a compound.
By understanding these calculations, chemists can discern the efficiency of mineral extraction processes and the availability of key elements within compounds.