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

The sample contains approximately \(5.585 \times 10^{13}\) silver atoms.

Step 1: Determine the mass of one silver atom

First, we need to find the mass of one silver atom. The atomic mass of silver (Ag) is approximately 107.87 amu (atomic mass units). To convert this into grams, note that 1 amu is equal to \(1.66053906660 \times 10^{-24}\) grams. Therefore, the mass of one silver atom is \(107.87 \times 1.66053906660 \times 10^{-24}\) g.

Step 2: Calculate the mass of one silver atom in grams

Multiply the atomic mass of silver by the conversion factor: \(107.87 \times 1.66053906660 \times 10^{-24} = 1.79031 \times 10^{-22}\) grams. This is the mass of a single silver atom.

Step 3: Calculate the number of silver atoms in the given mass

To find out how many atoms of silver are in a mass of \(1 \times 10^{-8}\) g, divide the mass of the sample by the mass of one silver atom: \( \frac{1 \times 10^{-8}}{1.79031 \times 10^{-22}} \approx 5.585 \times 10^{13}\) atoms.

Key Concepts

Grams to AMU Conversion

Understanding how to convert grams into atomic mass units (amu) is crucial in chemistry. The atomic mass unit is a smaller unit, better suited for expressing the minuscule masses of atoms and subatomic particles. One amu is defined as

• \(1.66053906660 \times 10^{-24}\) grams.

Given this conversion factor, it's possible to switch from atomic scale measurements in amus to more familiar mass measurements in grams.
For instance, if we know the atomic mass of silver (107.87 amu), we can convert it to grams by multiplying with the conversion factor. Hence, the process helps us contextualize atomic masses into the macroscopic world where grams are the common standard of measuring mass.

Atomic Mass Unit

The atomic mass unit (amu) is a fundamental concept in understanding atomic mass.
It represents the relative mass of atoms compared to a reference atom, which is carbon-12. For convenience,

• one amu is defined as exactly \( \frac{1}{12} \) of the mass of one carbon-12 atom.

This makes amu a very small unit suited for communicating the masses of atoms.
Instead of expressing atomic masses in grams, which would be impractically small, chemists use amus to describe the mass of an atom.

• Atomic masses in the periodic table, like silver's 107.87 amu, reflect this unit.

Understanding the amu allows chemists to perform calculations about molecular mass and chemical reactions with greater precision.

Mass of a Silver Atom

Knowing the mass of a silver atom can highlight the atomic scale and underscore why atomic mass units are essential.
Silver, with the chemical symbol Ag, has an atomic mass of 107.87 amu. By converting this atomic mass into grams, we can find the mass of a single silver atom.
The calculation uses the conversion factor

• \(107.87 \times 1.66053906660 \times 10^{-24} = 1.79031 \times 10^{-22}\) grams.

This reveals that even a single atom of a relatively heavy element like silver has an extremely small mass when considered in grams.
Such conversions allow scientists to relate atomic scale phenomena to macroscopic measurements, permitting insights into samples used in experiments that typical balances can measure, like the example given of a balance detecting masses in the order of \(10^{-8}\) grams.