Question

How many atoms are in each elemental sample? (a) \(3.4 \mathrm{~mol} \mathrm{Cu}\) (b) \(9.7 \times 10^{-3} \mathrm{~mol} \mathrm{C}\) (c) \(22.9 \mathrm{~mol} \mathrm{Hg}\) (d) \(0.215 \mathrm{~mol} \mathrm{Na}\)

Short answer

The number of atoms is: (a) \(2.047 \times 10^{24}\) atoms Cu, (b) \(5.843 \times 10^{21}\) atoms C, (c) \(1.379 \times 10^{25}\) atoms Hg, (d) \(1.2947 \times 10^{23}\) atoms Na.

Step-by-step solution

Calculate the number of atoms in a sample of copper (Cu)

Use Avogadro's number, which is approximately \(6.022 \times 10^{23}\) atoms per mol, to calculate the number of atoms in \(3.4 \mathrm{~mol} \mathrm{Cu}\). Multiply the number of moles by Avogadro's number: \(3.4 \mol \times 6.022 \times 10^{23} \frac{\text{atoms}}{\mol} = 2.047 \times 10^{24}\) atoms of Cu.

Calculate the number of atoms in a sample of carbon (C)

Use Avogadro's number to calculate the number of atoms in \(9.7 \times 10^{-3} \mathrm{~mol} \mathrm{C}\). Multiply the number of moles by Avogadro's number: \(9.7 \times 10^{-3} \mol \times 6.022 \times 10^{23} \frac{\text{atoms}}{\mol} = 5.843 \times 10^{21}\) atoms of C.

Calculate the number of atoms in a sample of mercury (Hg)

Use Avogadro's number to calculate the number of atoms in \(22.9 \mathrm{~mol} \mathrm{Hg}\). Multiply the number of moles by Avogadro's number: \(22.9 \mol \times 6.022 \times 10^{23} \frac{\text{atoms}}{\mol} = 1.379 \times 10^{25}\) atoms of Hg.

Calculate the number of atoms in a sample of sodium (Na)

Use Avogadro's number to calculate the number of atoms in \(0.215 \mathrm{~mol} \mathrm{Na}\). Multiply the number of moles by Avogadro's number: \(0.215 \mol \times 6.022 \times 10^{23} \frac{\text{atoms}}{\mol} = 1.2947 \times 10^{23}\) atoms of Na.

Key concepts

Mole to Atoms Conversion

Understanding how to convert moles to atoms is a fundamental aspect of chemistry, especially when dealing with chemical reactions and the synthesis of materials. A mole is a unit of measurement that expresses amounts of a chemical substance. It is one of the seven base units in the International System of Units (SI) and is defined as containing exactly 6.02214076×10²³ elementary entities, be they atoms, molecules, ions, or electrons. This number is known as Avogadro's number, and it's key to converting moles to atoms.

The conversion from moles to atoms is straightforward: you simply multiply the number of moles by Avogadro's number. For instance, to find how many atoms are in 3.4 moles of copper (Cu), you multiply:\[3.4 \, \text{mol} \times 6.022 \times 10^{23} \, \frac{\text{atoms}}{\text{mol}} = 2.047 \times 10^{24}\] atoms of Cu.

This calculation is an essential skill in chemistry because it allows scientists to work with the macroscopic amounts of substances and relate them to the number of particles involved in chemical processes.

Chemistry Stoichiometry

Chemistry stoichiometry involves the calculation of the reactants and products in chemical reactions. It is based on the conservation of mass where the quantity of each element does not change in a chemical reaction. Therefore, the mass and the mole relations in a balanced chemical equation must be constant. Stoichiometry allows us to predict the amounts of products formed in a reaction or the amount of reactants needed to create a desired amount of product.

When using stoichiometry for calculations, balance the equation and use mole ratios to convert between substances. For example, if we know the number of moles of sodium (Na) used in a reaction, and we have a balanced equation, we can use the mole ratios to find out how many moles of other substances are involved or produced in the reaction. From there, you can convert moles back into mass, volume, or number of particles (like atoms), depending on the information required for the problem at hand.

Moles of Elements

Moles of elements refer to the amount of a chemical element expressed in moles. Since elements can exist in various isotopic forms, chemists use the average atomic mass of these isotopes as it appears on the periodic table to calculate the molar mass—the mass of one mole of an element. The molar mass ties together the macroscopic scale (grams, liters, pieces) with the microscopic scale (atoms, molecules).

For example, mercury (Hg) has a molar mass of approximately 200.59 g/mol. If you have 22.9 moles of mercury, this amount is equivalent to a very large number of atoms. Using Avogadro's law, we can convert moles of an element to the number of atoms by multiplying the number of moles by Avogadro's number, giving us a very useful way to work with quantities that make sense on a human scale. The exact calculation for mercury would be: \[22.9 \, \text{mol} \times 6.022 \times 10^{23} \, \frac{\text{atoms}}{\text{mol}} = 1.379 \times 10^{25}\] atoms of Hg.

Working with moles of elements allows chemists to calculate the quantities required for reactions, to predict yields, and to analyze the empirical and molecular formulas of substances.