Problem 110 - General Chemistry: Principles and Modern Applications

Question

The four naturally occurring isotopes of strontium have the atomic masses 83.9134 u; 85.9093 u; 86.9089 u; and 87.9056 u. The percent natural abundance of the lightest isotope is \(0.56 \%\) and of the heaviest, \(82.58 \%\) Estimate the percent natural abundances of the other two. Why is this result only a rough approximation?

Step-by-Step Solution

Answer

The estimated percent natural abundances for these two isotopes of strontium are approximately 4.20% for the 85.9093 u isotope and 4.25% for the 86.9089 u isotope. The result is a rough approximation because the exact percent abundances depend on various factors, including the location where the strontium is found, and not merely on the atomic masses of the isotopes.

Step 1: Identify Given Values

The atomic masses for the isotopes are given as 83.9134 u, 85.9093 u, 86.9089 u, and 87.9056 u. The percent natural abundance of the lightest isotope (83.9134 u) is 0.56%, and of the heaviest (87.9056 u) is 82.58%.

Step 2: Calculate the Remaining Abundance

Knowing that the total natural abundances of all isotopes add up to 100%, you can calculate the remaining abundance to be distributed among the other two isotopes. That is \(100\% - 0.56\% - 82.58\% = 16.86\%\).

Step 3: Calculate the Proportions Compared to the Total Mass

Now, calculate the total atomic mass: \(83.9134 u + 85.9093 u + 86.9089 u + 87.9056 u = 344.6372 u\). Determine the proportion of the atomic mass for each of the two isotopes in question to the total mass. For the 85.9093 u isotope, the proportion is \(85.9093 u / 344.6372 u = 0.2491\). For the 86.9089 u isotope, the proportion is \(86.9089 u / 344.6372 u = 0.2522\).

Step 4: Calculate the Estimated Percent Natural Abundances

Multiply the proportions calculated in the previous step by the remaining percent abundance to estimate the percent natural abundances of the other two isotopes. For the 85.9093 u isotope, the estimate is \(0.2491 * 16.86 \% = 4.20 \%\). For the 86.9089 u isotope, the estimate is \(0.2522 * 16.86 \% = 4.25 \%\).

Key Concepts

Atomic Mass

Atomic mass is an important concept in understanding isotopes and their characteristics. It's the mass of an atom, typically measured in atomic mass units (u), which reflects both the number of protons and neutrons in an atom. Although electrons contribute to the mass of an atom, their mass is negligible compared to that of protons and neutrons.
For strontium isotopes, each isotope consists of a different number of neutrons, which results in varying atomic masses:

• \(83.9134\, ext{u}\)
• \(85.9093\, ext{u}\)
• \(86.9089\, ext{u}\)
• \(87.9056\, ext{u}\)

These masses are very close to one another, highlighting how similar isotopes can be in mass despite differences in neutron number. Small differences in atomic mass can markedly impact an element's average atomic mass, heavily influenced by each isotope's abundance in nature.

Percent Natural Abundance

Percent natural abundance is crucial for determining how prevalent a specific isotope is in a naturally occurring sample of an element. It tells us, in percentage terms, how much of each isotope is found compared to the other isotopes of the element.
For instance, in strontium:

• The lightest isotope (\(83.9134\, ext{u}\)) has a natural abundance of \(0.56\%\).
• The heaviest isotope (\(87.9056\, ext{u}\)) shows an abundance of \(82.58\%\).

The rest of the isotopes have unmeasured natural abundances that, when added to these, sum up to \(100\%\). However, these values are often rough estimates due to analytical limitations and the usage of averages. Even a small percentage variance impacts the calculated atomic weight of the element significantly.

Strontium Isotopes

Strontium has four naturally occurring isotopes, and each one has its specific mass and percent natural abundance. Each isotope has an essential role in the elemental composition of strontium's atomic mass.
Strontium isotopes include:

• \(^ {84}_{38} ext{Sr}\): 83.9134 u, with \(0.56\%\) abundance.
• \(^ {86}_{38} ext{Sr}\): 85.9093 u.
• \(^ {87}_{38} ext{Sr}\): 86.9089 u.
• \(^ {88}_{38} ext{Sr}\): 87.9056 u, with \(82.58\%\) abundance.

Each isotope's mass affects the overall atomic mass of strontium, and understanding the isotopes' distribution helps in diverse scientific fields, including radioisotope dating and nuclear physics. By evaluating the balance of isotope abundances, scientists can learn about strontium’s geological history and its role in various natural processes.