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 \(\stackrel{109}{47} \mathrm{Ag}\) 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 approximately 107.945 amu, which is calculated by taking the weighted average of its two isotopes, \(\stackrel{107}{47} \mathrm{Ag}\) and \(\stackrel{109}{47} \mathrm{Ag}\), with their percent abundances acting as weights.

Step 1: Convert percent abundances to decimal form

To convert the percent abundances to decimals, simply divide the percent value by 100. For \(\stackrel{107}{47} \mathrm{Ag}\) isotope: Percent abundance = 52% Decimal abundance = 52 / 100 = 0.52 For \(\stackrel{109}{47} \mathrm{Ag}\) isotope: Percent abundance = 48% Decimal abundance = 48 / 100 = 0.48

Step 2: Calculate the weighted average of the isotopes

To calculate the weighted average, multiply each isotope's mass by its decimal abundance, and then sum the results. Weighted average = (mass of \(\stackrel{107}{47} \mathrm{Ag}\) * decimal abundance of \(\stackrel{107}{47} \mathrm{Ag}\)) + (mass of \(\stackrel{109}{47} \mathrm{Ag}\) * decimal abundance of \(\stackrel{109}{47} \mathrm{Ag}\)) Weighted average = (106.905 amu * 0.52) + (108.905 amu * 0.48)

Step 3: Calculate the atomic mass of silver

Now, calculate the weighted average of the masses: Weighted average = (106.905 amu * 0.52) + (108.905 amu * 0.48) ≈ 55.5906 amu + 52.3544 amu ≈ 107.945 amu The atomic mass of silver is approximately 107.945 amu.

Key Concepts

Isotope Abundance

Understanding the concept of isotope abundance is crucial when calculating the atomic mass of an element like silver. Isotopes are atoms of the same element with the same number of protons but different numbers of neutrons, resulting in varying atomic masses.
For instance, silver has isotopes Agd, with an atomic mass of 106.905 atomic mass units (amu), and Agd, with an atomic mass of 108.905 amu. Their abundances in nature are expressed in percentages, which means Agd accounts for 52% of silver's natural occurrence, and Agd makes up the remaining 48%.
The task is to transform these percentages into decimal form to accurately represent their proportionality for further calculations. An understanding of this concept ensures that the student can proceed to compute the atomic mass using these abundances as accurate weights.

Weighted Average

When aiming to find the atomic mass of an element that has multiple isotopes, we employ the idea of a weighted average. This mathematical concept takes into account the varying masses of isotopes and their relative abundances to calculate an average mass that reflects the isotopes’ contributions to the element’s overall atomic mass.
The average isn't simply the arithmetic mean, but rather is weighted based on the decimal abundance of each isotope. So, even if one isotope is heavier, if it is less abundant, it contributes less to the final atomic mass of the element.

Formula:

Weighted average = (mass of isotope 1 * decimal abundance of isotope 1) + (mass of isotope 2 * decimal abundance of isotope 2)
By multiplying each isotope's mass by its corresponding decimal abundance and then summing up these values, we obtain the weighted average, which is a crucial step before reaching the final atomic mass value for the element.

Atomic Structure

When discussing isotopes and atomic masses, a fundamental understanding of atomic structure is essential. An atom consists of a nucleus containing protons and neutrons, with electrons in orbit around it. The number of protons (atomic number) defines the element, while the number of neutrons can vary, which gives us different isotopes of the same element. Each isotope has its unique mass, largely due to the mass of the neutrons.
In the case of silver (Ag), with the atomic number of 47, it has two common isotopes differing in their number of neutrons: Agd with 60 neutrons and Agd with 62 neutrons. This difference in neutron number not only influences the atomic mass but is also the reason we must calculate an accurate weighted average based on isotope abundance to determine the element's atomic mass as it would be found naturally.