Question

An uncommon mineral of boron is ulexite, \(\mathrm{NaCaB}_{5} \mathrm{O}_{9} \cdot 8 \mathrm{H}_{2} \mathrm{O}\). How many grams of boron can be produced from \(5.00 \times 10^{2} \mathrm{~kg}\) of ulexite-bearing ore if the ore contains \(0.032 \%\) ulexite by mass and the process has an \(88 \%\) yield?

Short answer

27.44 grams of boron can be produced from 500 kg of ulexite-bearing ore containing 0.032% ulexite by mass with an 88% yield.

Step-by-step solution

Determine the mass of pure ulexite in the ore

To find the mass of pure ulexite, calculate 0.032% of the total ore mass: \[ \text{Mass of ulexite} = \text{Total mass of ore} \times \frac{\text{Percentage of ulexite}}{100} = 5.00 \times 10^{2} \, \mathrm{kg} \times \frac{0.032}{100} = 0.16 \, \mathrm{kg} = 160 \, \mathrm{g} \]

Calculate the molar mass of ulexite

The molar mass of ulexite \(\mathrm{NaCaB}_{5} \mathrm{O}_{9} \cdot 8 \mathrm{H}_{2} \mathrm{O}\) is calculated by summing the molar masses of its constituent elements: \[ \text{Molar mass of ulexite} = (1 \times 22.99) + (1 \times 40.08) + (5 \times 10.81) + (9 \times 16.00) + (8 \times 2 \times 1.01) \, \text{g/mol} = 22.99 + 40.08 + 54.05 + 144 + 16.16 \, \text{g/mol} = 277.28 \, \text{g/mol} \]

Calculate the moles of ulexite

Use the molar mass to convert the mass of ulexite to moles: \[ \text{Moles of ulexite} = \frac{\text{Mass of ulexite}}{\text{Molar mass of ulexite}} = \frac{160 \, \mathrm{g}}{277.28 \, \text{g/mol}} = 0.577 \, \text{mol} \]

Calculate the moles of boron in ulexite

Ulexite contains 5 moles of boron for every 1 mole of ulexite. Therefore, the moles of boron are: \[ \text{Moles of boron} = \text{Moles of ulexite} \times 5 = 0.577 \, \text{mol} \times 5 = 2.885 \, \text{mol} \]

Calculate the mass of boron obtainable from the moles of boron

Convert the moles of boron into grams using the molar mass of boron (10.81 g/mol): \[ \text{Mass of boron} = \text{Moles of boron} \times \text{Molar mass of boron} = 2.885 \, \text{mol} \times 10.81 \, \text{g/mol} = 31.18 \, \text{g} \]

Account for the yield of the process

The actual amount of boron obtained is 88% of the theoretical amount: \[ \text{Actual mass of boron} = \text{Mass of boron} \times \frac{\text{Process yield}}{100} = 31.18 \, \text{g} \times \frac{88}{100} = 27.44 \, \text{g} \]

Key concepts

Chemical Yield Calculation

Understanding the chemical yield is essential as it helps to determine the efficiency of a reaction. Yield is the amount of product produced during a chemical reaction. The theoretical yield is the maximum amount of product that can be obtained, based on stoichiometric calculations, assuming that everything reacts to form the product. However, in practice, not all reactants turn into product due to side reactions or incomplete reactions, which gives us the actual yield.

To calculate the actual yield, we consider the percentage of the theoretical yield that is typically achieved in the process, known as the percent yield. The calculation is expressed as:
\[ \text{Actual yield} = \text{Theoretical yield} \times \frac{\text{Percent yield}}{100} \]
For instance, if the theoretical yield of boron is 31.18 grams and the process yield is 88%, the actual mass of boron would be obtained by multiplying the theoretical yield by 0.88. This distinction is critical in the industry as it helps to measure the efficiency and cost-effectiveness of the production process.

Molar Mass Calculation

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance. It is expressed in units of grams per mole (g/mol) and can be calculated by summing the atomic masses of all the atoms in a molecule. For instance, the molar mass of ulexite, \(\mathrm{NaCaB}_{5} \mathrm{O}_{9} \cdot 8 \mathrm{H}_{2} \mathrm{O}\), is calculated by adding the molar masses of sodium, calcium, boron, oxygen, and hydrogen as present in the chemical formula of ulexite.

Understanding how to compute the molar mass is vital for converting between mass and moles, which allows chemists to count molecules by weighing them, a practice crucial for reaction stoichiometry and yield calculations. Having the correct molar mass is essential for accurate calculations in chemistry.

Stoichiometry

Stoichiometry is the area of chemistry that pertains to the quantitative relationship between reactants and products in a chemical reaction. It is based on the conservation of mass where the total mass of reactants equals the total mass of products. Stoichiometry allows chemists to predict the amounts of substances consumed and produced in a given reaction.

When calculating the amount of boron that can be obtained from ulexite, stoichiometry tells us that one mole of ulexite will produce five moles of boron since the formula for ulexite includes five atoms of boron. This stoichiometric ratio is applied to convert between moles of ulexite and moles of boron by multiplying the moles of ulexite with this ratio, giving us the total moles of boron that can be produced.

This step is critical as it bridges the gap between the molecular scale (moles) and the real world scale (grams), enabling us to measure how much of a material we can expect from a reaction.