Question

Silver occurs naturally as \({ }^{107} \mathrm{Ag}\) and \({ }^{109} \mathrm{Ag} .\) Given the mass and abundance of silver-109 \((108.91\) amu and \(48.16 \%)\), what is the isotopic mass of silver-107?

Short answer

The isotopic mass of silver-107 is approximately 106.95 amu.

Step-by-step solution

Understanding the Problem

We are given two isotopes of silver: \({ }^{107} \mathrm{Ag}\) and \({ }^{109} \mathrm{Ag}\). Silver-109 has a mass of 108.91 amu and an abundance of 48.16%. Our task is to find the isotopic mass of silver-107.

Formulate the Weighted Average Equation

The average atomic mass of an element can be calculated using the formula: \( \text{Average Atomic Mass} = (\text{mass of } ^{107}\text{Ag})(\text{abundance of } ^{107}\text{Ag}) + (\text{mass of } ^{109}\text{Ag})(\text{abundance of } ^{109}\text{Ag}) \). The average atomic mass for silver is approximately 107.87 amu.

Define the Variables

Let \( x \) be the isotopic mass of silver-107. Since the total abundance must equal 100%, the abundance of silver-107 is \( 100\% - 48.16\% = 51.84\% \) or 0.5184 in decimal form.

Set Up the Equation

Using the average atomic mass formula and plugging in the known values: \[ 107.87 = (x)(0.5184) + (108.91)(0.4816) \].

Solve for the Unknown (Isotopic Mass of Silver-107)

First, calculate the second term: \( 108.91 \times 0.4816 = 52.453 \). Substitute this back into the equation: \[ 107.87 = (x)(0.5184) + 52.453 \]. Next, solve for \( x \): 1. Subtract 52.453 from both sides: \( 107.87 - 52.453 = x \times 0.5184 \). 2. This simplifies to \( 55.417 = x \times 0.5184 \).3. Finally, solve for \( x \) by dividing both sides by 0.5184: \( x = \frac{55.417}{0.5184} \approx 106.95 \) amu.

Conclusion

The isotopic mass of silver-107 is approximately 106.95 amu.

Key concepts

Average Atomic Mass

The average atomic mass is an essential concept when dealing with elements that have multiple isotopes. It represents the weighted mean of the masses of an element's isotopes based on their natural abundance. In other words, it gives us a single value to describe the mass of an element, even though that element might exist in different forms in nature. To calculate this average, we use a formula that takes into account both the mass and the relative abundance of each isotope. This equation is represented as:

• \[ \text{Average Atomic Mass} = (\text{mass of isotope 1}) \times (\text{abundance of isotope 1}) + (\text{mass of isotope 2}) \times (\text{abundance of isotope 2}) + \ldots \]

For example, when calculating the average atomic mass of silver, we consider its two isotopes, \\({ }^{107} \mathrm{Ag} \) and \({ }^{109} \mathrm{Ag} \), and use their isotope-specific masses and abundances in the calculation. This weighted average helps chemists and physicists understand more about the element's behavior and characteristics when found naturally.

Isotopic Abundance

Isotopic abundance is the percentage that a particular isotope contributes to the total amount of an element found on Earth. It tells us how many of each isotope's atoms are present compared to the total amount of that element. In calculations involving isotopic masses, it's important to convert these percentages into decimal form, as seen in the formula for average atomic mass.
For silver, the two naturally occurring isotopes are \\({ }^{107} \mathrm{Ag} \) and \({ }^{109} \mathrm{Ag} \). In our exercise, we know that silver-109 has an abundance of 48.16%, which represents the fraction it contributes to natural silver. To complete the process, we subtract this percentage from 100% to find the abundance of silver-107, which is 51.84%.
Utilizing their abundances is crucial because it allows for an accurate calculation of the average atomic mass. This concept simplifies understanding of materials around us, and it's essential to consider these abundances when working with isotopes in scientific research.

Silver Isotopes

Silver, a precious element often used in jewelry and electronics, naturally occurs with two stable isotopes: \\({ }^{107} \mathrm{Ag} \) and \({ }^{109} \mathrm{Ag} \). These isotopes have similar chemical properties but differ in terms of their atomic masses due to the different numbers of neutrons in their nuclei.

• Silver-107 has an isotopic mass of approximately 106.95 amu, based on the calculation from the provided exercise.
• Silver-109, on the other hand, has a mass of 108.91 amu and accounts for roughly 48.16% of the naturally occurring silver found on Earth.

The presence of these isotopes not only contributes to the weighted average atomic mass of silver but also plays an important role in various practical applications, from industrial use to radiometric dating in scientific studies. Understanding how isotopes differ and affect elemental properties helps scientists in analyzing chemical reactions and cycles, providing a clearer picture of the natural world.