Problem 20 - General Chemistry: Principles and Modern Applications

Question

Determine the mass percent \(O\) in the mineral malachite, \(\mathrm{Cu}_{2}(\mathrm{OH})_{2} \mathrm{CO}_{3}\)

Step-by-Step Solution

Answer

The mass percent of Oxygen in the mineral malachite is approximately 28.94%.

Step 1: Determine the Molar Mass of the Mineral

To calculate the molar mass of the mineral Malachite (\(\mathrm{Cu}_{2}(\mathrm{OH})_{2} \mathrm{CO}_{3}\)), the molar masses of its constituents must be added together. The molar mass of Copper (Cu) is approximately 63.55 g/mol, that of Oxygen (O) is 16.00 g/mol and Hydrogen (H) is 1.01 g/mol. Carbon (C) has a molar mass of 12.01 g/mol. Thus, the molar mass of Malachite is given by: \2(63.55 g/mol) + 2(16.00 g/mol + 1.01 g/mol) + 3(16.00 g/mol) + 12.01 g/mol = 221.12 g/mol.

Step 2: Calculate Total Molar Mass of Oxygen in the Mineral

There are 4 Oxygen atoms in one formula unit of Malachite: two in the hydroxide (\(\mathrm{OH}\)) ions and two in the carbonate (\(\mathrm{CO}_{3}\)) ion. Therefore, total molar mass of Oxygen in the mineral is: \4 * 16.00 g/mol = 64.00 g/mol.

Step 3: Calculate the Mass Percent of Oxygen

The mass percent of Oxygen can now be calculated from the molar mass of Oxygen and the total molar mass of the compound: \\(Mass\,Percent\,Oxygen = \frac{Total\,Oxygen\,Molar\,Mass}{Molar\,Mass\,of\,Malachite} * 100% = \frac{64.00\,g/mol}{221.12\,g/mol} * 100% ≈ 28.94% \)

Key Concepts

Molar Mass Determination

When determining the molar mass of a compound, we need to consider the molar mass of each element within that compound. This process involves checking the periodic table or a scientific reference for the atomic mass of each atom. For instance, in malachite, a mineral with the formula \(\mathrm{Cu}_{2}(\mathrm{OH})_{2} \mathrm{CO}_{3}\), we must account for each element's molar mass:

• Copper (Cu): 63.55 g/mol
• Oxygen (O): 16.00 g/mol
• Hydrogen (H): 1.01 g/mol
• Carbon (C): 12.01 g/mol

Next, multiply the molar mass of each element by the number of times it appears in the formula and then sum them up. For malachite, this calculation looks like 2(63.55 g/mol) for copper, 2(16.00 g/mol + 1.01 g/mol) for hydroxide, 3(16.00 g/mol) for carbonate, and 12.01 g/mol for carbon. All added together, the molar mass of malachite is 221.12 g/mol. Calculating this accurately is crucial for further chemical analysis.

Chemical Composition Analysis

Chemical composition analysis of a mineral involves determining the proportion of each element within the compound. This is essential in understanding the mineral's chemical properties and potential applications. In malachite, once we have calculated the molar mass, we can further dissect its chemical composition by determining each element's contribution to the total mass.
For instance, by examining the equation \(\mathrm{Cu}_{2}(\mathrm{OH})_{2} \mathrm{CO}_{3}\), we reckon that copper contributes 127.10 g/mol to the entire molar mass, which you arrive by calculating as 2 times the molar mass of copper (63.55 g/mol). Similarly, the hydroxyl group \((\mathrm{OH})\) contributes \(34.02\,g/mol\) and the carbonate group \((\mathrm{CO}_{3})\) contributes 60.01 g/mol.
This detailed analysis allows scientists and students alike to understand and predict the mineral's behavior in various environments and reactions.

Oxygen Content in Minerals

Understanding the oxygen content in minerals is an important aspect of mineral chemistry and various industrial applications. Each element within a mineral's structure has a critical role, and oxygen often comprises a significant portion due to its abundance.
In malachite \(\mathrm{Cu}_{2}(\mathrm{OH})_{2} \mathrm{CO}_{3}\), oxygen exists in two different groups, hydroxide and carbonate. The total oxygen content can be confirmed by multiplying four oxygen atoms by their molar mass, summing up to 64.00 g/mol. This corresponds to four oxygen atoms, with two each found in both the hydroxide and carbonate groups.

• 2 Oxygen in \((\mathrm{OH})\) = 2 * 16.00 g/mol
• 2 Oxygen in \((\mathrm{CO}_{3})\) = 2 * 16.00 g/mol

To find the mass percent of oxygen, we divide the total molar mass of oxygen by the molar mass of the mineral and multiply by 100, yielding approximately 28.94%. This percentage illustrates the prominence of oxygen in forming the mineral's structure and properties.