Question

Silver has only two natural isotopes, and they occur in about equal abundance. If \(109 \mathrm{Ag}\) is one of the isotopes, what is the other isotope given the atomic mass of silver is approximately 108 amu?

Short answer

The other isotope is \(^{107} \mathrm{Ag}\).

Step-by-step solution

Understand the Problem

Silver has two natural isotopes that occur in almost equal abundance. Given that one isotope is \(^{109} \mathrm{Ag}\) and the atomic mass of silver is approximately 108 amu, we need to find the mass number of the other isotope.

Set Up the Equation

If the atomic mass of silver is the weighted average of its isotopes, we can set up the equation:\[\frac{mass \ of \ 1st \ isotope + mass \ of \ 109 \mathrm{Ag}}{2} = 108\]Let \(x\) be the mass of the other isotope. This makes the equation:\[\frac{x + 109}{2} = 108\]

Solve for the Unknown

Solve the equation for \(x\):1. Multiply both sides by 2 to eliminate the fraction: \[ x + 109 = 216 \]2. Subtract 109 from both sides: \[ x = 216 - 109 \]3. Calculate the result: \[ x = 107 \]

Conclusion

The mass number of the other isotope is 107, so the other isotope is \(^{107} \mathrm{Ag}\).

Key concepts

Atomic Mass

The atomic mass of an element is a crucial concept in chemistry that relates to the average mass of the atoms of an element. It is expressed in atomic mass units (amu), with one amu being equivalent to one twelfth of the mass of a carbon-12 atom. Understanding atomic mass helps in identifying elements and understanding their isotopic composition. Unlike the mass number, which is a whole number, atomic mass is often a decimal number. This is because it reflects a weighted average of all the isotopes of an element found in nature.

In the case of silver, its atomic mass is approximately 108 amu, indicating that the weighted average of its naturally occurring isotopes is 108. Even though silver largely consists of two isotopes, the atomic mass gives us a sense of their collective average. This approach allows chemists to take into account both isotopes effectively in calculations and analysis.

Mass Number

Mass number is an essential term that refers to the total number of protons and neutrons in an atom's nucleus. This number is critical for identifying isotopes of a particular element. Since isotopes are atoms of the same element with different neutron counts, they have different mass numbers.

For example, in the context of silver isotopes, one isotope is represented as \(^{109}\mathrm{Ag}\). The mass number 109 tells us exactly how many protons and neutrons are present in the nucleus of this isotope: 47 protons (as silver's atomic number is 47) and 62 neutrons (109 - 47 = 62). The other natural isotope found in silver is \(^{107}\mathrm{Ag}\), with the mass number 107, which indicates 47 protons and 60 neutrons (107 - 47 = 60). Such distinctions help in comprehending the element's behavior and properties at an atomic level.

Weighted Average

Weighted average is a mathematical concept crucial in calculating atomic mass when different isotopes of an element exist. It considers the ratios in which different isotopes contribute to the overall composition of the element. The atomic mass of an element is not merely an average but a weighted average to account for the different abundances of its isotopes.

To understand this better, consider the equation used to find the unknown isotope mass in silver: \( \frac{x + 109}{2} = 108 \). Here, the weight or contribution of each isotope is considered equal since they are in nearly equal abundance. Solving this equation gives us \( x = 107 \), identifying the second isotope as \(^{107}\mathrm{Ag} \). This illustrates how weighted averages give accurate representations of the elements' masses in natural conditions, reflecting the true isotopic mix.