Question

\(0.25 \mathrm{~mol}\) of \(\mathrm{P}_{4}\) molecules contains ___ atoms. (a) \(1.764 \times 10^{23}\) (b) \(6.02 \times 10^{19}\) (c) \(6.023 \times 10^{23}\) (d) \(8.086 \times 10^{23}\)

Short answer

The correct answer is (c) \(6.023 \times 10^{23}\).

Step-by-step solution

Understand the Mole Concept

One mole of a substance contains Avogadro's number \(6.022 \times 10^{23}\) of molecules or atoms. For a given amount of substance in moles, you can find out how many molecules or atoms are in that sample by using this number.

Calculate Molecules in the Sample

Given \(0.25 \mathrm{~mol}\) of \(\mathrm{P}_4\) molecules, the number of \(\mathrm{P}_4\) molecules can be found using Avogadro's number. Multiply the moles by Avogadro's number: \(0.25 \times 6.022 \times 10^{23} = 1.5055 \times 10^{23}\) \(\mathrm{P}_4\) molecules.

Determine Atoms in Each Molecule

Each \(\mathrm{P}_4\) molecule contains 4 phosphorus atoms. To find total atoms, multiply the number of molecules by the number of atoms per molecule.

Calculate Total Number of Atoms

Multiply the \(1.5055 \times 10^{23}\) molecules by 4 atoms per molecule: \(1.5055 \times 10^{23} \times 4 = 6.022 \times 10^{23}\) atoms.

Select Closest Answer

The closest answer to \(6.022 \times 10^{23}\) atoms is option (c) \(6.023 \times 10^{23}\).

Key concepts

Mole Concept

The mole concept is a foundational idea in chemistry that provides a bridge between the atomic world and the macroscopic world we observe. It is a counting unit similar to a dozen but much more extensive. One mole of any substance contains exactly Avogadro's number of entities, which is approximately \(6.022 \times 10^{23}\) particles be they atoms, molecules, ions, or other entities.
For example, one mole of water (H_2O) contains \(6.022 \times 10^{23}\) water molecules. This unit is useful because it allows chemists to count particles by weighing them: when we say a mole of something, we know it has a specific mass directly related to the atomic or molecular weight of the substance.

• Avogadro's Number: A key component of the mole concept, it allows conversion between the amount of substance and the number of particles.
• Mass to Mole Conversion: By knowing the molar mass, it becomes possible to convert mass to moles, facilitating various chemical calculations.

Phosphorus Molecules

Phosphorus is an element that, in its most stable form, exists as tetrahedral molecules denoted P_4. These phosphorus molecules are unique structurally and play a significant role in various chemical reactions.
Within a P_4 molecule, there are exactly four phosphorus atoms, contributing to its tetrahedral geometrical shape. This form is highly stable and is a notable feature of elemental phosphorus.

• Phosphorus Tetrahedrons: Each P_4 molecule consists of four phosphorus atoms arranged in a tetrahedron.
• Application in Calculations: Knowing the molecular structure helps in determining the number of atoms when given moles of the compound.

Chemical Calculations

Chemical calculations often involve converting between different units and quantities, using known constants and relationships. Understanding how to use the mole concept, Avogadro's number, and molecule structure is crucial.
When given a number of moles, like with phosphorus P_4, calculations often involve determining how many molecules or atoms are present. The steps typically involve:

• Calculating the number of molecules using Avogadro's number.
• Determining the number of atoms per molecule based on the molecular structure (four atoms for each P_4).
• Multiplying both to find the total number of atoms.

These calculations are fundamental to formulating chemical reactions and understanding the quantities of products and reactants involved. These include critical applications in stoichiometry and yield predictions within chemical industries and academic settings.