Problem 123 - Chemistry and Chemical Reactivity

Question

Calculate the molar mass and the mass percent of each element in the blue solid compound \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{SO}_{4} \cdot \mathrm{H}_{2} \mathrm{O} .\) What is the mass of copper and the mass of water in 10.5 g of the compound?

Step-by-Step Solution

Answer

Molar mass: 245.80 g/mol. Mass percent of Cu: 25.84%, H₂O: 7.33%. Mass of Cu in 10.5 g: 2.71 g. Mass of H₂O in 10.5 g: 0.77 g.

Step 1: Calculate Molar Mass of the Compound

First, identify the molar masses of the individual elements. The atomic masses are approximately: Cu = 63.55 g/mol, N = 14.01 g/mol, H = 1.01 g/mol, S = 32.07 g/mol, O = 16.00 g/mol.- The compound \(\mathrm{Cu(NH}_3)_4 \mathrm{SO}_4 \cdot \mathrm{H}_2 \mathrm{O}\) contains 1 Cu, 4 N, 14 H, 1 S, and 5 O (4 from \(\mathrm{SO}_4\) and 1 from \(\mathrm{H}_2 \mathrm{O}\)).- Calculate molar mass: - \(\mathrm{Cu} = 63.55\) g/mol- \(\mathrm{N (4 \times 14.01) = 56.04\) g/mol}- \(\mathrm{H (14 \times 1.01) = 14.14\) g/mol}- \(\mathrm{S = 32.07\) g/mol}- \(\mathrm{O (5 \times 16.00) = 80.00\) g/mol}Add these up to find the molar mass of the compound: \[\mathrm{63.55 + 56.04 + 14.14 + 32.07 + 80.00 = 245.80\:g/mol}\]

Step 2: Compute Mass Percent of Each Element

To calculate the mass percent for each element, use the formula: \(\text{mass percent} = \left( \frac{\text{atomic mass of element} \times \text{number of atoms}}{\text{molar mass of compound}} \right) \times 100\%\).- Copper: \[\left( \frac{63.55}{245.80} \right) \times 100 = 25.84\%\] - Nitrogen: \[ \left( \frac{56.04}{245.80} \right) \times 100 = 22.80\%\] - Hydrogen: \[ \left( \frac{14.14}{245.80} \right) \times 100 = 5.76\%\] - Sulfur: \[ \left( \frac{32.07}{245.80} \right) \times 100 = 13.04\%\] - Oxygen: \[ \left( \frac{80.00}{245.80} \right) \times 100 = 32.56\%\]

Step 3: Calculate Mass of Copper in 10.5 g of Compound

Use the mass percent of copper to find its mass in 10.5 g of the compound.- \(\text{Mass of Cu} = \left( \frac{25.84}{100} \right) \times 10.5 = 2.71 \: \text{g}\)

Step 4: Calculate Mass of Water in 10.5 g of Compound

Identify that the water part is represented by two hydrogens and one oxygen in the structure.\(\text{Molar mass of water (\(\mathrm{H_2O}\))} = 18.02 \: \text{g/mol}\)\(\text{Mass percent of } \mathrm{H_2O} = \left( \frac{18.02}{245.80} \right) \times 100 = 7.33\%\)- \(\text{Mass of } \mathrm{H_2O} = \left( \frac{7.33}{100} \right) \times 10.5 = 0.77 \: \text{g}\)

Key Concepts

Mass Percent Composition

When analyzing a compound like \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{SO}_{4} \cdot \mathrm{H}_{2} \mathrm{O} \), it is important to determine the mass percent composition for each element. This means calculating how much of the compound's total mass is made up by each element. Here’s how you do it:

• First, figure out the molar mass of the entire compound, which is a sum of the molar masses of all constituent elements combined.
• To find the mass percent of an element, take its total mass in the compound and divide it by the compound's molar mass, then multiply by 100 to convert it to a percentage.

This calculation helps you understand the proportion of each element within the compound. For example, in our compound, copper holds a mass percent composition of roughly 25.84%, which indicates a significant presence in the compound.

Copper Mass in Compound

To find the mass of copper in a given sample of the compound, you need to use the mass percent composition of copper. Knowing that copper accounts for around 25.84% of the compound, you can calculate the copper content in any mass of the compound.

Suppose you have 10.5 grams of \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{SO}_{4} \cdot \mathrm{H}_{2} \mathrm{O} \). You multiply this total mass by the mass percent of copper:

• Mass of Copper = \(( \frac{25.84}{100} ) \times 10.5\)

Calculating this formula reveals that the mass of copper in the sample is about 2.71 grams. This shows how the mass percent composition can be practical in determining the quantity of an element in a specific amount of compound.

Water Mass in Compound

Water, being an integral part of many compounds, also contributes to the mass composition. In the compound \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{SO}_{4} \cdot \mathrm{H}_{2} \mathrm{O} \), water is represented by the formula \(\mathrm{H}_2\mathrm{O}\) within the molecular structure. The molar mass of water is 18.02 g/mol.

To find the mass percentage of water in this compound, calculate its proportion in the molar mass:

• Mass Percent of Water = \( ( \frac{18.02}{245.80} ) \times 100\)

This gives us a mass percent of 7.33%, meaning water constitutes that fraction of the total mass. If you have 10.5 grams of the compound, you extend this to find the water mass using:

• Mass of Water = \( ( \frac{7.33}{100} ) \times 10.5\)

Resulting in approximately 0.77 grams of water in the sample, allowing deeper insights into how water links the structure and influences its properties.

Molecular Formula Analysis

Understanding the molecular formula, such as \(\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4} \mathrm{SO}_{4} \cdot \mathrm{H}_{2} \mathrm{O} \), is crucial for breaking down complex compounds. Each element has specific contributions to the compound's overall structure and properties.
The molecular formula tells us how many of each type of atom is present:

• 1 Copper (Cu)
• 4 Nitrogen (N)
• 14 Hydrogen (H)
• 1 Sulfur (S)
• 5 Oxygen (O)

Understanding these ratios helps determine how elements bond and interact, influencing both chemical behavior and applications. As such, molecular formula analysis is not just about counting atoms; it’s about connecting this data to chemical reactions, possible uses, and the compound's role within broader chemical processes.