Problem 77 - Chemistry Matter and Change

Question

Atomic Mass of Silver Silver has two isotopes: \(\stackrel{107}{47} \mathrm{Ag}\) which has a mass of 106.905 amu and a percent abundance of 52.00\(\%\), and 47 \(\stackrel{109}{47} \mathrm{Ag}\), which has a mass of 108.905 amu and an percent abundance of 48.00\(\%\) What is the atomic mass of silver?

Step-by-Step Solution

Answer

The atomic mass of silver is 107.865 amu.

Step 1: Understand Atomic Mass Calculation

The atomic mass of an element with multiple isotopes can be calculated using the formula: \[\text{Atomic Mass} = (\text{Fraction of Isotope 1} \times \text{Mass of Isotope 1}) + (\text{Fraction of Isotope 2} \times \text{Mass of Isotope 2})\] Here, we have two isotopes of silver, and we will use their given mass and abundance to find the atomic mass.

Step 2: Convert Percent Abundance to Fraction

The percent abundance of each isotope must be converted into a decimal (fraction) by dividing by 100. - Fraction of \( ^{107}\text{Ag} \) = \( \frac{52.00}{100} = 0.52 \) - Fraction of \( ^{109}\text{Ag} \) = \( \frac{48.00}{100} = 0.48 \)

Step 3: Calculate Contribution of Each Isotope

Now, compute the contribution of each isotope to the atomic mass. Use the formula: - Contribution of \( ^{107}\text{Ag} \): \[ 0.52 \times 106.905 = 55.5906 \text{ amu} \] - Contribution of \( ^{109}\text{Ag} \): \[ 0.48 \times 108.905 = 52.2744 \text{ amu} \]

Step 4: Add Contributions to Find Atomic Mass

Finally, add the contributions from both isotopes to find the total atomic mass of silver: \[ \text{Atomic Mass of Silver} = 55.5906 + 52.2744 = 107.865 \text{ amu} \]

Step 5: Conclusion

By summing the contributions of each isotope based on their fractional abundance and mass, we find that the atomic mass of silver is 107.865 amu.

Key Concepts

Isotopes

Isotopes are variants of a chemical element that have the same number of protons but different numbers of neutrons. This results in different atomic masses for the same element. The number of protons, also known as the atomic number, determines the chemical properties of the element and thus remains consistent across isotopes. For instance, silver has two isotopes, \( ^{107}\text{Ag} \) and \( ^{109}\text{Ag} \), which differ in neutron count but not in proton count.

• Isotopes of a given element have virtually identical chemical behavior.
• They can differ significantly in their nuclear properties, such as stability and radioactivity.

Understanding isotopes is essential for calculating the atomic mass of elements, which is based on the weighted average of the isotopes considering their different abundances.

Percent Abundance

Percent abundance refers to the proportion of each isotope present in a naturally occurring sample of an element, expressed as a percentage. This figure is crucial for determining the atomic mass. For example, if we know that \( ^{107}\text{Ag} \) has a percent abundance of 52.00%and \( ^{109}\text{Ag} \) a percent abundance of 48.00%, we understand that over half of the silver on Earth consists of the lighter isotope.

• Percent abundance can be converted into a fraction by dividing by 100.
• These fractions are used to calculate the contribution of each isotope to the element's overall atomic mass.

This approach allows us to consider the natural occurrence of each isotope, rather than simply averaging the isotopic masses.

amu (atomic mass unit)

Atomic Mass Unit (amu) is a standard unit of mass that quantifies mass on an atomic or molecular scale. One amu is defined as one-twelfth of the mass of a carbon-12 atom. This unit is used because the masses of atoms and subatomic particles are extremely small, making it difficult to use conventional units like grams or kilograms. Thus, in atomic mass calculations, such as those for silver and its isotopes, amu provides a much more convenient measure.

• Often, the atomic mass of an element listed on the periodic table is given in amu.
• It allows for precise calculations of isotopic contributions to the overall atomic mass of an element.

When calculating the atomic mass of silver from its isotopes, we multiply each isotope's mass (measured in amu) by its fractional abundance and sum these products, resulting in the weighted average atomic mass presented in amu.