Help me on this question??

Question

The relative atomic mass of an element with atomic number 10 is 20.2. The element consists of two isotopes, the nuclei of which contain 10 and 12 neutrons respectively. The ratio of the relative abundance of the lighter to heavier atoms is ______

(1) 1 : 9
(2) 9 : 1
(3) 1 : 4
(4) 4 : 1

The answer is (2) 9 : 1... i don't understand...please show me your workings.. and explain too..thanks.

I still don't understand..why can't it be 4 : 1?? sorry for causing you any trouble...

Answer 1

Atomic mass one isotope: p + n = 10 + 10 = 20 amu
Atomic mass of the other isotope: p + n = 10 + 12 = 22 amu.

Since this element is consist of the mixture of two naturally occuring isotopes of 20 and 22, the sum of the relative abundances of the isotopes should be 100%.

If the relative abundance of the isotope-20 is X% and
If the relative abundance of the isotope-22 is Y% , then;

X% + Y% = 100%

The relative atomic mass is obtained by using these percentages, but they are not known.

20(X%) + 22(Y%) = 20.2 = 20(0.0X) + 22(0.0Y) = 20.2

X% = 100% - Y% or 0.0X = 1 - 0.0Y
Substituting this value of X in the above equation;

20(1 - 0.0Y) + 22(0.0Y) = 20.2

20 - 0.20Y + 0.22Y = 20.2
0.02Y = 20.2 - 20 = 0.2
Y = 0.2 / 0.02 = 10%

X = 100 - 10 = 90%

This means that the element is composed of 10% by mass of isotope-22 and 90% by mass of isotope-20.

Mass ratio, lighter to heavier : 90 : 10 = 9 : 1

Answer 2

You can do the detailed weighted average, but it is just as easy to think it out.

Isotope one has a mass of 20 (10 protons and 10 neutrons)
Isotope two has a mass of 22 (10 protons and 12 neutrons)

If there was a 50/50 mix, the average mass would be 21. But you have an average mass of 20.2, so there is much more of isotope 20 than the 22. Since the average is 20.2, it has to be a 9:1 ratio.