Problem 123 - Chemistry The Molecular Science

Question

Which member of each pair has the greater number of atoms? Explain why. (a) \(1 \mathrm{~mol} \mathrm{Cl}\) or \(1 \mathrm{~mol} \mathrm{Cl}_{2}\) (b) 1 molecule \(\mathrm{O}_{2}\) or \(1 \mathrm{~mol} \mathrm{O}_{2}\) (c) 1 nitrogen atom or 1 nitrogen molecule (d) \(6.032 \times 10^{23}\) fluorine molecules or \(1 \mathrm{~mol}\) fluorine molecules (e) \(20.3 \mathrm{~g} \mathrm{Ne}\) or \(1 \mathrm{~mol} \mathrm{Ne}\) (f) 1 molecule \(\mathrm{Br}_{2}\) or \(159.8 \mathrm{~g} \mathrm{Br}_{2}\) (g) \(107.9 \mathrm{~g} \mathrm{Ag}\) or \(9.6 \mathrm{~g} \mathrm{Li}\) (h) \(58.9 \mathrm{~g} \mathrm{Co}\) or \(58.9 \mathrm{~g} \mathrm{Cu}\) (i) \(1 \mathrm{~g}\) calcium or \(6.022 \times 10^{23}\) calcium atoms (j) \(1 \mathrm{~g}\) chlorine atoms or \(1 \mathrm{~g}\) chlorine molecules

Step-by-Step Solution

Answer

(a) \(1 \text{ mole of } \text{Cl}_2\), (b) \(1 \text{ mole of } \text{O}_2\), (c) nitrogen molecule, (d) \(6.032 \times 10^{23}\) molecules, (e) \(20.3 \text{ g Ne}\), (f) \(159.8 \text{ g Br}_2\), (g) \(9.6 \text{ g Li}\), (h) \(58.9 \text{ g Co}\), (i) \(6.022 \times 10^{23}\) atoms, (j) \(1 \text{ g Cl atoms}\).

Step 1: Analyze Part (a)

1 mol of Cl has Avogadro's number of atoms, which is \(6.022 \times 10^{23}\) Cl atoms. 1 mol of \(\text{Cl}_2\) also has Avogadro's number of molecules, but each Cl molecule consists of 2 Cl atoms, totaling \(2 \times 6.022 \times 10^{23}\) Cl atoms. Therefore, \(1 \text{ mole of } \text{Cl}_2\) contains more atoms.

Step 2: Analyze Part (b)

1 molecule of \(\text{O}_2\) contains 2 oxygen atoms. 1 mol of \(\text{O}_2\) contains \(6.022 \times 10^{23}\) molecules, each with 2 atoms, resulting in a total of \(2 \times 6.022 \times 10^{23}\) atoms. Thus, \(1 \text{ mole of } \text{O}_2\) contains more atoms.

Step 3: Analyze Part (c)

1 nitrogen atom is just a single atom. 1 nitrogen molecule, typically \(\text{N}_2\), consists of 2 nitrogen atoms. Therefore, \(1 \text{ nitrogen molecule}\) has more atoms.

Step 4: Analyze Part (d)

\(6.032 \times 10^{23}\) fluorine molecules have slightly more than Avogadro's number of atoms compared to 1 mol of fluorine molecules, which is exactly \(6.022 \times 10^{23}\). Thus, \(6.032 \times 10^{23} \text{ molecules}\) have more atoms.

Step 5: Analyze Part (e)

20.3 g of Ne, where the molar mass of Ne is approximately \(20.18 \text{ g/mol}\), equates to slightly more than 1 mol of Ne. Thus, \(20.3 \text{ g Ne}\) contains more atoms than \(1 \text{ mole of Ne}\).

Step 6: Analyze Part (f)

1 molecule of \(\text{Br}_2\) contains 2 atoms. The molar mass of \(\text{Br}_2\) is approximately \(159.8 \text{ g/mol}\), so \(159.8 \text{ g Br}_2\) is 1 mole, which contains \(6.022 \times 10^{23}\) molecules or \(2 \times 6.022 \times 10^{23}\) atoms. Therefore, \(159.8 \text{ g Br}_2\) contains more atoms.

