Question

The total number of protons in \(10 \mathrm{~g}\) of calcium carbonate is \(\left(\mathrm{N}_{0}=6.023 \times 10^{25}\right)\) (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 is approximately \(3.01 \times 10^{24}\), option (a).

Step-by-step solution

Determine Molar Mass

Calcium carbonate (CaCO_3) is composed of 1 calcium (Ca), 1 carbon (C), and 3 oxygen (O) atoms. Calculate each component's mass: - Calcium: approximately 40 g/mol - Carbon: approximately 12 g/mol - Oxygen: approximately 16 g/mol each, so 3 oxygens are 48 g/mol Adding them together, the molar mass of CaCO extsubscript{3} is 40 + 12 + 48 = 100 g/mol.

Calculate Moles of Calcium Carbonate

Given that the mass of calcium carbonate is 10 g, use the molar mass to find the number of moles:\[\text{Moles of }CaCO_3 = \frac{\text{Mass of } CaCO_3}{\text{Molar mass of } CaCO_3} = \frac{10 \, ext{g}}{100 \, ext{g/mol}} = 0.1 \, ext{mol}\]

Calculate Number of Molecules

Use Avogadro's number to determine the number of molecules in 0.1 moles of calcium carbonate.\[\text{Number of molecules} = 0.1 \, ext{mol} \times 6.023 \times 10^{23} \, \text{molecules/mol} = 6.023 \times 10^{22} \, \text{molecules}\]

Determine Protons per Molecule

Each CaCO3 molecule has calcium (20 protons), carbon (6 protons), and three oxygen atoms (8 protons each), giving:- Calcium: 20 protons- Carbon: 6 protons- Oxygen: 3 atoms \( \times \, 8 \text{ protons} = 24 \text{ protons} \)Total protons per CaCO3 molecule = 20 + 6 + 24 = 50 protons.

Calculate Total Number of Protons

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

Compare to Options

The calculated total number of protons is approximately \(3.0115 \times 10^{24}\), which closely matches option (a) \(3.01 \times 10^{24}\).

Key concepts

Molar Mass

Molar mass is the mass of a given substance (chemical element or chemical compound) divided by the amount of substance, usually expressed in grams per mole (g/mol). It is a crucial concept in chemistry, especially when performing stoichiometric calculations, which involve predicting the quantities of reactants and products in chemical reactions.
To calculate the molar mass of a compound like calcium carbonate (CaCO_3), you need to sum the atomic masses of each atom in the formula.

• Calcium (Ca) has an atomic mass of approximately 40 g/mol.
• Carbon (C) has an atomic mass of approximately 12 g/mol.
• Oxygen (O) has an atomic mass of approximately 16 g/mol. Since there are three oxygen atoms in a calcium carbonate molecule, you multiply 16 by 3 to get 48 g/mol.

By adding these together—40 + 12 + 48—you find that the molar mass of calcium carbonate is 100 g/mol. Knowing this allows you to convert between mass and moles, facilitating the calculation of other chemical properties.

Avogadro's Number

Avogadro's Number is a fundamental constant, represented as 6.022 x 10^{23}, which defines the number of particles in one mole of a substance. It is essential for converting between the macroscopic scale of lab measurements and the microscopic scale of atoms and molecules.
This constant helps chemists understand how many molecules or atoms are in a given sample when the mass and molar mass are known. For example, with calcium carbonate, once you calculate that you have 0.1 moles in 10 grams, you can use Avogadro's Number to determine the number of molecules.
The calculation goes as follows:
- Multiply the moles (0.1) by Avogadro's Number (6.022 x 10^{23}) to get the number of molecules. - This will equal approximately 6.022 x 10^{22} molecules. Understanding this relation allows you to predict how many individual molecules of a compound are present in a sample, which is vital for chemical analysis and reactions.

Protons in Molecules

To count the total number of protons in a molecule like calcium carbonate (CaCO_3), you need to know the number of protons in each atom and how many of each type of atom are present in the molecule.
Each element's atomic number corresponds to the number of protons found in its nucleus, and these numbers are essential for calculating the total proton count in a molecular formula.
Here’s the breakdown:

• Calcium (Ca) has an atomic number of 20, thus 20 protons.
• Carbon (C) has an atomic number of 6, thus 6 protons.
• Oxygen (O) has an atomic number of 8, so each atom contributes 8 protons, for 3 oxygen atoms, it totals to 24 protons.

This gives a total of 50 protons per calcium carbonate molecule: 20 from calcium, 6 from carbon, and 24 from oxygen.
Given that you now know each CaCO_3 molecule contains 50 protons, you can calculate the total number of protons in a sample by multiplying the number of molecules by the protons per molecule, providing invaluable insight into the compound's composition.