Question

Which contains the greatest number of oxygen atoms? (1) \(1 \mathrm{~g}\) of \(\mathrm{O}_{2}\) (2) \(1 \mathrm{~g}\) of \(\mathrm{O}_{2}\) (3) \(1 \mathrm{~g}\) of \(\mathrm{O}_{3}\) (4) All have the same number of atoms

Short answer

All have approximately the same number of oxygen atoms.

Step-by-step solution

Find the molar mass of each molecule

Determine the molar mass of \( \text{O}_2 \) and \( \text{O}_3 \). The molar mass of \( \text{O}_2 \) is \[ 2 \times 16 = 32 \text{ g/mol} \]. For \( \text{O}_3 \), the molar mass is \[ 3 \times 16 = 48 \text{ g/mol} \].

Calculate moles of each gas

Calculate the number of moles in \( 1 \text{ g} \) of each gas using the formula \[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} \]. \- For \( 1 \text{ g} \text{of} \text{O}_2 \): \[ \text{moles of} \text{O}_2 = \frac{1}{32} = 0.03125 \text{ mol} \] \- For \( 1 \text{ g of } \text{O}_3 \): \[ \text{moles of} \text{O}_3 = \frac{1}{48} = 0.02083 \text{ mol} \]

Determine the number of oxygen atoms in each mole

Each mole of \text{O}_2 \ contains \[ 2 \times \text{ Avogadro's number } (6.022 \times 10^{23} \text{ atoms/mol}) \] of oxygen atoms. Each mole of \text{O}_3 \ contains \[ 3 \times \text{ Avogadro's number } (6.022 \times 10^{23} \text{ atoms/mol}) \] of oxygen atoms.

Calculate total number of oxygen atoms

- For \( 1 \text{ g} \text{of} \text{O}_2 \): \[ 0.03125 \text{ mol} \times 2 \times 6.022 \times 10^{23} = 3.759 \times 10^{22} \text{ atoms} \] \- For \( 1 \text{ g of } \text{O}_3 \): \[ 0.02083 \text{ mol} \times 3 \times 6.022 \times 10^{23} = 3.760 \times 10^{22} \text{ atoms} \]

Compare the number of oxygen atoms

Compare the number of oxygen atoms calculated for \text{O}_2 \ and \text{O}_3. The numbers are approximately equal: \[ 3.759 \times 10^{22} \text{ atoms for } \text{O}_2 \] and \[ 3.760 \times 10^{22} \text{ atoms for } \text{O}_3 \].

Conclusion

Since the number of atoms in all cases is very close, they can be considered approximately equal.

Key concepts

Molar Mass Determination

Understanding molar mass is crucial in chemistry because it lets us convert between the mass of a substance and the number of molecules or atoms it contains. Molar mass is the mass of one mole of a substance, measured in grams per mole (g/mol). To find the molar mass, simply sum the atomic masses of all the atoms in the molecule. For example, the atomic mass of an oxygen atom (\text{O}) is roughly 16 g/mol. Hence, the molar mass of \text{O}_2 (which has two oxygen atoms) is \[ 2 \times 16 = 32 \text{ g/mol} \]. Similarly, for \text{O}_3 (ozone), which consists of three oxygen atoms, the molar mass would be \[ 3 \times 16 = 48 \text{ g/mol} \]. This step is essential to proceed with our calculations since knowing the molar mass allows us to find the number of moles given a certain mass of a substance.

Moles Calculation

The concept of moles bridges the gap between the atomic scale and the everyday scale we can observe. One mole of any substance contains exactly Avogadro's number of entities, which is approximately \[ 6.022 \times 10^{23} \]. To find the number of moles in a given sample, you use the formula: \[ \text{moles} = \frac{\text{mass}}{\text{molar mass}} \]. For instance, if we have 1 gram of \text{O}_2 and know its molar mass is 32 g/mol, the number of moles is: \[ \frac{1}{32} = 0.03125 \text{ mol} \]. Similarly, for 1 gram of \text{O}_3 with a molar mass of 48 g/mol, we get: \[ \frac{1}{48} = 0.02083 \text{ mol} \]. This calculation is vital as it helps us move forward to find the actual number of atoms or molecules within the sample.

Avogadro's Number

Avogadro's number is a cornerstone in chemistry, providing the link between the amount of substance (in moles) and the number of particles it contains. Avogadro's number is \[ 6.022 \times 10^{23} \] particles per mole. So, 1 mole of \text{O}_2 contains \[ 2 \times 6.022 \times 10^{23} \] oxygen atoms because each molecule has two oxygen atoms. Calculating the number of atoms in a given mass involves converting the mass to moles and then multiplying by Avogadro's number. For example, for 0.03125 mol of \text{O}_2: \[ 0.03125 \text{ mol} \times 2 \times 6.022 \times 10^{23} = 3.759 \times 10^{22} \text{ atoms} \]. Similarly, for 0.02083 mol of \text{O}_3: \[ 0.02083 \text{ mol} \times 3 \times 6.022 \times 10^{23} = 3.76 \times 10^{22} \text{ atoms} \]. Knowing Avogadro's number allows us to quantify the incredibly large number of atoms in even the tiniest samples.