Problem 83 - General Chemistry: Principles and Modern Applications

Question

The mineral spodumene has the empirical formula \(\mathrm{LiAlSi}_{2} \mathrm{O}_{6} .\) Given that the percentage of lithium- 6 atoms in naturally occuring lithium is \(7.40 \%,\) how many lithium- 6 atoms are present in a 518 g sample of spodumene?

Step-by-Step Solution

Answer

The number of lithium- 6 atoms present in a 518 g sample of spodumene is approximately \(1.883 \times 10^{23}\) atoms.

Step 1: Determine gram-molecular weight of Spodumene

Using the periodic table, we find the atomic weights of Li, Al, Si, and O: \[Li = 6.94\, g/mol\]; \[Al = 26.98\, g/mol\]; \[Si = 28.09\, g/mol\]; \[O = 16.00\, g/mol\]. Using the empirical formula \(LiAlSi_{2}O_6\), the gram-molecular weight calculates to \[6.94\, g/mol + 26.98\, g/mol + 2(28.09\, g/mol) + 6(16.00\, g/mol) = 154.22\, g/mol\].

Step 2: Calculate the grams of Lithium in 518 g sample

The fractional part by weight of Lithium in the empirical formula equals to 6.94 g out of the total 154.22 g. Hence, the grams of Lithium in a 518 g sample of Spodumene calculates to \[(6.94\, g / 154.22\, g) * 518\, g = 29.37\, g\].

Step 3: Calculate the moles of Lithium

From the knowledge of basic Chemistry, we can say that, one mole equals the atomic weight expressed in grams. Hence, the moles of Lithium in a 29.37 g is \[29.37\, g / 6.94\, g/mol = 4.23\, mol\].

Step 4: Calculate the total number of Lithium atoms

Using Avogadro's number, \(6.022 \times 10^{23}\), which states that there are \(6.022 \times 10^{23}\) atoms/ions in one mole of any substance. So, the total number of Lithium atoms in 4.23 mol is \[4.23 \times 6.022 \times 10^{23} = 2.545 \times 10^{24}\].

Step 5: Calculate the number of Lithium-6 atoms

Since only 7.40% of all naturally occurring Lithium atoms are Lithium-6, the total number of Lithium-6 atoms in our Spodumene sample will be \[0.0740 * (2.545 \times 10^{24}) = 1.883 \times 10^{23}\].

Key Concepts

Atomic Weight

When we talk about atomic weight, we refer to the average mass of an element's atoms, typically expressed in atomic mass units (amu). Atomic weight combines the masses of an element's isotopes, weighted by their natural abundance.
To find the atomic weight, you use the periodic table, where each element has a specific atomic mass. For instance, lithium (Li) has an atomic weight of 6.94 amu. In chemical calculations, such as determining the empirical formula or atomic ratios in compounds, this number is essential.

• Atomic weights are used to calculate the gram-molecular weight of compounds. This involves adding up the atomic weights of each element in the compound based on their proportions.
• In case of the compound \(\text{LiAlSi}_2\text{O}_6\), knowing the individual atomic weights means you can calculate its molecular weight, enabling further conversions and calculations in chemistry.

The relationship between atomic weight, molar mass, and mole concept is a foundational aspect of chemistry.

Mole Concept

The mole concept is a fundamental idea in chemistry, used to describe the amount of substance. A mole is a quantity that helps in scaling atomic and molecular masses to a macroscopic level, making chemistry calculations more straightforward.
The mole allows chemists to count particles, such as atoms or molecules, by relating the number to Avogadro’s number, which equals approximately \(6.022 \times 10^{23}\) entities per mole.

• The mole concept helps convert mass into number of particles. For example, given the molar mass of lithium, you can figure out the number of moles present in a certain mass of lithium.
• In our exercise, we used the molar mass of lithium to calculate the number of moles in the sample, which is a crucial step in determining the number of atoms present.

This concept plays a crucial role in understanding the composition and reactions of substances.

Avogadro's Number

Avogadro's number, \(6.022 \times 10^{23}\), is a constant that represents the number of atoms, molecules, ions, or other elementary entities in one mole of a substance.
This number is very large because it is meant to scale up the tiny atomic level to amounts that can be measured easily.

• Avogadro’s number enables chemists to convert moles to particles. For example, using Avogadro's number, you turn moles of lithium into the number of lithium atoms.
• In the given problem, we calculated the total number of lithium atoms in the sample. Further, we used the percentage of lithium-6 isotopes to find the specific number of lithium-6 atoms, thanks to Avogadro's number.

Understanding Avogadro’s number is key to performing accurate conversions and computations in chemistry.