Problem 31 - 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: \(0.50 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}, 23 \mathrm{~g} \mathrm{Na}, 6.0 \times 10^{23} \mathrm{~N}_{2}\) molecules.

Step-by-Step Solution

Answer

The samples can be ranked in order of increasing number of atoms as follows: 23 g of Na (\(6.022 \times 10^{23}\) atoms) < \(6.0 \times 10^{23}\) N2 molecules (\(1.20 \times 10^{24}\) atoms) < 0.50 mol H2O (\(1.81 \times 10^{24}\) atoms).

Step 1: Calculate the number of atoms in 0.50 mol H2O.

Using the Avogadro's constant, which is \(6.022 \times 10^{23}\) atoms/mol, we can calculate the number of atoms in 0.50 mol H2O. Since there are three atoms in each H2O molecule (two H atoms and one O atom), the calculation is as follows: Number of atoms in 0.50 mol H2O = (0.50 mol) * (6.022 x 10^23 atoms/mol) * 3 = \(1.81 \times 10^{24}\) atoms

Step 2: Calculate the number of atoms in 23 g of Na.

Using the atomic weight of Na (23 g/mol), we can first calculate the number of moles and then convert it into the number of atoms using the Avogadro's constant: Number of moles of Na = (23 g) / (23 g/mol) = \(1 \mathrm{~mol}\) Number of atoms in 23 g of Na = (1 mol) * (6.022 x 10^23 atoms/mol) = \(6.022 \times 10^{23}\) atoms

Step 3: Calculate the number of atoms in \(6.0 \times 10^{23}\) N2 molecules.

Since there are two N atoms in each N2 molecule, we can calculate the number of atoms as follows: Number of atoms in \(6.0 \times 10^{23}\) N2 molecules = (\(6.0 \times 10^{23}\) molecules) * (2 atoms/molecule) = \(1.20 \times 10^{24}\) atoms

Step 4: Rank the samples in order of increasing numbers of atoms.

Based on our calculations, the samples have the following numbers of atoms: 1. 23 g of Na: \(6.022 \times 10^{23}\) atoms 2. \(6.0 \times 10^{23}\) N2 molecules: \(1.20 \times 10^{24}\) atoms 3. 0.50 mol H2O: \(1.81 \times 10^{24}\) atoms Thus, the order of increasing numbers of atoms is: 23 g of Na < \(6.0 \times 10^{23}\) N2 molecules < 0.50 mol H2O

Key Concepts

Avogadro's Constant

Understanding Avogadro's constant is fundamental to comprehending the mole concept and comparing quantities of atoms in different substances. Avogadro's constant, denoted as \(6.022 \times 10^{23}\), represents the number of units (atoms, molecules, or ions) in one mole of a substance. In the context of our exercise,

when we talk about \(0.50 \text{ mol} \text{ H}_2\text{O}\) or \(23 \text{ g Na}\), we're referring to amounts that are related to each other through Avogadro's constant. For instance, if we have one mole of any substance, we inherently have \(6.022 \times 10^{23}\) units of that substance, be they atoms in elemental sodium (Na) or molecules in water (H2O).

Practical Use and Calculations

Avogadro's constant lets us translate between the microscopic world of atoms and the macroscopic world we can measure in a lab. For instance, if a student needs to find the number of atoms in a given sample, they'd use Avogadro's constant as a conversion factor. This constant is a bridge and makes stoichiometry and molecular calculations possible and meaningful in chemistry.

Atomic Weights

The atomic weight of an element, sometimes called its relative atomic mass, is the average mass of its atoms compared to 1/12th of the mass of a carbon-12 atom. It's a dimensionless quantity, commonly expressed in atomic mass units (amu) or grams per mole (g/mol). This value is crucial when comparing different elements and their amounts in terms of moles.

In our exercise, for example, sodium (Na) has an atomic weight of approximately 23 g/mol. This number indicates that one mole of sodium weighs 23 grams and contains \(6.022 \times 10^{23}\) atoms of sodium. When a problem, such as the one we're considering, provides a mass of an element, we need the atomic weight to transition from grams to moles, and then to the number of atoms using Avogadro's constant.

How Atomic Weights Facilitate Comparisons

Atomic weights enable the comparison between different elements and their respective amounts in a reaction. Without this knowledge, it would be nearly impossible to conduct precise chemical calculations or to compare one substance to another quantitatively, as we are doing between samples of Na, N2, and H2O.

Mole Concept

The mole is a cornerstone concept in chemistry that measures quantity. One mole is defined as exactly \(6.022 \times 10^{23}\) elementary entities, which may be atoms, molecules, ions, or other particles. This number, Avogadro's constant, is selected to provide an easy bridge between the atomic scale and the scale of chemical reactions at the macroscopic level.

When confronted with exercises involving different substances, the mole concept allows us to count particles by weighing them. For instance, in the provided exercise, we are not directly counting atoms or molecules; rather, we use the weight of Na and the given number of moles for H2O to infer the number of atoms present.

The mole is particularly helpful because it is independent of the type of substance. It merely represents a quantity, much like a 'dozen' represents twelve items, regardless of what those items are. This feature of the mole concept is why we can provide a straightforward ordering of various substances based on the number of atoms, as we did with our samples of Na, N2 molecules, and H2O.