# A sample of rhenium (Re) is #"37.40% Re-185"# and #"62.60% Re-187"#. What are the atomic masses of these isotopes? The sum of the masses of the two isotopes is #"371.9087 u"#. Find the mass of the individual isotopes.

##### 1 Answer

The atomic weight of Re-185 is 185.0 u, and the atomic weight of Re-187 is 187.0 u.

#### Explanation:

The atomic weight of Rhenium (Re) as found on the periodic table of the elements is 186.207. Let x = the atomic weight of Re-185 and (371.9087 - x) = the atomic weight of Re-187.

**Known/Given:**

atomic weight of Re = 186.207 u

x = atomic weight of Re-185 in u

(371.9087 - x) = atomic weight of Re-187 in u

abundance of Re-185 = 37.40% = 0.3740

abundance of Re-187 = 62.60% = 0.6260

**Unknown:**

atomic masses of Re-185 and Re-187

**Solution:**

Note: In order to reduce rounding errors, I will include guard digits and round to the proper number of significant figures at the end.

**1. Multiply x times the abundance of Re-185 and multiply (371.9087 - x) times the abundance of Re-187.**

Re-185: (0.3740)(x) = 0.3740x

Re-187: (0.6260)(371.9087 - x) = 232.8148462 - 0.6260x

**2. Add the results and set them equal to 186.207.**

0.3740x + 232.8148462 - 0.6260x = 186.207

**3. Solve for x by subtracting 232.8148462 from both sides and then divide both sides by -0.2520.**

0.3740x + 232.8148462 - 0.6260x - 232.8148462 = 186.207 -

232.8148462

0.3740x - 0.6260x = -46.6078462

-0.2520x = -46.6078462

-0.2520x/-0.2520x = -46.6078462/-0.2520

x = 184.9517706 u

**4. Atomic weights of Re-185 and Re-187.**

x = 185.0 u = the atomic weight of Re-185 (rounded to four significant figures due to significant figures rule for multiplication and division.).

(371.9087 - 184.9517706) = 186.9569294 = 187.0 u = the atomic weight of Re-187 (rounded to one decimal place due to significant figures rule for addition and subtraction...Notice 184.9517706 was subtracted, not 185.0, because rounding is done after all calculations)

**The atomic weight of Re-185 is 185.0 u, and the atomic weight of Re-187 is 187.0 u.**