Problem 28 - Chemistry in 30 days

Question

Which of the following has the highest number of atoms ? (a) \(7.5 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}\) (b) \(250 \mathrm{~g}\) of \(\mathrm{H}_{2} \mathrm{SO}_{5}\) (c) \(200 \mathrm{ml} \mathrm{O}_{3}\) at NTP (d) \(1.5 \mathrm{~mol}\) of \(\mathrm{OsO}_{4}\)

Step-by-Step Solution

Answer

(a) \(7.5 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}\) has the highest number of atoms.

Step 1: Calculate Avogadro's Number of Atoms in Molecules

Avogadro's number is \(6.022 \times 10^{23}\), indicating the number of atoms or molecules in a mole. To find the number of atoms, multiply the moles of molecules by the number of atoms per molecule.(a) \(\text{Atoms in } \mathrm{H}_{2} \mathrm{O}: 7.5 \text{ mol} \times 3 \text{ atoms/molecule} \). (b) \( \text{Atoms in } \mathrm{H}_{2} \mathrm{SO}_{5}: \frac{250 ext{g}}{(2 + 32 + 5\times16)\text{g/mol}} \times 8 \text{ atoms/molecule} \).(d) \(\text{Atoms in } \mathrm{OsO}_{4}: 1.5 \text{mol} \times 5 \text{ atoms/molecule}\).

Step 2: Calculate the Number of \(H_2O\) Atoms

For \(7.5 \mathrm{~mol}\) of \(\mathrm{H}_{2} \mathrm{O}\):- Each water molecule has 3 atoms.- Number of \(\mathrm{H}_{2} \mathrm{O}\) molecules = \(7.5 \times 6.022 \times 10^{23}\).- Total atoms = \(7.5 \times 3 \times 6.022 \times 10^{23} = 1.356 \times 10^{25} \text{ atoms} \).

Step 3: Calculate the Number of \(H_2SO_5\) Atoms

Molar mass of \(\mathrm{H}_{2} \mathrm{SO}_{5} = 2 + 32 + 5\times16 = 132 \text{ g/mol}.\)- Moles = \(\frac{250 \text{ g}}{132 \text{ g/mol}} = 1.89 \text{ mol}.\)- Each molecule has 8 atoms.- Number of \(\mathrm{H}_{2} \mathrm{SO}_{5}\) molecules = \(1.89 \times 6.022 \times 10^{23}\).- Total atoms = \(1.89 \times 8 \times 6.022 \times 10^{23} = 9.11 \times 10^{24} \text{ atoms}\).

Step 4: Calculate the Number of \(O_3\) Atoms

At NTP, 1 mol of a gas occupies 22.4 L.- For \(200 \mathrm{~ml} = 0.2 \text{ L}\), moles of \(\mathrm{O}_3\) = \(\frac{0.2}{22.4} = 0.00893 \text{ mol}.\)- Each \(\mathrm{O}_3\) has 3 atoms.- Total atoms = \(0.00893 \times 3 \times 6.022 \times 10^{23} = 1.61 \times 10^{22} \text{ atoms}\).

Step 5: Calculate the Number of \(OsO_4\) Atoms

For \(1.5 \mathrm{~mol}\) of \(\mathrm{OsO}_{4}\):- Each molecule has 5 atoms.- Number of \(\mathrm{OsO}_{4}\) molecules = \(1.5 \times 6.022 \times 10^{23}\).- Total atoms = \(1.5 \times 5 \times 6.022 \times 10^{23} = 4.52 \times 10^{24} \text{ atoms}\).

Step 6: Compare the Number of Atoms

Compile the total atoms computed from each part:(a) \(\mathrm{H}_{2} \mathrm{O}\) has \(1.356 \times 10^{25} \text{ atoms}.\)(b) \(\mathrm{H}_{2} \mathrm{SO}_{5}\) has \(9.11 \times 10^{24} \text{ atoms}.\)(c) \(\mathrm{O}_3\) has \(1.61 \times 10^{22} \text{ atoms}.\)(d) \(\mathrm{OsO}_{4}\) has \(4.52 \times 10^{24} \text{ atoms}.\)Thus, \(\mathrm{H}_{2} \mathrm{O}\) has the highest number of atoms.

Key Concepts

Moles

A mole is a unit that chemists use to measure the amount of substance. It's like a bridge between the atomic world and the world we can measure.
One mole of any substance contains Avogadro's number of entities, which is approximately \(6.022 \times 10^{23}\). This could be atoms, molecules, or other particles.
It's similar to how a dozen always refers to 12 items, a mole always refers to Avogadro's number of particles.

• If you have 1 mole of water, \( H_2O \), you have \(6.022 \times 10^{23}\) water molecules.
• Each of those water molecules contains 3 atoms—2 hydrogen and 1 oxygen.

So, 1 mole of water contains \(1.807 \times 10^{24}\) atoms. That's a lot of trillions! But that's why moles are so useful—they make dealing with these huge numbers manageable.

Atoms in Molecules

Understanding atoms in molecules helps you know the complexity of substances. Each molecule is made up of atoms held together by chemical bonds.

• A molecule of water \( H_2O \) consists of 2 hydrogen atoms and 1 oxygen atom, totaling 3 atoms.
• A molecule of sulfuric acid \( H_2SO_5 \) includes 2 hydrogen, 1 sulfur, and 5 oxygen atoms, making up 8 atoms.
• Another example is osmium tetroxide \( OsO_4 \), which contains 1 osmium atom and 4 oxygen atoms, totaling 5 atoms.

By understanding how many and what type of atoms compose a molecule, chemists can predict properties and behaviors of substances. This also allows for calculating the total number of atoms in larger samples, critical for chemical reactions.

Molar Mass

Molar mass is the mass of one mole of a particular substance, expressed in grams per mole (g/mol). It's a crucial concept because it allows for converting between the mass of a substance and the moles of a substance.
To calculate molar mass, add up the atomic masses of all the atoms in a molecule's formula. For example:

• Water \( H_2O \): 2 hydrogen atoms \( 2 \times 1 \text{ g/mol} \) + 1 oxygen atom \( 16 \text{ g/mol} \) = 18 g/mol.
• For \( H_2SO_5 \): 2 hydrogen \( 2 \times 1 \text{ g/mol} \) + 1 sulfur \( 32 \text{ g/mol} \) + 5 oxygen \( 5 \times 16 \text{ g/mol} \) = 132 g/mol.

Knowing the molar mass helps you convert the direct physical measurements you're taking (like grams) into the chemical quantities (moles) needed for chemical equations and reactions.

Number of Atoms

Calculating the number of atoms in a substance is fundamental for chemical calculations. Here's how it works:
Start with the number of moles you have, multiply by Avogadro's number to get the number of molecules, and then multiply by the number of atoms per molecule.

• Suppose you have 1 mole of \( H_2O \). That’s \( 6.022 \times 10^{23} \) molecules of water.
• Each molecule of \( H_2O \) has 3 atoms, so you’d have \( 1.807 \times 10^{24} \) atoms in total.

This process lets us understand the scale of substances we're working with, from tiny grams of sugar to liters of gases.
It’s especially important in stoichiometry, where the number of atoms dictates how substances react in a balanced chemical reaction.