Problem 32 - Chemistry The Central Science

Question

Without doing any detailed calculations (but using a periodic table to give atomic weights, rank the following samples in order of increasing numbers of atoms: 42 gof NaHCO \(_{3}, 1.5 \mathrm{mol} \mathrm{CO}_{2}, 6.0 \times 10^{24} \mathrm{Ne}\) atoms.

Step-by-Step Solution

Answer

In short, we rank the samples based on increasing number of atoms by first calculating the number of atoms in each sample: NaHCO3 has \(1.805 \times 10^{24}\) atoms, CO2 has \(2.708 \times 10^{24}\) atoms, and Ne has \(6.0 \times 10^{24}\) atoms. Therefore, the ranking is: NaHCO3 < CO2 < Ne.

Step 1: Calculate the number of moles of NaHCO3

To find the number of atoms in the given mass of NaHCO3, first, we need to find the number of moles present. We can use the formula: Number of moles = (Given mass) / (Molar mass) The molar mass of NaHCO3 can be calculated as follow (from the periodic table): Na (1 x 22.99) + H (1 x 1.008) + C (1 x 12.01) + O (3 x 16.00) = 84.01 g/mol Now, we can find the number of moles of NaHCO3 in 42 g: Number of moles = 42 g / 84.01 g/mol = 0.5 mol

Step 2: Calculate the number of atoms in NaHCO3

To find the number of atoms in NaHCO3, we'll use Avogadro's number (6.022 x 10^23 atoms/mol) and multiply it by the number of moles. Also, remember that NaHCO3 has 6 atoms in each molecule (Na, H, C, and three O's). Number of atoms in NaHCO3 = 0.5 mol × 6.022 × 10^23 atoms/mol × 6 atoms/molecule = 1.805 × 10^24 atoms

Step 3: Calculate the number of atoms in CO2

We have been given 1.5 moles of CO2, and each molecule of CO2 has 3 atoms (1 C and 2 O). Therefore, we can directly calculate the number of atoms in CO2 by multiplying the given moles, Avogadro's number, and the atoms in each molecule. Number of atoms in CO2 = 1.5 mol × 6.022 × 10^23 atoms/mol × 3 atoms/molecule = 2.708 × 10^24 atoms

Step 4: Compare the number of atoms and rank the samples

Now that we have the number of atoms for all samples, we can compare and rank them: 1. NaHCO3: 1.805 × 10^24 atoms 2. CO2: 2.708 × 10^24 atoms 3. Ne: 6.0 × 10^24 atoms Based on the increasing number of atoms, the order is: NaHCO3 < CO2 < Ne.

Key Concepts

Molar mass

Every substance has a unique molar mass, which is the mass of one mole of that substance. It's measured in grams per mole (g/mol). The molar mass is calculated using the atomic weights of the elements from the periodic table.
For example, in the compound NaHCO₃ (baking soda):

• Sodium (Na) contributes about 22.99 g/mol.
• Hydrogen (H) adds 1.008 g/mol.
• Carbon (C) adds 12.01 g/mol.
• Oxygen (O), with three atoms, contributes 48.00 g/mol (3 x 16.00).

All these values come together to give the molar mass of NaHCO₃ as approximately 84.01 g/mol.
By knowing the molar mass, you can convert between the mass of a substance and the number of moles, using the equation:\[\text{Number of Moles} = \frac{\text{Given Mass}}{\text{Molar Mass}}\]This conversion is a crucial step in determining the number of atoms in a given sample.

Avogadro's number

Avogadro's number is an essential tool in chemistry, providing a bridge between the microscopic world of atoms and the macroscopic world we observe. It is defined as the number of atoms, ions, or molecules in one mole of a substance, which is approximately 6.022 x 10^23.
This number tells us how many particles are in a mole, allowing us to convert between moles and the actual number of particles. For example, if you have 0.5 moles of NaHCO₃, the number of molecules is calculated as:\[0.5 \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mole} = 3.011 \times 10^{23} \text{ molecules}\]Each molecule contains 6 atoms (Na, H, C, and three O), so multiplying by 6 gives the total number of atoms in the sample. Understanding Avogadro's number helps us scale these microscopic calculations to real-world quantities.

Periodic table

The periodic table is an invaluable tool for chemists, providing information on the atomic structure and properties of elements. Each element's atomic number, symbol, and atomic weight are essential for calculating molar mass and understanding chemical behavior.
Here's how you might use the periodic table when working with a compound like CO₂:

• Carbon (C) has an atomic weight of 12.01 g/mol.
• Oxygen (O) has an atomic weight of 16.00 g/mol, and there are two oxygen atoms in CO₂.

The total molar mass of CO₂ is calculated as:\[12.01 \text{ g/mol (C)} + 2 \times 16.00 \text{ g/mol (O)} = 44.01 \text{ g/mol}\]Using the periodic table, you can quickly obtain these values to facilitate calculations in stoichiometry, enhancing your understanding of chemical reactions and compositions.