Problem 100 - Chemistry and Chemical Reactivity

Question

The elements silver, molybdenum, and sulfur combine to form \(\mathrm{Ag}_{2} \mathrm{MoS}_{4} .\) What is the maximum mass of \(\mathrm{Ag}_{2} \mathrm{MoS}_{4}\) that can be obtained if \(8.63 \mathrm{g}\) of silver, 3.36 g of molybdenum, and 4.81 g of sulfur are combined? (Hint: What is the limiting reactant?)

Step-by-Step Solution

Answer

The maximum mass of \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \) that can be obtained is 15.40 g.

Step 1: Calculate Molar Masses

First, determine the molar masses of each element in \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \).- Silver (Ag): 107.87 g/mol. Since there are 2 Ag atoms, \( 2 \times 107.87 = 215.74 \) g/mol.- Molybdenum (Mo): 95.95 g/mol.- Sulfur (S): 32.07 g/mol. There are 4 S atoms, \( 4 \times 32.07 = 128.28 \) g/mol.The total molar mass of \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \) is \( 215.74 + 95.95 + 128.28 = 439.97 \) g/mol.

Step 2: Calculate Moles of Each Element

Convert the mass of each element to moles.- Moles of Ag: \( \frac{8.63}{107.87} = 0.08 \) moles.- Moles of Mo: \( \frac{3.36}{95.95} = 0.035 \) moles.- Moles of S: \( \frac{4.81}{32.07} = 0.15 \) moles.

Step 3: Determine Limiting Reactant

The chemical formula \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \) requires 2 moles of Ag, 1 mole of Mo, and 4 moles of S per formula unit. - For Ag: \( \frac{0.08}{2} = 0.04 \) reaction units.- For Mo: \( 0.035 \) reaction units.- For S: \( \frac{0.15}{4} = 0.0375 \) reaction units.The limiting reactant is Mo with 0.035 reaction units, being the smallest value.

Step 4: Calculate Maximum Mass of \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \)

Using the limiting reactant (Mo), calculate the maximum mass of \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \) that can form.Each reaction unit produces 1 mole of \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \), so the moles of the compound formed are 0.035.Maximum mass = moles of compound \( \times \) molar mass.\( 0.035 \times 439.97 = 15.40 \) g is the maximum mass of \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \) that can be obtained.

Key Concepts

Molar Mass Calculation

Calculating molar mass is essential in chemistry to understand how much one mole of a substance weighs. Molar mass is the sum of the atomic masses of all atoms in a molecule. For a compound like \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \), you need to:

• Find the atomic mass of each element: Silver (Ag) is 107.87 g/mol, Molybdenum (Mo) is 95.95 g/mol, and Sulfur (S) is 32.07 g/mol.
• Multiply the atomic mass by the number of atoms of each element in the formula: For Ag, it's \( 2 \times 107.87 = 215.74\) g/mol, and for S, it's \( 4 \times 32.07 = 128.28 \) g/mol.
• Add them together: \( 215.74 + 95.95 + 128.28 = 439.97 \) g/mol for \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \).

This calculation helps us understand the weight of one mole of the compound, which is key for converting between grams and moles.

Chemical Formula Analysis

Analyzing a chemical formula reveals the proportion and type of atoms in a compound. The subscript numbers in \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \) tell us how many atoms of each element are present.

Here's what is represented:

• There are 2 silver (Ag) atoms.
• There is 1 molybdenum (Mo) atom.
• There are 4 sulfur (S) atoms.

This analysis helps determine how much of each constituent is needed for the reaction to proceed. It's like following a recipe where each ingredient must be measured accurately to achieve the desired result.

Stoichiometry

Stoichiometry is used to calculate the quantities of reactants and products in a chemical reaction. It begins with a balanced chemical equation and uses mole ratios to find the amount of substance needed or produced.

In the case of \( \mathrm{Ag}_{2} \mathrm{MoS}_{4} \), the ratios are:

• 2 moles of Ag are needed for every mole of the compound.
• 1 mole of Mo is needed.
• 4 moles of S are needed.

Stoichiometry helps identify the limiting reactant in a reaction by comparing the ratios of reactants consumed to their availability. Molybdenum is the limiting reactant in our example, as it runs out first and dictates the maximum amount of product formed.

Mole Concept

The mole is a fundamental concept that connects the atomic scale to the macroscopic scale. It allows chemists to count atoms by weighing them, as one mole of any substance contains Avogadro's number of particles, \(6.022 \times 10^{23}\).

When given a mass of a substance, like 8.63 g of silver, the number of moles is found using the molar mass:

\[ \text{Moles of Ag} = \frac{8.63}{107.87} = 0.08 \text{ moles} \]

This conversion is repeated for all elements involved. The mole concept is crucial for converting between mass and number of particles, enabling precise calculations in chemical reactions. It provides the link between a measurable quantity and the number of atoms or molecules involved.