Problem 150 - Introductory Chemistry Atoms First

Question

The beryllium mineral beryl is made up of \(5.03 \%\) by mass Be, \(10.04 \%\) by mass Al, \(31.35 \%\) by mass \(\mathrm{Si}\), and \(53.58 \%\) by mass \(\mathrm{O}\). If the molar mass is approximately \(538 \mathrm{~g} / \mathrm{mol}\), what is the molecular formula for beryl?

Step-by-Step Solution

Answer

The molecular formula for beryl is Be3Al2Si6O18.

Step 1: Convert the mass percentages to grams

First, assume that there is 100 grams of beryl. This way the mass percentages can be converted directly to grams: - Be: \(5.03 \% * 100 \mathrm{~g} = 5.03 \mathrm{~g}\) - Al: \(10.04 \% * 100 \mathrm{~g} = 10.04 \mathrm{~g}\) - Si: \(31.35 \% * 100 \mathrm{~g} = 31.35 \mathrm{~g}\) - O: \(53.58 \% * 100 \mathrm{~g} = 53.58 \mathrm{~g}\)

Step 2: Determine the moles of each element

Now, we will divide each mass by its respective molar mass to find the moles: - Moles of Be: \(\frac{5.03 \mathrm{~g}}{9.012 \mathrm{~g/mol}} = 0.558 \mathrm{~mol}\) - Moles of Al: \(\frac{10.04 \mathrm{~g}}{26.982 \mathrm{~g/mol}} = 0.372 \mathrm{~mol}\) - Moles of Si: \(\frac{31.35 \mathrm{~g}}{28.085 \mathrm{~g/mol}} = 1.116 \mathrm{~mol}\) - Moles of O: \(\frac{53.58 \mathrm{~g}}{16.00 \mathrm{~g/mol}} = 3.349 \mathrm{~mol}\)

Step 3: Determine the simplest mole ratio

To find the simplest whole-number mole ratio, divide all the mole values by the smallest mole value: - Ratio of Be: \(\frac{0.558 \mathrm{~mol}}{0.372 \mathrm{~mol}} = 1.5\) - Ratio of Al: \(\frac{0.372 \mathrm{~mol}}{0.372 \mathrm{~mol}} = 1\) - Ratio of Si: \(\frac{1.116 \mathrm{~mol}}{0.372 \mathrm{~mol}} = 3\) - Ratio of O: \(\frac{3.349 \mathrm{~mol}}{0.372 \mathrm{~mol}} = 9\) However, the ratio of Be is not a whole number. In order to convert all ratios to whole numbers, multiply everything by 2: - Ratio of Be: \(1.5 * 2 = 3\) - Ratio of Al: \(1 * 2 = 2\) - Ratio of Si: \(3 * 2 = 6\) - Ratio of O: \(9 * 2 = 18\)

Step 4: Write the molecular formula

Now that we have the whole number mole ratios for each element, we can write the molecular formula for beryl: Be3Al2Si6O18 So, the molecular formula for beryl is Be3Al2Si6O18.

Key Concepts

Exploring Beryllium Mineral

Beryl is a fascinating mineral primarily made up of beryllium. It's often used as a gemstone and has variants like emerald and aquamarine. The mineral's structure is not just made of beryllium; it includes aluminum (Al), silicon (Si), and oxygen (O) in specific proportions. Understanding the composition of beryl is important because the presence of beryllium contributes to its uniqueness. By knowing the overall composition through mass percentages, one can determine the mineral’s molecular formula, which reflects the exact number of atoms of each element in the smallest unit of beryl.

Understanding Molar Mass

Molar mass is a critical concept in chemistry that refers to the mass of one mole of a given substance. The molar mass is expressed in grams per mole (g/mol). For beryl, it is approximately 538 g/mol. This value is useful when you want to convert mass into moles, allowing you to compare how much substance is present quantitatively. To calculate molar mass, you sum up the atomic masses of all atoms in the molecular formula of the substance. This calculation helps in understanding how much of each element is needed to create a specific amount of the compound, like beryl.

Decoding Mass Percentage

Mass percentage is a way to express how much of a certain element exists in a compound compared to the total mass. For instance, the mass percentage of beryl includes:

• 5.03% of Beryllium (Be)
• 10.04% of Aluminum (Al)
• 31.35% of Silicon (Si)
• 53.58% of Oxygen (O)

These percentages tell us how the molar composition of beryl is distributed. By assuming a 100 g sample, the mass percentages can be directly used to find the mass in grams, simplifying calculations needed for determining the moles of each element contributing to the compound's structure and properties.

Calculating Moles of Elements

Knowing the number of moles of each element in a compound can tell us how many atoms or molecules are present. To find moles, you divide the mass of an element by its molar mass. For beryl:

• Beryllium: 5.03 g divided by 9.012 g/mol gives 0.558 mol
• Aluminum: 10.04 g divided by 26.982 g/mol gives 0.372 mol
• Silicon: 31.35 g divided by 28.085 g/mol gives 1.116 mol
• Oxygen: 53.58 g divided by 16.00 g/mol gives 3.349 mol

These calculations are fundamental because they allow us to establish the ratio of elements, which is necessary to derive the correct molecular formula of a compound. This whole-number mole ratio is vital to construct a formula that accurately represents a compound's actual structure.