Question

The total number of protons in \(10 \mathrm{~g}\) of calcium carbonate is \(\left(\mathrm{N}_{0}=6.023 \times 10^{23}\right)\) : (a) \(3.01 \times 10^{24}\) (b) \(4.06 \times 10^{24}\) (c) \(30.1 \times 10^{24}\) (d) \(3.01 \times 10^{23}\)

Short answer

The total number of protons in 10 g of calcium carbonate is \(3.01 \times 10^{24}\), option (a).

Step-by-step solution

Determine Molar Mass

Calcium carbonate (\(\text{CaCO}_3\)) has one calcium atom, one carbon atom, and three oxygen atoms. The atomic masses are approximately: \(\text{Ca} = 40\), \(\text{C} = 12\), and \(\text{O} = 16\). The molar mass of calcium carbonate is calculated as follows: \(40 + 12 + 3 \times 16 = 100 \, \text{g/mol}\).

Find Number of Moles

Use the molar mass to find the number of moles in 10 g of calcium carbonate: \[\text{Number of moles} = \frac{10 \, \text{g}}{100 \, \text{g/mol}} = 0.1 \, \text{mol}\].

Calculate Total Number of Molecules

Use Avogadro’s number \(N_0 = 6.023 \times 10^{23}\) to determine the total number of \(\text{CaCO}_3\) molecules in 0.1 moles: \[\text{Total Molecules} = 0.1 \, \text{mol} \times 6.023 \times 10^{23} = 6.023 \times 10^{22}\, \text{molecules}\].

Calculate Number of Protons per Molecule

In \(\text{CaCO}_3\), Calcium has 20 protons, Carbon has 6 protons, and each Oxygen has 8 protons. Therefore, total protons per molecule are \(20 + 6 + 3 \times 8 = 50\).

Calculate Total Number of Protons

Multiply the number of protons per molecule by the total number of molecules: \[\text{Total Protons} = 50 \times 6.023 \times 10^{22} = 3.0115 \times 10^{24}\].

Select Correct Option

Comparing the calculated total number of protons \(3.0115 \times 10^{24}\) to the given options, the closest option is (a) \(3.01 \times 10^{24}\).

Key concepts

Avogadro's Number

Avogadro's Number is a fundamental concept in chemistry that allows us to quantize particles at the atomic and molecular level. Simply put, it is the number of atoms, molecules, or other particles in one mole of a substance. This number is universally accepted as approximately \(6.023 \times 10^{23}\). Avogadro's Number provides a bridge between the microscopic world of atoms and molecules and the macroscopic quantities we measure in the lab.
This means that if you have one mole of a substance, it contains exactly Avogadro's Number of entities, whether they are atoms, ions, or molecules.
Understanding Avogadro's Number is crucial because it allows chemists to convert between the mass of a sample and the number of atoms or molecules it contains. For example, if you have 0.1 moles of a substance like calcium carbonate, you can use Avogadro’s Number to find the total number of molecules in that sample: \(0.1 \, \text{mol} \times 6.023 \times 10^{23} = 6.023 \times 10^{22}\) molecules. This concept allows calculations that take us from the molecular scale to practical quantities that can be weighed and measured.

Molar Mass

Molar Mass is the mass of one mole of a substance, typically measured in grams per mole (g/mol). It serves as a conversion factor between the amount of substance (in moles) and its mass (in grams). To calculate the molar mass of a compound, add together the atomic masses of each element in the molecular formula.
For example, if we want to find the molar mass of calcium carbonate \(\text{CaCO}_3\), we consider the molar masses of its constituent atoms: calcium \(\text{(Ca)}\) is \(40 \, \text{g/mol}\), carbon \(\text{(C)}\) is \(12 \, \text{g/mol}\), and oxygen \(\text{(O)}\) is \(16 \, \text{g/mol}\). Since \(\text{CaCO}_3\) includes one calcium atom, one carbon atom, and three oxygen atoms, its molar mass is calculated as follows:
- \(40 + 12 + 3 \times 16 = 100 \, \text{g/mol}\).
This value is used to convert the mass of the substance to the amount of substance in moles, allowing for further chemical calculations, such as determining how many moles there are in 10 grams of calcium carbonate: \(\frac{10 \, \text{g}}{100 \, \text{g/mol}} = 0.1 \, \text{mol}\).

Protons in Compounds

Understanding the number of protons in a compound is essential for identifying the composition and reactivity of the substance. Protons are subatomic particles found in the nucleus of an atom, and each element has a unique number of protons, which is its atomic number.
In a compound like calcium carbonate \((\text{CaCO}_3)\), you can determine the total number of protons by adding the protons from each atom within the molecule:
- Calcium has 20 protons
- Carbon has 6 protons
- Each Oxygen atom has 8 protons
Given there are three oxygen atoms, the total number of protons per molecule is calculated by adding these together: \(20 + 6 + 8 \times 3 = 50\). This means each molecule of calcium carbonate has 50 protons.
To find the total number of protons in a sample, multiply the number of protons per molecule by the total number of molecules in the sample. In the exercise, we computed the total number of molecules in 10 grams of calcium carbonate as \(6.023 \times 10^{22}\) molecules. Therefore, in this sample, the total number of protons is: \(50 \times 6.023 \times 10^{22} = 3.0115 \times 10^{24}\). This calculation shows how we can bridge the micro world of atoms with the macro world of visible materials by understanding their composition at an atomic level.