Question

The three stable isotopes of magnesium are 24Mg, 25Mg, and 26Mg. The sum of the abundances of the three isotopes is 1.00. The abundance of 24Mg is 78.99%. Which equation represents the abundance of 25Mg if x = the abundance of 25Mg and y = the abundance of 26Mg?

Answer 1

Answer :

The abundance of $^{25}\textrm{Mg}$ isotope is, 10.2 %

The abundance of $^{26}\textrm{Mg}$ isotope is, 10.9 %

Explanation :

Average atomic mass of an element is defined as the sum of masses of each isotope each multiplied by their natural fractional abundance.

Formula used to calculate average atomic mass follows:

$\text{Average atomic mass }=\sum_{i=1}^n\text{(Atomic mass of an isotopes)}_i\times \text{(Fractional abundance})_i$   .....(1)

Let the fractional abundance of $^{25}\textrm{Mg}$ isotope be 'x' and the fractional abundance of $^{26}\textrm{Mg}$ isotope will be 'y'

For $^{24}\textrm{Mg}$ isotope:

Mass of $^{24}\textrm{Mg}$ isotope = 23.9850 amu

Fractional abundance of $^{24}\textrm{Mg}$ isotope = 78.99 % = 0.78899

For $^{25}\textrm{Mg}$ isotope:

Mass of $^{25}\textrm{Mg}$ isotope = 24.9858 amu

Fractional abundance of $^{25}\textrm{Mg}$ isotope = x

For $^{26}\textrm{Mg}$ isotope:

Mass of $^{26}\textrm{Mg}$ isotope = 25.9826 amu

Fractional abundance of $^{26}\textrm{Mg}$ isotope = y

Average atomic mass of magnesium = 24.3050 amu

Putting values in equation 1, we get:

$24.3050=[(23.9850\times 0.78899)+(24.9858\times x)+(25.9826\times y)]$      .............(1)

and,

$0.78899+x+y=1.00$     .............(2)

or,

$y=1.00-0.78899-x$

$y=0.21101-x$         ............(3)

Putting equation 3 in 1, we get:$24.3050=[(23.9850\times 0.78899)+(24.9858\times x)+(25.9826\times (0.21101-x))]$

By solving the term, we get the value of 'x'.

$x=0.101839$

Now put the value of 'x' in equation 3, we get the value of 'y'.

$y=0.21101-x$

$y=0.21101-0.101839$

$y=0.109171$

Fractional abundance of $^{25}\textrm{Mg}$ isotope = x = 0.101839 × 100 = 10.2 %

Fractional abundance of $^{26}\textrm{Mg}$ isotope = y = 0.109171 × 100 = 10.9 %