Question

Measurements show that \(1.0 \mathrm{~g}\) of iron (Fe) contains \(1.1 \times 10^{22} \mathrm{Fe}\) atoms. How many Fe atoms are in \(4.9 \mathrm{~g}\) of Fe, which is the total amount of iron in the body of an average adult?

Short answer

5.39 x 10^22 Fe atoms.

Step-by-step solution

Understand the Given Values

We are given that 1.0 grams of iron (Fe) contains \(1.1 \times 10^{22}\) Fe atoms. We need to find out how many Fe atoms are in 4.9 grams of iron.

Set Up a Proportion

The problem involves a direct proportion between the mass of iron and the number of Fe atoms. We can set up the proportion as follows: \(\frac{1.0 \text{ g of Fe}}{1.1 \times 10^{22} \text{ Fe atoms}} = \frac{4.9 \text{ g of Fe}}{x \text{ Fe atoms}}\)where \(x\) is the number of Fe atoms in 4.9 g of iron.

Solve the Proportion for x

To solve for \(x\), we multiply both sides of the proportion by \(x \text{ Fe atoms}\) and \(1.1 \times 10^{22} \text{ Fe atoms per gram}\) as follows:\[x = \frac{4.9 \text{ g} \times 1.1 \times 10^{22} \text{ atoms}}{1.0 \text{ g}}\]This allows us to calculate the number of atoms in 4.9 g.

Perform the Calculation

Calculate \(x\) using the equation: \[x = 4.9 \times 1.1 \times 10^{22} = 5.39 \times 10^{22}\]Therefore, there are \(5.39 \times 10^{22}\) Fe atoms in 4.9 grams of iron.

Key concepts

Atomic Structure

Atoms are the fundamental building blocks of matter, and they define the chemical element itself. Every atom is composed of a central nucleus, housing protons and neutrons, surrounded by electrons that orbit the nucleus in various energy levels.
These protons are positively charged, whereas electrons carry a negative charge. In a neutral atom, the number of protons equals the number of electrons.
For example, an iron atom (Fe) possesses 26 protons in its nucleus. The electrons orbiting these protons dictate the atom's chemical behavior. Atomic structure is vital in stoichiometry because the quantity of atoms present in a sample can influence the mass of the sample and its chemical reactions.
Understanding atomic structure is crucial to comprehend how the mass of iron (Fe) involves the sheer number of its atoms, as seen in the calculations in the exercise, where each gram of iron consists of a multitude of Fe atoms.

Mole Concept

The mole is an essential concept in chemical calculations. It acts as a bridge between the macroscopic world that we observe and the atomic particles that we can't see.
A mole constitutes Avogadro's number, which is approximately \(6.022 \times 10^{23}\), representing the number of constituent particles, usually atoms or molecules, found in one mole of a substance.
The mole concept allows scientists to convert between the mass of a substance and the number of atoms or molecules it contains.

• Atomic Mass Unit (AMU): Typically, the atomic mass of an element listed on the periodic table refers to one mole of that element.
• Molecular Mass: The mass of one mole of a given molecule measured in grams per mole (g/mol).

This concept is practical, as seen in the exercise where the direct calculation is simplified by first understanding how many atoms fit into a certain mass, thereby making the proportion calculation straightforward and manageable.

Dimensional Analysis

Dimensional analysis, also known as the factor-label method or unit factor method, is a systematic conversion process between different units of measure, helping clarify relationships in calculations.
It involves multiple steps, often setting up conversion factors so that units you wish to cancel out are eliminated and what's left are the desired units.
For example, converting grams into the number of atoms involves understanding that there exists a specific relationship between these two quantities. This is apparent in the solution, where we convert the mass of iron into the equivalent number of atoms using the proportionality derived from known values.

• Start by identifying the unit conversion required.
• Set up a conversion factor such that unwanted units cancel out, leaving the desired unit.
• Apply conversion factors formed by known relationships between measurements such as grams and atoms.

Dimensional analysis is fundamental in stoichiometry as it enables the conversion of masses to moles and consequently particles, allowing for clearer resolution of problems involving large or small quantities such as the number of atoms in a given mass.