Step 7: Analyze Part (g)

The molar mass of Ag is \(107.9 \text{ g/mol}\), so \(107.9 \text{ g}\) is 1 mole of Ag, containing \(6.022 \times 10^{23}\) atoms. The molar mass of Li is \(6.94 \text{ g/mol}\), so \(9.6 \text{ g}\) is about \(1.383 \) moles of Li, which is \(1.383 \times 6.022 \times 10^{23}\) atoms. Therefore, \(9.6 \text{ g Li}\) has more atoms.

Step 8: Analyze Part (h)

The molar mass of Co is \(58.9 \text{ g/mol}\), and of Cu is \(63.55 \text{ g/mol}\). Thus, \(58.9 \text{ g Co}\) corresponds to 1 mole of Co, while \(58.9 \text{ g Cu}\) is less than 1 mole due to the larger molar mass. Therefore, \(58.9 \text{ g Co}\) contains more atoms.

Step 9: Analyze Part (i)

1 g of calcium has a molar mass of \(40.08 \text{ g/mol}\), giving \(1/40.08\) mol or about 0.025 moles, which is less than Avogadro's number of atoms. Therefore, \(6.022 \times 10^{23}\) calcium atoms have more atoms.

Step 10: Analyze Part (j)

1 g of Cl atoms has a molar mass of \(35.45 \text{ g/mol}\), providing \(1/35.45\) moles. 1 g of \(\text{Cl}_2\) has a molar mass of \(70.90 \text{ g/mol}\), providing \(1/70.90\) moles, with each molecule containing 2 atoms. After calculating, it becomes clear \(1 \text{ g of Cl atoms}\) has more atoms.

Key Concepts

Mole Concept

The mole concept is fundamental in chemistry and helps us understand the quantity of substance involved in chemical reactions and equations. A mole, defined by Avogadro's Number, is

• 6.022 \times 10^{23}

representative particles, be it atoms, molecules, or ions. This massive number is used to convert between atoms and grams in chemical calculations.
In real-world terms, the mole provides a bridge between the atomic scale and our everyday macroscopic realm. For instance:

• 1 mole of hydrogen atoms contains approximately 1 gram of hydrogen.
• 1 mole of carbon has a mass of about 12 grams.

When comparing quantities, as seen in exercises like whether 1 mole of \(\text{Cl}_2\) or chlorine atoms contains more atoms, the mole concept helps ascertain that 1 mole of indivisible atoms possesses fewer actual atoms compared to a mole of diatomic molecules where each includes 2 atoms.

Atomic Structure

Understanding atomic structure is crucial in evaluating the count of atoms in different substances. An atom is the smallest unit of an element, maintaining its properties, and consists of protons, neutrons, and electrons. When atoms bond together, they form molecules, the chemical behavior depending on the atomic structure.
In the given exercise examples:

• A single nitrogen atom (\(\text{N}\)) holds fewer atoms compared to a nitrogen molecule (\(\text{N}_2\)), which embraces two nitrogen atoms.
• The atomic structure of elements and their combination forms molecules, affecting how we calculate the number of atoms in larger chemical entities.

Through understanding atomic structure, we can better grasp why a substance like \(\text{Cl}_2\) relates to twice the atomic count of \(\text{Cl}\) atoms, despite having the same number of moles.

Molecular Composition

Molecular composition refers to the exact formulation of molecules—detailing which and how many atoms of each element are involved. Each molecule is a specific combination that dictates its properties and chemical behavior.
For example, molecular composition tells us:

• "1 mole of \(\text{O}_2\) contains 2 atoms of oxygen per molecule," so it results in \(2 \times 6.022 \times 10^{23}\) total atoms per mole.
• By understanding molecular composition, we can determine that molecules like \(\text{Br}_2\) have a greater number of atoms per mole (\(2 \times 6.022 \times 10^{23}\)) compared to individual bromine atoms or a lesser number of moles.

This systematic breakdown helps identify characteristics such as mass calculation and the behavior of a substance during physical or chemical change, influencing predictions and reactions in chemical thermodynamics.