Question

Without doing any detailed calculations (but using a periodic table to give atomic weights), rank the following samples in order of increasing numbers of atoms: \(0.2 \mathrm{~mol} \mathrm{PCl}_{5}\), molecules, \(80 \mathrm{~g} \mathrm{Fe}_{2} \mathrm{O}_{3}, 3.0 \times 10^{23}\) CO molecules.

Short answer

The order of increasing number of atoms in the given samples is as follows: \(3.0 \times 10^{23}\) CO molecules < 0.2 mol PCl5 < 80 g Fe2O3.

Step-by-step solution

Convert moles to atoms for PCl5

To find the number of atoms in 0.2 mol of PCl5, we'll first calculate the number of molecules and then multiply it by the total atoms in one PCl5 molecule. We know that the number of molecules in 1 mole is \(6.022 \times 10^{23}\) molecules (Avogadro's number). Number of PCl5 molecules = \(0.2 \mathrm{~mol} \times 6.022 \times 10^{23} \mathrm{~molecules/mol}\) There are 6 atoms in 1 PCl5 molecule (1 P and 5 Cl), so we'll multiply the number of molecules by 6 to find the total number of atoms. Number of atoms in PCl5 = \(0.2 \mathrm{~mol} \times 6.022 \times 10^{23} \mathrm{~atoms/mol} \times 6\)

Convert grams to atoms for Fe2O3

To find the number of atoms in 80 g of Fe2O3, first, we need to convert grams to moles and then multiply it by the Avogadro's number and total atoms in one Fe2O3 molecule. The molar mass of Fe2O3 is \(2 \times 55.85 + 3 \times 16 = 159.7\) g/mol. Number of moles of Fe2O3 = \(80 \mathrm{~g} \div 159.7 \mathrm{~g/mol}\) Next, we'll calculate the number of molecules and then multiply it by the total atoms in one Fe2O3 molecule (5 atoms: 2 Fe and 3 O). Number of atoms in Fe2O3 = \((80 \mathrm{~g} \div 159.7 \mathrm{~g/mol}) \times 6.022 \times 10^{23} \mathrm{~atoms/mol} \times 5\)

Find atoms in CO molecules

The number of CO molecules is already provided, so we just need to multiply it by the total atoms in one CO molecule (2 atoms: 1 C and 1 O). Number of atoms in CO molecules = \(3.0 \times 10^{23} \mathrm{~molecules} \times 2\)

Compare and Rank

Now we compare the total number of atoms in each sample and rank them accordingly: 1. Number of atoms in PCl5: \(0.2 \mathrm{~mol} \times 6.022 \times 10^{23} \mathrm{~atoms/mol} \times 6\) 2. Number of atoms in Fe2O3: \((80 \mathrm{~g} \div 159.7 \mathrm{~g/mol}) \times 6.022 \times 10^{23} \mathrm{~atoms/mol} \times 5\) 3. Number of atoms in CO molecules: \(3.0 \times 10^{23} \mathrm{~molecules} \times 2\) When comparing these values, we find that the order of increasing number of atoms is: \(3.0 \times 10^{23}\) CO molecules < 0.2 mol PCl5 < 80 g Fe2O3

Key concepts

Avogadro's Number

Avogadro's number is a fundamental constant used in chemistry to describe the quantity of molecules or atoms in one mole of a substance. This number is approximately \(6.022 \times 10^{23}\).
This large value helps chemists translate atomic-scale measurements into quantities that are more tangible on a human scale. Understanding this number is key to converting moles to actual counts of molecules or atoms in a sample. For example, if you have 1 mole of any substance, it contains \(6.022 \times 10^{23}\) molecules or atoms of that substance.

Avogadro's number is crucial when you need to calculate amounts of atoms or molecules in a given sample, such as in the problem above. By knowing the number of moles, you can multiply by Avogadro's number to determine how many individual particles you are dealing with.

Mole Concept

The mole is a unit of measurement in chemistry that serves as a bridge between the atomic world and the macroscopic world. One mole of any substance contains \(6.022 \times 10^{23}\) particles, which could be atoms, molecules, ions, etc.
This simplification makes it easier for chemists to work with the large numbers of atoms and molecules typically involved in a given chemical reaction.To convert moles into atoms or molecules, you use the formula:

• Number of molecules = Number of moles \(\times 6.022 \times 10^{23}\)

This conversion allows us to determine the actual size of a sample at an atomic level by starting with a measured quantity in moles. The mole concept is invaluable for stoichiometry, allowing calculations to be based on the number of atoms, which then dictates reaction amounts and product outputs.

Molecular Mass

Molecular mass is the sum of the atomic masses of all atoms in a molecule. To calculate it, sum the average masses of the constituent atoms as listed on the periodic table.
For instance, the molecular mass of \(Fe_2O_3\), iron(III) oxide, is \(159.7\) g/mol.

Knowing the molecular mass allows you to convert between grams and moles. For example:

• Number of moles = Mass in grams / Molecular mass

In our provided exercise, knowing the molecular mass allowed us to convert 80 g of \(Fe_2O_3\) into moles. This conversion is crucial for determining how many molecules—and thus, atoms—are in a sample.

Atom Counting

Atom counting is a method used to determine the number of atoms in a sample. This process involves several steps, typically converting first to moles, and then using Avogadro's number to calculate the number of atoms.
Each molecule's chemical formula helps identify how many atoms are present per molecule.For example, to count atoms in \(PCl_5\) we must first identify that it consists of \(1\) phosphorus atom and \(5\) chlorine atoms, totaling \(6\) atoms per molecule.
When solving problems like counting the number of atoms in \(0.2\) moles of \(PCl_5\), you calculate:

• Number of atoms = Number of moles \(\times 6.022 \times 10^{23}\) molecules/mol \(\times 6\) atoms/molecule

Atom counting ensures you can determine exact atomic quantities, an essential process for accurate chemical calculations.