Problem 32 - Chemistry: The Central Science

Question

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

Step-by-Step Solution

Answer

The ranking of the samples in order of increasing numbers of atoms is: \(\mathrm{NaHCO}_{3} \rightarrow \mathrm{CO}_2 \rightarrow \mathrm{Ne}\).

Step 1: Finding the molar mass of the compounds

First, we need to find the molar mass of the compounds using the atomic weights from the periodic table. For Sodium Bicarbonate \(\mathrm{NaHCO}_{3}\): Molar mass of \(\mathrm{Na} = 22.99 \mathrm{~g/mol}\)\ Molar mass of \(\mathrm{H} = 1.01 \mathrm{~g/mol}\)\ Molar mass of \(\mathrm{C} = 12.01 \mathrm{~g/mol}\)\ Molar mass of \(\mathrm{O} = 16.00 \mathrm{~g/mol}\) The molar mass of \(\mathrm{NaHCO}_{3}\) will be the sum of the molar mass of its constituent atoms: Molar mass of \(\mathrm{NaHCO}_{3} = 22.99 + 1.01 + 12.01 + 3\cdot16.00 = 84.01 \mathrm{~g/mol}\)

Step 2: Converting given amounts to moles or atoms

Now we'll convert the given amounts of each sample into either moles or number of atoms. For \(\mathrm{NaHCO}_{3}:\) Given mass = \(42 \mathrm{~g}\)\ Molar mass = \(84.01 \mathrm{~g/mol}\) Number of moles of \(\mathrm{NaHCO}_{3}\) will be: \(n = \frac{\text{Given mass}}{\text{Molar mass}} = \frac{42}{84.01} \approx 0.5 \textrm{ moles}\) For Carbon Dioxide \(\mathrm{CO_2}\): Number of moles are given: \(1.5\ \mathrm{mol}\) For Ne: Number of atoms are given: \(6.0\times10^{24}\ \mathrm{Ne}\) atoms.

Step 3: Converting moles to atoms

Now we need to convert the number of moles of both \(\mathrm{NaHCO}_{3}\) and \(\mathrm{CO}_2\) to atoms. For \(\mathrm{NaHCO}_{3}\): We got \(0.5\) moles in step 2. To find the number of atoms, we need to multiply the number of moles with the Avogadro's number, which is \(6.022\times10^{23}\ \mathrm{atoms/mol}\): Number of \(\mathrm{NaHCO}_3\) atoms = \(0.5\ \mathrm{mol} \times 6.022\times10^{23}\ \mathrm{atoms/mol} = 3.011\times10^{23}\ \mathrm{NaHCO_3}\) atoms. For each molecule of \(\mathrm{NaHCO}_{3}\), there are \(6\) atoms. Thus, total number of atoms in \(42\ \mathrm{g}\) of \(\mathrm{NaHCO}_{3}\): \(3.011\times10^{23} \times 6 \approx 1.81 \times 10^{24}\) atoms. For \(\mathrm{CO}_2\): We got \(1.5\ \mathrm{mol}\) in step 2. To find the number of atoms, we need to multiply the number of moles with the Avogadro's number: Number of \(\mathrm{CO}_2\) atoms = \(1.5\ \mathrm{mol} \times 6.022\times10^{23}\ \mathrm{atoms/mol} = 9.033\times 10^{23}\ \mathrm{CO_2}\) atoms. For each molecule of \(\mathrm{CO}_{2}\), there are \(3\) atoms. Thus, total number of atoms in \(1.5\ \mathrm{mol}\) of \(\mathrm{CO}_2\): \(9.033\times 10^{23}\times 3 \approx 2.71 \times 10^{24}\) atoms.

Step 4: Comparing the number of atoms and ranking

Now we have the number of atoms for each sample: 1. \(\mathrm{NaHCO}_{3}\): \(1.81 \times 10^{24}\) atoms 2. \(\mathrm{CO}_2\): \(2.71 \times 10^{24}\) atoms 3. \(\mathrm{Ne}\): \(6.0 \times 10^{24}\) atoms Thus, the ranking of the samples in order of increasing numbers of atoms is: \(\mathrm{NaHCO}_{3} \rightarrow \mathrm{CO}_2 \rightarrow \mathrm{Ne}\)

Key Concepts

Molar Mass Calculation

When it comes to understanding the mole concept, molar mass calculation is a fundamental skill. The molar mass is the weight of one mole of a substance, typically expressed in grams per mole (g/mol). To calculate molar mass, add up the atomic weights of all the atoms in a compound's chemical formula. Here's how it's done:

You can find the atomic weights for each element on the periodic table. For a compound like Sodium Bicarbonate (NaHCO₃), the calculation would be as follows:

• Molar mass of Na = 22.99 g/mol
• Molar mass of H = 1.01 g/mol
• Molar mass of C = 12.01 g/mol
• Molar mass of O = 16.00 g/mol (times 3 because there are three oxygen atoms)

So, the molar mass of NaHCO₃ would be the sum of the molar masses of its constituent elements:
Molar mass of NaHCO₃ = 22.99 + 1.01 + 12.01 + 3(16.00) = 84.01 g/mol.

Keep in mind that the molar mass can be used to convert between the mass of a sample and the number of moles, which is essential for stoichiometry.

Avogadro's Number

Another pillar of the mole concept is Avogadro's number, which is 6.022 x 10²³. This gigantic number represents the number of particles found in one mole of a substance. Whether these particles are atoms, molecules, or ions, one mole of them will equate to Avogadro's number.

To convert from moles to particles (or vice versa), you multiply (or divide) by Avogadro's number, providing a bridge between the macroscopic world we can measure and the atomic scale. For example, if you have 0.5 moles of a substance, the number of atoms or molecules would be:0.5 moles x 6.022 x 10²³ atoms/mole. This concept allows us to count actual atoms or molecules in a given sample, connecting the conceptual with the quantitative aspects of chemistry.

Atomic Weights

Atomic weights, sometimes referred to as atomic masses, are crucial when dealing with chemical substances. Each element has its atomic weight listed on the periodic table and it reflects the weighted average mass of the atoms that compose that element. These weights take into account the different isotopes of an element and their respective abundances.

Atomic weights are the building blocks for calculations such as molar mass or determining the ratios of elements in compounds. They are also used for converting masses of substances to moles, an essential process in stoichiometry. For instance, if you know the atomic weight of carbon is 12.01 g/mol, you can calculate how many moles are in a 24.02-gram sample of carbon (which would be 2 moles, because 24.02 divided by 12.01 equals 2).

Chemical Formula

Understanding a chemical formula is vital as it conveys a wealth of information about the substance it represents. A chemical formula identifies the elements in a compound and the number of atoms of each element in one molecule of that compound. The notation uses the element symbols from the periodic table and numerical subscripts to indicate the number of atoms.

Take H₂O as an example – the '2' tells us there are two hydrogen atoms for each oxygen atom in water. The formula not only indicates the composition of the substance but also allows the calculation of the molar mass by summing the atomic weights of each element as indicated by the formula. In more complex compounds, understanding the formula is essential for predicting the compound's properties and reactivity, and for performing stoichiometric calculations in chemical reactions.