Problem 115 - Chemistry Matter and Change

Question

Balance Precision A sensitive electronic balance can detect masses of \(1 \times 10^{-8}\) g. How many atoms of silver would be in a sample having this mass?

Step-by-Step Solution

Answer

There are approximately 5.58 × \(10^{13}\) atoms of silver in the sample with a mass of \(1 \times 10^{-8}\) g.

Step 1: Calculate the molar mass of silver

According to the periodic table, the atomic mass of silver (Ag) is 107.87 g/mol. Therefore, the molar mass of silver is also 107.87 g/mol.

Step 2: Calculate the number of moles in the given sample

To calculate the number of moles (n) in the given sample, we can use the formula: n = mass of sample / molar mass of silver The mass of the sample is given as \(1 \times 10^{-8}\) g. Using the molar mass of silver, this can be calculated as: n = \(\frac{1 \times 10^{-8}}{107.87} \)

Step 3: Simplify the expression for the number of moles

Now, let's simplify the expression for the number of moles: n = \(\frac{1 \times 10^{-8}}{107.87} \) ≈ 9.27 × \(10^{-11}\) moles

Step 4: Find the number of silver atoms

To find the number of silver atoms, we need to multiply the number of moles by Avogadro's number (6.022 × \(10^{23}\) atoms/mol): Number of atoms = n × Avogadro's number Number of atoms = (9.27 × \(10^{-11}\) moles) × (6.022 × \(10^{23}\) atoms/mol)

Step 5: Calculate the number of silver atoms

Finally, let's calculate the number of silver atoms: Number of atoms = (9.27 × \(10^{-11}\) moles) × (6.022 × \(10^{23}\) atoms/mol) ≈ 5.58 × \(10^{13}\) atoms So, there are approximately 5.58 × \(10^{13}\) atoms of silver in the sample.

Key Concepts

Understanding Atomic Mass

Atomic mass refers to the mass of a single atom, typically expressed in atomic mass units (amu) or grams per mole (g/mol) due to its position on the periodic table. Each element on the periodic table has its own atomic mass, which helps scientists determine the molar mass of substances.
A molar mass is the weight of one mole (or approximately 6.022 × 10²³ particles) of a given chemical element or compound. For silver (Ag), the atomic mass is 107.87 g/mol. This means that one mole of silver atoms weighs 107.87 grams. This value is crucial when converting between grams and moles, and it is derived from the average mass of all the isotopes of an element based on their natural abundances.
Understanding atomic mass helps bridge the micro and macro perspectives in chemistry, allowing calculations involving grams and moles, facilitating practical laboratory measurements.

The Magic of Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics, representing the number of atoms, ions, or molecules in one mole of a substance. It is approximately 6.022 × 10²³ atoms/mol, named after the Italian scientist Amedeo Avogadro.

• This number allows chemists to count out atoms and molecules by weighing them, providing a bridge between the microscopic atomic level and the macroscopic level we interact with daily.
• For example, in the problem concerning silver atoms, once we determine the number of moles, Avogadro's number is used to calculate the number of individual atoms. Multiplying the number of moles by Avogadro's number yields the total number of atoms.

This concept is foundational in stoichiometry, the study of the quantified relationships of substance quantities in chemical reactions.

Importance of Precision Measurement

Precision measurement is critical in the field of chemistry for making exact calculations. When working with atomic and molecular scales, even small errors can significantly impact results. A precision balance with the ability to detect masses as small as 1 × 10^-8 grams demonstrates how finely tuned scientific instruments can be.

• Precision allows scientists to accurately measure materials, aiding in the repeatability of experiments and reliability of data.
• In our example, it highlights the sensitivity required to detect and use such tiny masses, ensuring correct mole calculations for substances like silver.

High precision in measuring tools is vital because working with quantities of matter at an atomic level means dealing with incredibly small mass units. As a result, precision not only reduces errors but also ensures clarity in scientific communications and findings.