Problem 46 - Chemistry An Atoms First Approach

Question

What number of Fe atoms and what amount (moles) of Fe atoms are in \(500.0 \mathrm{g}\) of iron?

Step-by-Step Solution

Answer

There are approximately 8.949 moles of iron (Fe) atoms and approximately \(5.385 \times 10^{24}\) individual Fe atoms in 500.0 grams of iron.

Step 1: Find the molar mass of iron (Fe)

To solve for the number of moles, we first need to find the molar mass of iron. One mole of iron has a mass of 55.845 grams (taken from the periodic table). So, we have: Molar mass of Fe = 55.845 g/mol

Step 2: Calculate the moles of iron

Next, we'll use the given mass of iron (500.0 g) to calculate the number of moles of iron by dividing the mass by the molar mass. This can be done using the formula: Moles of Fe = (Mass of Fe) / (Molar mass of Fe) Substitute the known values into the formula: Moles of Fe = (500.0 g) / (55.845 g/mol) Now, perform the calculation: Moles of Fe ≈ 8.949 mol

Step 3: Determine the number of Fe atoms

Now that we know the number of moles of iron, we can find the number of Fe atoms using Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms/mol. The formula to find the number of atoms is: Number of atoms = (Moles of substance) × (Avogadro's number) Substitute the known values into the formula: Number of Fe atoms = (8.949 mol) × (\(6.022 \times 10^{23}\) atoms/mol) Now, perform the calculation: Number of Fe atoms ≈ \(5.385 \times 10^{24}\) atoms

Step 4: Final Answers:

The given mass of iron (500.0 g) contains approximately 8.949 moles of iron (Fe) atoms and approximately \(5.385 \times 10^{24}\) individual Fe atoms.

Key Concepts

Molar Mass of Iron

Understanding the concept of molar mass is crucial in chemistry, particularly when dealing with elements like iron (Fe). The molar mass is essentially the weight of one mole of a substance, which can be found on the periodic table. For iron, this molar mass is 55.845 grams per mole. This means that one mole of iron atoms weighs 55.845 grams.

When working with larger amounts of a substance, knowing the molar mass allows you to convert between grams and moles, which can be thought of as the 'chemistry version' of counting by dozens or pieces. This conversion is vital for stoichiometric calculations, where understanding the amount of substance is key to predicting reaction yields and products.

Avogadro's Number

Avogadro's number is another fundamental constant in chemistry and is defined as the number of units in one mole of any substance. Its numerical value is approximately \(6.022 \times 10^{23}\).

In the context of iron or any other element, Avogadro's number lets us quantify just how many atoms we're talking about when we say 'one mole'. So, when you calculate the number of moles of Fe in a given sample, you multiply that quantity by Avogadro's number to find out the exact number of iron atoms present. This is invaluable for understanding the scale of chemical reactions at the atomic level.

Stoichiometry

Stoichiometry could be likened to the recipe for a cake, but instead of cups and teaspoons, chemists use moles to measure their 'ingredients'. It's a branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction.

When solving stoichiometric problems, using a balanced chemical equation is often the starting point. From there, you can calculate how much of one substance will react with a given amount of another substance, or how much product will be formed. In the case of iron, understanding its stoichiometry is essential, especially in iron-related industries like steelmaking where precise measurements are critical for quality and safety.

Atomic Structure

The atomic structure applies to our discussion about moles and Avogadro's number because it provides a basis for understanding why substances have the molar masses that they do.

Every iron atom consists of protons, neutrons, and electrons. The number of protons (which equals the number of electrons) defines the element's atomic number. Iron, with an atomic number of 26, has 26 protons. The mass of an atom is predominantly in its nucleus (protons and neutrons), and it's this mass that contributes to the molar mass of the element. Knowing the atomic structure helps us appreciate the reason behind the unique molar mass of each element, including iron, which in turn is used in moles and Avogadro's number calculations to determine quantities in chemical reactions.