Silicon has three naturally occurring isotopes (Si-28,Si-29,Si-30). The mass and natural abundance of Si-28 are 27.9769 amu and 92.2% respectively. The mass and natural abundance of Si-29 are...
Silicon has three naturally occurring isotopes (Si-28,Si-29,Si-30). The mass and natural abundance of Si-28 are 27.9769 amu and 92.2% respectively. The mass and natural abundance of Si-29 are 28.9765 amu and 4.67% respectively. Find the mass and natural abundance of Si-30.
Please show me the step by step process, I would be grateful!
First, let me list down the given information:
Si-28, 27.9769 amu, 92.20%
Si-29, 28.9765 amu, 4.67%
Si has an atomic weight of 28.0855 amu (while this is not given, the information should be readily available in the periodic table. this is a crucial bit of information as you will see in the solution below.)
The problem mentions that there are three isotopes. Hence, Si-28, Si-29, and Si-30 are all the isotopes of silicon and should make up 100% of the silicon in the world. This allows us to calculate the abundance of Si-30 - the sum of abundance of all isotopes should be 100%:
Abundance of Si-30 = 100 - 92.2 - 4.67 = 3.13
Hence, 3.13% of all silicon reserves is silicon-30.
The atomic weight of an element (listed in a reference such as the periodic table) is they weighted average of the masses of its isotopes. Hence, 28.0855 amu, the atomic weight of silicon, is a weighted average of the individual masses of its isotopes - that is, the mass multiplied by its abundance.
Hence (let m be atomic mass and a be abundance):
mSi = mSi28*aSi28 + mSi29*aSi29 + mSi30*aSi30
28.0855 = 27.9769*0.922 + 28.9756*0.0467 + 0.0313X.
Here, X is the atomic mass of silicon-30, which we do not know yet.
Doing the calculation, you end up with;
0.0313X = 0.93763768
X = 29.9565
This means that Silicon-30 has an atomic mass of 29.9565 amu (which makes sense cause this rounds up to 30, the basis of the names of the isotopes).
Note: you can check if this is right by calculating the atomic mass of silicon based on the abundance and masses of its isotopes.