Question

The element rhenium (Re) has two naturally occurring isotopes, ${}^{185}\mathrm{Re}$ and ${}^{187}\mathrm{Re},$ with an average atomic mass of 186.207 u. Rhenium is $62.60\mathrm{\%}$ is ${}^{187}\mathrm{Re},$, and the atomic mass of ${}^{187}$Re is 186.956 u. Calculate the mass of ${}^{185}\mathrm{Re}$.

Short answer

The mass of the isotope ${}^{185}\mathrm{Re}$ is 185.458 u.

Step-by-step solution

Since the total percentages of both isotopes must add up to 100%, the percentage of ${}^{185}\mathrm{Re}$ is the difference between 100% and the percentage of ${}^{187}\mathrm{Re}$. Percentage of ${}^{185}\mathrm{Re}=100\mathrm{\%}-\mathrm{\%}of{}^{187}\mathrm{Re}=100\mathrm{\%}-62.60\mathrm{\%}=37.40\mathrm{\%}$ #Step 2: Convert the percentages to decimal form#

We need to convert the percentage values to their decimal equivalents for further calculations. Decimal of ${}^{185}\mathrm{Re}=\frac{37.40}{100}=0.374$ Decimal of ${}^{187}\mathrm{Re}=\frac{62.60}{100}=0.626$ #Step 3: Plug in the known values into the weighted average formula#

Now we need to rewrite the weighted average formula with all the available information. $$186.207=0.374\times massof{}^{185}\mathrm{Re}+0.626\times 186.956$$ #Step 4: Solve for the mass of ${}^{185}\mathrm{Re}$#

To solve for the mass of ${}^{185}\mathrm{Re}$, we first need to isolate it on one side of the equation. $$massof{}^{185}\mathrm{Re}=\frac{186.207-0.626\times 186.956}{0.374}$$ Now, calculate the value: $$massof{}^{185}\mathrm{Re}=\frac{186.207-117.277}{0.374}=\frac{68.93}{0.374}=185.458u$$ #Conclusion# The mass of the isotope ${}^{185}\mathrm{Re}$ is 185.458 u.

Key concepts

Weighted Average Formula

The weighted average formula is essential in isotope calculations, especially when determining the average atomic mass of an element with multiple isotopes. This formula considers both the mass of each isotope and its abundance in nature. In mathematical terms, the weighted average formula is expressed as follows:
$$\text{Average Atomic Mass}=(\text{fraction of isotope 1}\times \text{mass of isotope 1})+(\text{fraction of isotope 2}\times \text{mass of isotope 2})+\dots$$
Here, the fraction refers to the percentage of each isotope converted to decimal form. The average atomic mass provides a more accurate reflection of an element's mass by taking into account the mass and relative abundances of its isotopes. When utilizing this formula, ensure that the total sum of the fractions equals 1, representing 100% of the isotopes considered.

Atomic Mass

Atomic mass refers to the mass of a specific isotope of an element. It is usually expressed in atomic mass units (u), where 1 u is approximately 1/12 of the mass of a carbon-12 atom. Each isotope of an element has its own distinct atomic mass because it differs in the number of neutrons.
Knowing the atomic mass of isotopes is crucial for calculations involving the element's average atomic mass. In our exercise, the isotopes of rhenium, ${}^{185}\mathrm{Re}$ with unknown mass, and ${}^{187}\mathrm{Re}$ with a mass of 186.956 u, exhibit different atomic masses. Determining these masses allows chemists to calculate values crucial for scientific exploration and industry applications.
Understanding atomic mass helps in grasping how tiny changes in neutron numbers can significantly impact the physical properties of elements.

Naturally Occurring Isotopes

Elements in nature often exist as a mixture of isotopes, called naturally occurring isotopes. Each isotope of an element has the same number of protons but differs in the number of neutrons, resulting in different atomic masses.
In the case of rhenium, it has two naturally occurring isotopes: ${}^{185}\mathrm{Re}$ and ${}^{187}\mathrm{Re}$. Their natural abundance—or how frequently they appear naturally—impacts the calculated average atomic mass of the element.
The presence of multiple isotopes of elements helps to explain certain physical and chemical properties. Naturally occurring isotopes play a vital role in fields like geology, medicine, and energy production. Comprehending their existence aids scientific understanding at both micro and macro levels.

Average Atomic Mass

The average atomic mass is a significant concept in chemistry, reflecting the weighted mean mass of an element's isotopes as they appear naturally. It provides a more nuanced understanding of an element than a single isotope's mass. Calculated using the abundance and atomic mass of each naturally occurring isotope, we utilize the weighted average formula to determine this value.
Rhenium, for example, has an average atomic mass of 186.207 u due to its isotopes ${}^{185}\mathrm{Re}$ and ${}^{187}\mathrm{Re}$. This value indicates how the isotopes contribute collectively to the element's overall mass in a natural sample.
Knowing the average atomic mass is crucial for various chemical calculations, including stoichiometry and molecular weight determination. It assists scientists in predicting how elements behave in different reactions and applications.