Question

Sulfur trioxide, \(\mathrm{SO}_{3},\) is made industrially in enormous quantities by combining oxygen and sulfur dioxide, SO \(_{2} .\) What amount (moles) of \(\mathrm{SO}_{3}\) is represented by \(1.00 \mathrm{kg}\) of sulfur trioxide? How many molecules? How many sulfur atoms? How many oxygen atoms?

Short answer

12.49 moles of \( \mathrm{SO_3} \); \( 7.52 \times 10^{24} \) molecules; \( 7.52 \times 10^{24} \) sulfur atoms; \( 2.26 \times 10^{25} \) oxygen atoms.

Step-by-step solution

Understand the Problem

We need to find out how many moles, molecules, and atoms make up 1 kg of \( \mathrm{SO_3} \). To do this, we'll first calculate the number of moles in that mass.

Calculate Moles of \( \mathrm{SO_3} \)

To find the number of moles, use the formula: \( \text{moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \). The molar mass of \( \mathrm{SO_3} \) is calculated as follows: \( \text{Sulfur (S)} = 32.07 \, \text{g/mol} \) and \( \text{Oxygen (O)} = 16.00 \, \text{g/mol} \times 3 = 48.00 \, \text{g/mol} \). Therefore, \( \text{Molar Mass of } \mathrm{SO_3} = 32.07 + 48.00 = 80.07 \, \text{g/mol} \). Given: 1 kg = 1000 g. Moles = \( \frac{1000 \text{ g}}{80.07 \text{ g/mol}} \approx 12.49 \text{ moles} \).

Calculate Number of Molecules

To find the number of molecules, multiply the number of moles by Avogadro's number \( (6.022 \times 10^{23} \text{ molecules/mol}) \). Therefore: \( 12.49 \text{ moles} \times 6.022 \times 10^{23} \approx 7.52 \times 10^{24} \text{ molecules} \).

Calculate Number of Sulfur Atoms

Each molecule of \( \mathrm{SO_3} \) contains one sulfur atom. Therefore, the number of sulfur atoms is equal to the number of molecules. Thus, there are \( 7.52 \times 10^{24} \text{ sulfur atoms} \).

Calculate Number of Oxygen Atoms

Each molecule of \( \mathrm{SO_3} \) has three oxygen atoms. Therefore, multiply the number of molecules by 3: \( 7.52 \times 10^{24} \text{ molecules} \times 3 = 2.26 \times 10^{25} \text{ oxygen atoms} \).

Key concepts

Molar Mass

Molar mass is a fundamental concept when dealing with chemical compounds and reactions. It enables us to convert between the mass of a substance and the amount in moles. Molar mass is defined as the mass of one mole of a substance, generally expressed in grams per mole (g/mol). For any compound, you can calculate its molar mass by summing the molar masses of its constituent atoms retrieved from the periodic table.

Taking sulfur trioxide ( SO_{3} ) as an example, it's composed of one sulfur atom and three oxygen atoms. The molar mass of sulfur is approximately 32.07 g/mol, and for an oxygen atom, it is about 16.00 g/mol. To find the molar mass of SO_{3} , you calculate as follows:

32.07 ext{ g/mol} ext{ (sulfur)} + (3 imes 16.00 ext{ g/mol} ext{ (oxygen)}) = 80.07 ext{ g/mol} .

This information is critical for further calculations, such as determining the number of moles in a given mass of SO_{3} .

Avogadro's Number

Avogadro's number is an essential tool in chemistry that allows us to convert between moles and individual particles, such as atoms or molecules. It is numerically represented as 6.022 imes 10^{23} and signifies the number of atoms, ions, or molecules in one mole of a substance.

Avogadro's number connects the macroscopic world, where we can measure quantities of substances in grams, with the microscopic world, where we count molecules or atoms. When you have calculated the amount in moles, multiplying by Avogadro's number gives you the actual number of molecules or atoms in that sample.

For example, if you have found that 1 kg of SO_{3} contains approximately 12.49 moles, by multiplying this by Avogadro's number: 12.49 ext{ moles} imes 6.022 imes 10^{23} = 7.52 imes 10^{24} molecules. This enormous quantity is manageable due to Avogadro's number.

Sulfur Atoms

Understanding the composition of a molecule helps us determine the number of atoms of each element present. In the case of SO_{3} , each molecule contains one sulfur atom. This means the number of sulfur atoms in a mass of sulfur trioxide is directly equivalent to the number of SO_{3} molecules.

After finding the total number of SO_{3} molecules in our sample, which is 7.52 imes 10^{24} molecules, we can deduce that there are thus 7.52 imes 10^{24} sulfur atoms too.

This direct relationship simplifies the process of calculating the number of each type of atom in a molecular substance. Keeping track of these quantities is crucial when you're involved in processes like balancing chemical equations or estimating reactant products.

Oxygen Atoms

Oxygen atoms in a molecule of SO_{3} must also be established to understand the full scope of its composition. Each SO_{3} molecule contains three oxygen atoms. Hence, to determine the total number of oxygen atoms, you need to multiply the total number of molecules by three.

Returning to our example, where you found 7.52 imes 10^{24} SO_{3} molecules in 1 kg of sulfur trioxide, you can compute the oxygen atoms as follows:

7.52 imes 10^{24} ext{ molecules} imes 3 = 2.26 imes 10^{25} ext{ oxygen atoms} .

Recognizing how many oxygen atoms constitute a compound is vital in processes like stoichiometry, reaction balancing, and calculating reactant/product ratios.