Why isn’t the average atomic mass of silver given on the periodic table simply 108?
a. Calculate the average mass in grams of an individual atom of silver.
b. Calculate the mass in grams of 10 trillion silver atoms.
c. Calculate the mass in grams of 5.00 × 1045 silver atoms.
The correct answer and explanation is :
Why the Average Atomic Mass of Silver Is Not 108:
The average atomic mass of silver (Ag) is not exactly 108 because of the presence of isotopes. Silver has two naturally occurring isotopes: Ag-107 and Ag-109. The atomic mass given on the periodic table is a weighted average of the masses of these isotopes based on their relative abundances. The average atomic mass of silver on the periodic table is approximately 107.87 g/mol instead of 108, which reflects the combination of both isotopes in their natural proportions.
Solution:
a. Calculate the Average Mass in Grams of an Individual Atom of Silver:
To find the mass of an individual atom, we use the concept of moles and Avogadro’s number.
The formula is:
[
\text{Mass of one atom} = \frac{\text{Atomic mass (g/mol)}}{\text{Avogadro’s number}}
]
Using the atomic mass of silver, 107.87 g/mol, and Avogadro’s number, 6.022 \times 10^{23} atoms/mol, we can calculate the mass of one silver atom.
[
\text{Mass of one silver atom} = \frac{107.87 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{atoms/mol}}
]
[
\text{Mass of one silver atom} \approx 1.79 \times 10^{-22} \, \text{grams}
]
b. Calculate the Mass in Grams of 10 Trillion Silver Atoms:
Now, to find the mass of 10 trillion (or ( 10^{13} )) silver atoms, we multiply the mass of one atom by the number of atoms:
[
\text{Mass of 10 trillion atoms} = (1.79 \times 10^{-22} \, \text{grams}) \times 10^{13} \, \text{atoms}
]
[
\text{Mass of 10 trillion atoms} \approx 1.79 \times 10^{-9} \, \text{grams}
]
c. Calculate the Mass in Grams of ( 5.00 \times 10^{45} ) Silver Atoms:
For this calculation, we multiply the mass of one silver atom by the number of atoms, ( 5.00 \times 10^{45} ):
[
\text{Mass of } 5.00 \times 10^{45} \, \text{atoms} = (1.79 \times 10^{-22} \, \text{grams}) \times 5.00 \times 10^{45} \, \text{atoms}
]
[
\text{Mass of } 5.00 \times 10^{45} \, \text{atoms} \approx 8.95 \times 10^{24} \, \text{grams}
]
Explanation:
- Why isn’t the atomic mass of silver 108?
The atomic mass of silver is an average derived from the relative abundances of its isotopes. Ag-107 has a greater abundance than Ag-109, and their average gives 107.87 g/mol instead of 108. - Atomic mass calculations:
To calculate the mass of individual atoms, we use Avogadro’s number. This large number reflects the number of atoms in a mole, allowing us to scale from the atomic level to practical quantities. - Mass of many atoms:
By multiplying the mass of a single atom by the number of atoms, we can calculate the mass of various quantities of atoms, whether it’s a few trillion or an astronomically large number like ( 5.00 \times 10^{45} ) atoms.
These steps show how atomic mass relates to both individual atoms and larger amounts of matter, providing insight into the scale of atomic mass and the incredibly small masses involved.