Problem 68 - Chemistry and Chemical Reactivity

Question

A semiconducting material is composed of \(52 \mathrm{g}\) of Ga, \(9.5 \mathrm{g}\) of \(\mathrm{Al}\), and \(112 \mathrm{g}\) of As. Which element has the largest number of atoms in this material?

Step-by-Step Solution

Answer

Arsenic (As) has the largest number of atoms.

Step 1: Determine Molar Mass of Each Element

To find the number of atoms, we first need the molar mass of each element. The molar masses are approximately: - Ga: 69.72 \( \text{g/mol} \)- Al: 26.98 \( \text{g/mol} \)- As: 74.92 \( \text{g/mol} \)

Step 2: Calculate Moles of Each Element

Using the molar masses, calculate the number of moles of each element using the formula:\[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} \]- Moles of Ga: \[ \frac{52}{69.72} \approx 0.746 \text{ moles} \]- Moles of Al: \[ \frac{9.5}{26.98} \approx 0.352 \text{ moles} \]- Moles of As: \[ \frac{112}{74.92} \approx 1.495 \text{ moles} \]

Step 3: Convert Moles to Number of Atoms

Convert the moles of each element to atoms using Avogadro's number \(6.022 \times 10^{23}\) atoms/mol:- Atoms of Ga: \[ 0.746 \times 6.022 \times 10^{23} \approx 4.49 \times 10^{23} \]- Atoms of Al: \[ 0.352 \times 6.022 \times 10^{23} \approx 2.12 \times 10^{23} \]- Atoms of As: \[ 1.495 \times 6.022 \times 10^{23} \approx 9.00 \times 10^{23} \]

Step 4: Compare the Number of Atoms

Compare the number of atoms calculated for each element:- Ga has \(4.49 \times 10^{23}\) atoms.- Al has \(2.12 \times 10^{23}\) atoms.- As has \(9.00 \times 10^{23}\) atoms.Arsenic (\(\mathrm{As}\) has the largest number of atoms in this material.

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry that helps quantify the mass of a chemical element or compound per mole. It's expressed in grams per mole (g/mol). For any given element, the molar mass is determined by the relative atomic mass found on the periodic table, typically rounded to two decimal places.
For example, in the given problem, we calculated the molar mass of Gallium (Ga), Aluminum (Al), and Arsenic (As):

• Gallium (Ga): 69.72 g/mol
• Aluminum (Al): 26.98 g/mol
• Arsenic (As): 74.92 g/mol

These values indicate how much one mole of each element weighs. Properly calculating and using molar mass is essential in converting between mass and the amount of substance in moles, which is often needed in chemistry calculations.

Moles

The concept of a mole is central to chemistry calculations as it allows chemists to relate microscopic atomic levels to macroscopic quantities. One mole of any substance contains Avogadro's number of entities, which could be atoms, molecules, ions, etc. To calculate moles, you use the formula:\[ \text{Moles} = \frac{\text{Mass (g)}}{\text{Molar Mass} \text{ (g/mol)}}\]This formula shows the relationship between the mass of a substance and the number of moles present. For this particular exercise, let's revisit the calculations:

• Gallium: \( \text{Moles} = \frac{52}{69.72} \approx 0.746 \text{ moles} \)
• Aluminum: \( \text{Moles} = \frac{9.5}{26.98} \approx 0.352 \text{ moles} \)
• Arsenic: \( \text{Moles} = \frac{112}{74.92} \approx 1.495 \text{ moles} \)

This calculation reveals how many moles of each element are present in the sample based on their respective masses and allows the further conversion to the number of atoms.

Avogadro's Number

Avogadro's number is a constant that is key to translating between the microscopic world of atoms and the macroscopic world we can measure. Its value is approximately \(6.022 \times 10^{23}\) atoms/mol. This enormous number tells us that one mole of any element or molecule contains exactly that many atoms or molecules.
In this problem, after finding the number of moles, you can determine the number of atoms in each element by multiplying the number of moles by Avogadro's number:

• Atoms of Ga: \(0.746 \times 6.022 \times 10^{23} \approx 4.49 \times 10^{23}\)
• Atoms of Al: \(0.352 \times 6.022 \times 10^{23} \approx 2.12 \times 10^{23}\)
• Atoms of As: \(1.495 \times 6.022 \times 10^{23} \approx 9.00 \times 10^{23}\)

Avogadro's number thus bridges a critical gap in chemistry by allowing us to convert moles to actual numbers of atoms or molecules, facilitating deeper understanding and further chemical calculations.

Semiconductors

Semiconductors are materials that have electrical conductivity between that of an insulator and a conductor. They are essential in the electronic and technological industries as they are used to manufacture integrated circuits and various electronic components. The semiconductor in this exercise is made of elements Gallium (Ga), Aluminum (Al), and Arsenic (As), known collectively as a III-V semiconductor.
These semiconductors uniquely exhibit properties that change with temperature, composition, and impurity levels, making them versatile materials in designing transistors, diodes, solar cells, and even LEDs (light-emitting diodes). The amount and type of atoms in a semiconductor material can significantly impact its electronic properties, which is why determining the number of atoms can be crucial in the process of doping, where small amounts of impurities are added to change its conductivity.