Question

Without doing any detailed calculations (but using a periodic table to give atomic weights), rank the following samples in order of increasing number of atoms: \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\), \(23 \mathrm{~g} \mathrm{Na}, 6.0 \times 10^{23} \mathrm{~N}_{2}\) molecules.

Short answer

The samples ranked in increasing order of number of atoms are: \(23 \mathrm{~g} \mathrm{Na}\) < \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) < \(6.0 \times 10^{23} \mathrm{~N}_{2}\) molecules.

Step-by-step solution

Calculate the number of moles for each sample

To compare the number of atoms, we first need to convert each sample into moles using the atomic weights and the Avogadro constant (6.022 x 10^23). 1. For \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\): Already given in moles, no conversion needed. 2. For \(23 \mathrm{~g} \mathrm{Na}\): Use the atomic weight of Na (\(22.99 \)) to convert the mass (g) to moles: \(\dfrac{23 \mathrm{~g}}{22.99 \mathrm{~g/mol}} \approx 1 \mathrm{~mol}\) Na. 3. For \(6.0 \times 10^{23} \mathrm{~N}_{2}\) molecules: Convert number of molecules to moles using Avogadro's constant (\(6.022 \times 10^{23}\) molecules/mol): \(\dfrac{6.0 \times 10^{23} \mathrm{~molecules}}{6.022 \times 10^{23} \mathrm{~molecules/mol}} \approx 1 \mathrm{~mol} \mathrm{N}_{2}\).

Calculate the number of atoms for each sample

Now, we can find the number of atoms in each sample by multiplying the number of moles by the number of atoms per mole: 1. For \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\): There are three atoms per mole (\(2\) hydrogen atoms and \(1\) oxygen atom), so \(0.50 × 3 = 1.50\) moles of atoms. 2. For \(1 \mathrm{~mol} \mathrm{Na}\): There is one atom of Na per mole, so \(1 × 1 = 1\) mole of atoms. 3. For \(1 \mathrm{~mol} \mathrm{N}_{2}\): There are two atoms of N per mole, so \(1 × 2 = 2\) moles of atoms.

Rank the samples by increasing number of atoms

Using the moles of atoms from step 2, we can now rank the samples in order of increasing number of atoms: 1. \(1 \mathrm{~mol} \mathrm{Na}\) (1 mole of atoms) 2. \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) (1.50 moles of atoms) 3. \(1 \mathrm{~mol} \mathrm{N}_{2}\) (2 moles of atoms) So, the final ranking in order of increasing number of atoms is: \(23 \mathrm{~g} \mathrm{Na}\) < \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) < \(6.0 \times 10^{23} \mathrm{~N}_{2}\) molecules.

Key concepts

Atomic Weight

The concept of atomic weight is central to chemistry and measuring how much of a substance there is. At its core, atomic weight represents the average mass of atoms of an element, measured in atomic mass units (amu). This reflects the distribution of isotopes in nature, which are atoms with the same number of protons but different numbers of neutrons.
To determine the number of moles of an element in a given mass, atomic weight becomes crucial because it allows conversion from mass to the number of moles. The formula used is:

• Moles = \( \frac{\text{mass of sample (g)}}{\text{atomic weight (g/mol)}} \)

This is used in calculations when the problem presents a weight in grams, such as in the exercise with 23 g of Na (sodium). Knowing that Na has an atomic weight of approximately 22.99 g/mol allows us to find that 23 g corresponds to about 1 mole of Na.
Understanding atomic weight helps streamline calculations and gives meaningful context to how substances are measured and compared in chemistry.

Avogadro's Constant

Avogadro's Constant is a key concept when dealing with amounts of substances in chemistry. Named after Amedeo Avogadro, this constant is approximately \(6.022 \times 10^{23}\), which refers to the number of atoms or molecules in one mole of a substance.
It serves as a bridge between the atomic scale and the quantities we handle in the laboratory. For instance, when given a quantity in terms of molecules, such as \(6.0 \times 10^{23} \) molecules of \( \text{N}_2 \), Avogadro’s Constant allows for conversion to moles by:

• Moles = \( \frac{\text{number of molecules}}{6.022 \times 10^{23} \text{ molecules/mol}} \)

This conversion is essential because many chemical equations and reactions operate on the mole level, rather than on the molecular level directly. Hence, Avogadro's Constant is critical for calculating how much of a substance is involved in chemical reactions, helping to standardize measurements across various chemical processes.

Number of Atoms

The number of atoms in a substance can fundamentally influence how it behaves and reacts. When dealing with molecules, it is vital to consider how many atoms are present to deduce how much substance you're working with.
For example, water (\(\text{H}_2\text{O}\)) has three atoms per molecule - two hydrogen atoms and one oxygen atom. Therefore, in 0.50 moles of water, the number of atoms is \(0.50 \text{ moles} \times 3 \text{ atoms/molecule} = 1.50 \text{ moles of atoms}\).
Similarly, with nitrogen gas (\(\text{N}_2\)), there are two atoms of nitrogen per molecule. Calculating the number of atoms involves knowing the moles and the atoms per mole relationship:

• Number of Atoms = Moles \( \times \) Atoms per Molecule

This calculation method helps us understand chemical species not just by weight or mass, but by the fundamental building blocks, which are the atoms themselves. Such knowledge is vital in understanding the nature of chemical reactions and molecular interactions.