Question

An ore contains \(2.296 \%\) of the mineral argentite, \(\mathrm{Ag}_{2} \mathrm{~S}\), by mass. How many grams of this ore would have to be processed in order to obtain \(1.00 \mathrm{~g}\) of pure solid silver? \((\mathrm{Ag}=108)\) (a) \(1.148 \mathrm{~g}\) (b) \(0.026 \mathrm{~g}\) (c) \(50 \mathrm{~g}\) (d) \(2.296 \mathrm{~g}\)

Short answer

To obtain 1.00 g of pure silver, 50 g of this ore would need to be processed. The correct answer is (c) 50 g.

Step-by-step solution

Calculate the mass percent of silver in argentite

First, calculate the molar mass of argentite (Ag2S) which is the sum of twice the atomic mass of silver (Ag) plus the atomic mass of sulfur (S). For silver (Ag), the atomic mass is 108. Therefore, the molar mass of Ag2S is computed as (2 * 108) + 32 (atomic mass of sulfur) which equals 248 grams/mole. Now, to find the mass percent of silver in argentite, divide the mass contributed by silver (2 * 108) by the molar mass of Ag2S and multiply by 100% to get the percentage.

Determine the mass of Ag in the ore that contains the mineral

To find out how much ore is needed for 1.00 g of silver, take note that 2.296% of the ore by mass is argentite. First, find the mass percent of silver in argentite from Step 1. Multiply this value by the percentage of argentite in the ore (2.296%). This gives you the mass percent of silver in the ore.

Calculate the mass of ore required

To extract 1.00 g of pure silver, you would need to process enough ore that contains 1.00 g of silver. Divide 1.00 g by the mass percent of silver in the ore to find out the mass of the ore that needs to be processed.

Key concepts

Molar Mass Calculation

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance, typically in grams per mole (\text{g/mol}). To calculate the molar mass of a compound, like argentite (\text{Ag}_2\text{S}), you need to sum the atomic masses of all the atoms present in the molecule.

For argentite, this involves adding twice the atomic mass of silver (\text{Ag}), which is given as 108 atomic mass units (amu), with the atomic mass of sulfur (\text{S}), which is approximately 32 amu. Therefore, the molar mass of argentite is calculated as follows: \[ \text{Molar mass of } \text{Ag}_2\text{S} = (2 \times \text{mass of } \text{Ag}) + (\text{mass of } \text{S}) = (2 \times 108) + 32 = 248 \text{ g/mol} \]
This calculation is essential as it serves as a bridge between the microscopic world of atoms and the macroscopic world of grams that can be measured in a laboratory.

Mass Percent Composition

The mass percent composition indicates the percentage by mass of each element in a compound. In the context of argentite, the mass percent composition of silver can be determined by dividing the total mass of silver in the molecule by the molar mass of the compound and then multiplying by 100 to convert it to a percentage.

With the molar mass of argentite previously calculated as 248 \text{g/mol}, and knowing that each argentite molecule contains two silver atoms, the mass percent composition of silver is calculated as follows: \[ \text{Percent by mass of } \text{Ag} = \frac{2 \times \text{mass of } \text{Ag}}{\text{Molar mass of } \text{Ag}_2\text{S}} \times 100\% = \frac{2 \times 108}{248} \times 100\% \]
Understanding mass percent composition is vital, as it allows chemists to predict the quantity of each element present in a sample of a given compound.

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. For practical applications, such as extracting a pure element from a mineral, stoichiometry helps determine the exact amount of starting material needed.

When considering the extraction of silver from an ore containing argentite, stoichiometry involves using the mass percent of silver in the mineral and the percentage of argentite within the ore to calculate how much ore is required to obtain a specific amount of pure silver. It includes setting up ratios and using conversion factors to go from grams of ore to grams of silver. This process is key in industries for maximizing yield and cost efficiency.

Chemistry of Minerals

Understanding the chemistry of minerals is essential for processes like the extraction of metals from ores. Minerals, like argentite, which is a source of silver, are naturally occurring chemical compounds or elements. They are characterized by a specific chemical composition and crystal structure.

Argentite's composition directly affects how it's processed. Its chemical behavior determines the methods used to extract silver, such as roasting in the presence of oxygen to yield silver and sulfur dioxide. The purity of extracted silver can also depend on the presence of other minerals and the precise chemical processing steps taken. Therefore, a thorough understanding of a mineral's chemistry is crucial for efficient extraction and pure results.