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Chapter 3: Problem 39

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

Short Answer

Expert verified
The mass of \(^{185}\mathrm{Re}\) is approximately 184.744 u.

Step by step solution

01

Determine the abundance of each isotope

We are given that rhenium is \(62.60\%\) \(^{187}\mathrm{Re}\). Since there are only two naturally occurring isotopes, we can find the abundance of \(^{185}\mathrm{Re}\). The abundance of \(^{185}\mathrm{Re}\) is: \[100\% - 62.60\% = 37.40\%\] Now, convert the percentages to decimal form: \[\frac{62.60}{100} = 0.6260\] \[\frac{37.40}{100} = 0.3740\]
02

Use the weighted average formula for atomic mass

Average atomic mass = (mass of \(^{185}\mathrm{Re}\) × abundance of \(^{185}\mathrm{Re}\)) + (mass of \(^{187}\mathrm{Re}\) × abundance of \(^{187}\mathrm{Re}\)) We have the average atomic mass of rhenium (186.207 u), the abundance of each isotope, and the atomic mass of \(^{187}\mathrm{Re}\) (186.956 u). We are solving for the mass of \(^{185}\mathrm{Re}\), denoted as x. Plug in the values into the formula: \[186.207 = (x \times 0.3740) + (186.956 \times 0.6260)\]
03

Solve for the mass of \(^{185}\mathrm{Re}\)

Solve the equation from Step 2 to find the mass of \(^{185}\mathrm{Re}\): \[186.207 = (x \times 0.3740) + (186.956 \times 0.6260)\] First, find the result of the second term: \[186.956 \times 0.6260 = 117.134776\] Then rearrange the equation to solve for x: \[x \times 0.3740 = 186.207 - 117.134776\] Now, find the result of the subtraction: \[186.207 - 117.134776 = 69.072224\] Finally, divide by the abundance of \(^{185}\mathrm{Re}\) (0.3740) to find the mass of \(^{185}\mathrm{Re}\): \[x = \frac{69.072224}{0.3740}\] \[x = 184.744\] Thus, the mass of \(^{185}\mathrm{Re}\) is approximately 184.744 u.

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Most popular questions from this chapter

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