Problem 4 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

Assume that sodium atoms are spheres of radius \(0.2 \mathrm{~nm}\) and that they are lined up side by side. How many miles, in length, is the line of atoms present in a \(1.15 \mathrm{mg}\) sample of sodium? \(\left(N_{\mathrm{A}}=6 \times 10^{23}\right)\) (a) \(1.2 \times 10^{10}\) (b) \(1.2 \times 10^{8}\) (c) \(7.5 \times 10^{8}\) (d) \(7.5 \times 10^{6}\)

Step-by-Step Solution

Answer

The line of sodium atoms has length \(7.5 \times 10^{6}\) miles, thus option (d) is correct.

Step 1: Calculate the number of sodium atoms

To find the number of sodium atoms in a sample, use Avogadro's number \(N_A = 6 \times 10^{23}\) atoms per mole. First, determine the molar mass of sodium which is approximately 23 grams per mole. Convert the mass of the sample from milligrams to grams: \(1.15 \text{ mg} = 0.00115 \text{ g}\). The number of moles of sodium is \(\frac{0.00115 \text{ g}}{23 \text{ g/mol}}\), and the number of sodium atoms is the number of moles multiplied by Avogadro's number.

Step 2: Calculate the total length of sodium atoms

Each sodium atom is a sphere with a radius of \(0.2 \text{ nm}\) (nanometers). The diameter of a sodium atom, therefore, is \(2 \times 0.2 \text{ nm} = 0.4 \text{ nm}\). Since the atoms are lined up side by side, the length of one atom will be the diameter. To find the total length in nanometers, multiply the number of atoms by the diameter of each atom.

Step 3: Convert to miles

To convert the total length from nanometers to miles, use the conversion factors: \(1 \text{ nm} = 10^{-9} \text{ m}\) and \(1 \text{ m} = 0.000621371 \text{ mi}\). First, convert the length from nanometers to meters and then from meters to miles to get the final length in miles.

Key Concepts

Chemical Calculations

Chemical calculations form the foundation of understanding relationships in chemistry, especially when it comes to working with atoms and molecules on a macroscopic scale.
The exercise provided involves calculating the length of a line of sodium atoms in a 1.15 mg sample. To perform this chemical calculation, we must apply Avogadro's number, which is a fundamental constant used to convert between the number of particles (atoms, molecules, etc.) and moles, which is a standard unit in chemistry for quantifying substance amounts.
Understanding Avogadro's number is crucial: it allows for the translation of the microscopic world (individual atoms and molecules) into quantities that can be measured and used in the laboratory. Furthermore, recognizing that the molar mass (the mass of one mole of a chemical element or compound) is integral to these calculations, helps one to bridge the gap between mass and the number of particles.

Molar Mass

The molar mass is an essential concept in chemical calculations because it relates the mass of a substance to the number of particles it contains. It is the mass of one mole of a given substance and is usually reported in grams per mole (g/mol).
For sodium (Na), with an atomic number of 11, the molar mass is approximately 23 g/mol; this is determined using the atomic weight of sodium on the periodic table. In our exercise, the mass of sodium is given in milligrams, which must first be converted to grams, to then use the molar mass for finding the number of moles. The process of calculating moles from mass and molar mass is a fundamental routine in chemistry, arranging the groundwork for further calculations, such as determining the number of atoms using Avogadro's number.

Unit Conversion

Unit conversion is a pivotal skill in chemistry, enabling scientists to convert measurements from one unit to another. In the exercise, we deal with multiple unit conversions: from milligrams to grams, from grams to moles, from the number of sodium atoms in a mole to actual numbers of atoms in our sample, and finally, from nanometers to miles for the length of the chain of atoms.
Precision in unit conversion is a must, as incorrect conversions can lead to significant errors in the outcome. For such conversions, we use conversion factors that define the relationship between different units. For example, understanding that 1 nm equals 10^-9 meters and 1 meter equals approximately 0.000621371 miles allows us to navigate from the scale of individual sodium atoms to the practical units of length used in everyday life. These steps are fundamental in transforming the very small scale of atoms to a comprehensible and tangible scale, as seen in this exercise.