Given:
Sum of masses of two isotopes = 371.9087 u
Re-185 natural abudance = 37.40%
Re-187 natural abudance = 62.60%
Known:
atomic weight of Re = 186.207 u
Atomic mass of Re-185:
To find the atomic mass of Re-185, take the total mass given and subtract atomic weight.
abundance of Re-185 = 37.40% = 0.3740
(371.9087 - x) = atomic weight of Re-187 in u
To find mass of Re-187:
abundance of Re-187 = 62.60% = 0.6260
Solution:
Step 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
Step 2. Add the results and set them equal to 186.207.
0.3740x + 232.8148462 - 0.6260x = 186.207
Step 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
Step 4. Atomic weights of Re-185 and Re-187.
x = 185.0 u = the atomic weight of Re-185
(371.9087 - 184.9517706) = 186.9569294 = 187.0 u = the atomic weight of Re-187
Therefore the atomic weight of Re-185 is 185.0 u, and the atomic weight of Re-187 is 187.0 u.