Question

If an average sodium atom weighs 22.99 amu, how many sodium atoms are contained in \(1.98 \times 10^{13}\) amu of sodium? What will \(3.01 \times 10^{23}\) sodium atoms weigh?

Short answer

There are \(8.61 \times 10^{11}\) sodium atoms in \(1.98 \times 10^{13}\) amu of sodium. The weight of \(3.01 \times 10^{23}\) sodium atoms is \(6.92 \times 10^{24}\) amu.

Step-by-step solution

Find the number of sodium atoms in a given mass of sodium atoms

Given an average sodium atom weighs 22.99 amu, we can find the total number of sodium atoms in \(1.98 \times 10^{13}\) amu of sodium by dividing the given mass by the mass of a single sodium atom. Number of sodium atoms = \(\frac{\text{(mass of sodium)}}{\text{(mass of a single sodium atom)}}\)

Calculate the number of sodium atoms

Now we can plug in the given values and solve for the number of sodium atoms. Number of sodium atoms = \(\frac{1.98 \times 10^{13}\, \text{amu}}{22.99\, \text{amu}}\) Number of sodium atoms = \(8.61 \times 10^{11}\) So, there are \(8.61 \times 10^{11}\) sodium atoms in \(1.98 \times 10^{13}\) amu of sodium.

Find the weight of a specified number of sodium atoms

Now, we need to find the weight of \(3.01 \times 10^{23}\) sodium atoms. To do this, we can multiply the number of sodium atoms by the mass of a single sodium atom. Mass of sodium = Number of sodium atoms × Mass of a single sodium atom

Calculate the weight of the specified number of sodium atoms

Now we can plug in the given values and solve for the mass of sodium. Mass of sodium = \(3.01 \times 10^{23}\) sodium atoms × 22.99 amu Mass of sodium = \(6.92 \times 10^{24}\) amu So, \(3.01 \times 10^{23}\) sodium atoms weigh \(6.92 \times 10^{24}\) amu.

Key concepts

Molar Mass of Sodium

Understanding the molar mass of an element is essential when studying chemistry. The molar mass is the mass of one mole of a substance. A mole is defined as a measurement that accounts for a specific number of particles, just like a dozen refers to 12 items. For sodium (Na), which is an element on the periodic table, the molar mass is calculated from its atomic mass. The atomic mass is often listed in atomic mass units (amu), and for sodium, it is approximately 22.99 amu.

Let's put this in perspective with a real-life comparison: if the atomic mass is the weight of a single 'grain' of an element, the molar mass would be the total weight of a 'bucketful' of these grains that contains Avogadro's number of them - a standard quantity in chemistry. To find the molar mass of sodium, we use its atomic weight in grams per mole, resulting in a molar mass of approximately 22.99 g/mol. This means that one mole of sodium atoms, which contains Avogadro's number of atoms, weighs about 22.99 grams.

Avogadro's Number

Avogadro's number, named after the scientist Amedeo Avogadro, is a critical constant in chemistry. This number, which is approximately 6.022 x 10²³, represent the number of particles, often atoms or molecules, in one mole of a substance. This colossal number allows chemists to count out atoms and molecules in a practical way for laboratory measurements and calculations.

Why is this important?

For a substance like sodium, if we have one mole of it, we effectively have 6.022 x 10²³ sodium atoms which can be manipulated and accounted for during reactions and in calculating the mass of sodium samples. Avogadro's number is the bridge that connects the microscopic world of atoms and molecules to the macroscopic world we can measure, describe, and utilize in experiments and industry.

Atomic Mass Calculation

The subject of atomic mass calculation is central for transforming a qualitative understanding of chemical elements to a quantitative one. Atomic mass is often expressed in atomic mass units (amu), which symbolizes the mass of a single atom of an element. The atomic mass can be found on the periodic table for each element and is equivalent to the weight of one mole of the substance in grams.

For calculations involving the atomic mass, knowing that 1 amu is defined as one-twelfth of the mass of a carbon-12 atom, which is approximately equal to 1.66 x 10^-24 grams, is essential. In practice, to calculate the mass of a sample, you simply multiply the number of atoms by the atomic mass of the element. Conversely, to find the number of atoms in a given mass, you divide the total mass by the atomic mass of a single atom, as shown in the exercise involving sodium atoms. This foundational concept is used in countless calculations across all fields of chemistry.