Problem 5 - Chemistry Matter and Change

Question

How many moles contain each of the following? a. \(5.75 \times 10^{24}\) atoms Al \(\quad\) b. \(2.50 \times 10^{20}\) atoms Fe

Step-by-Step Solution

Answer

a. 9.55 moles of Aluminum (Al) b. \(4.15 \times 10^{-4}\) moles of Iron (Fe)

Step 1: Identify the given information

The given information is the number of atoms for each element: a. 5.75 x 10^24 atoms of Aluminum (Al) b. 2.50 x 10^20 atoms of Iron (Fe)

Step 2: Identify the conversion factor

The conversion factor between the number of atoms and moles is Avogadro's number: 1 mole = 6.022 x 10^23 atoms

Step 3: Calculate the number of moles for Aluminum (Al)

Using the conversion factor, we convert the number of Aluminum atoms to moles: Number of moles = Number of atoms / Avogadro's number Number of moles of Al = \( \frac{5.75 \times 10^{24}}{6.022 \times 10^{23}} \)

Step 4: Calculate the number of moles for Iron (Fe)

Using the conversion factor, we convert the number of Iron atoms to moles: Number of moles = Number of atoms / Avogadro's number Number of moles of Fe = \( \frac{2.50 \times 10^{20}}{6.022 \times 10^{23}} \)

Step 5: Simplify the calculations

Now, we simplify the calculations for both elements: Number of moles of Al ≈ 9.55 moles Number of moles of Fe ≈ 4.15 x 10^-4 moles Thus, the number of moles for each element are: a. 9.55 moles of Aluminum (Al) b. 4.15 x 10^-4 moles of Iron (Fe)

Key Concepts

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that links the macroscopic scale of substances to the microscopic world of atoms and molecules. It is defined as the number of constituent particles (usually atoms or molecules) present in one mole of a substance. The value of Avogadro's number is precisely defined as
\(6.022 \times 10^{23}\) particles per mole. This makes it incredibly convenient for chemists to use Avogadro's number in various calculations, particularly in molar conversions between atoms and moles. For instance, when we are given the number of atoms of an element, we can easily calculate how many moles of the element we have by dividing by Avogadro's number, as seen in the provided solution for aluminum and iron.

Molar Conversion

Molar conversion involves the process of converting between grams, moles, and the number of atoms or molecules in a substance. It relies on the following three important relationships:

1. The molar mass of a substance, which is the mass in grams of one mole of that substance.
2. The mole concept, which represents a set amount of particles, specifically Avogadro's number of particles.
3. Avogadro's number, as described previously, connects the number of particles with the moles of a substance.

To perform a molar conversion, you can use the provided steps to divide the total number of atoms by Avogadro's number, thereby finding the amount in moles. This conversion is crucial in stoichiometry calculations and understanding the proportions needed for chemical reactions.

Stoichiometry

Stoichiometry is the section of chemistry that concerns the calculation of relative quantities of reactants and products in chemical reactions. It is based on the conservation of mass, where the total mass of the reactants equals the total mass of the products. Stoichiometry calculations often involve balancing chemical equations and using molar ratios from the balanced reaction to determine how much of each reactant is needed or how much product will be formed.
In straightforward terms, if you picture a recipe for a cake that requires a certain number of eggs for a set amount of flour, stoichiometry is the 'recipe book' for finding out how much of each chemical is required in a reaction. The solutions to the original exercise use stoichiometric principles to convert number of atoms to moles. It is also key in determining how much of one substance will react with another, ensuring that chemical reactions are as efficient as possible.

Atomic Structure

Atomic structure refers to the composition and arrangement of particles within an atom. Atoms consist of a central nucleus made of protons and neutrons, surrounded by electrons in various energy levels or shells. Understanding atomic structure is vital for analyzing how atoms will interact during chemical reactions.
Each element on the periodic table has a unique atomic structure based on its number of protons (atomic number) and electrons. In the process of calculating moles from the number of atoms, as demonstrated in the exercise, it is implied that each atom retains its identity and number of protons and electrons. In chemistry, this atomic individuality is essential when it comes to determining the characteristics and reactions of different elements.