Question

A hypothetical element has an atomic weight of 48.68 amu. It consists of three isotopes having masses of 47.00 amu, 48.00 amu, and 49.00 amu. The lightest-weight isotope has a natural abundance of 10.0%. What is the percent abundance of the heaviest isotope?

Answer 1

78 %

Step-by-step explanation:

The atomic mass is the weighted average of the atomic masses of each isotope.

In a weighted average, we multiply each value by a number representing its relative importance.

In this problem, the percent abundance represents the relative importance of each isotope.

Data:

X-47: mass = 47.00 u; abundance = 10.0 % = 0.100

X-48: mass = 48.00 u

X-49: mass = 49.00 u

Calculations:

Let x = abundance of X-49

Then 0.900 - x = abundance of X-48

`\begin{array}{cccc}\\\textbf{Isotope} & \textbf{Mass/u} & \textbf{Abundance} & \textbf{Contribution/u}\\\text{X-47} & 47.00 & 0.100 & 4.700\\\text{X-48} & 48.00 & 0.900 - x & 48.00(0.900 - x)\\\text{X-49} & 49.00 & x & 49.00x\\& \text{TOTAL} & = & \mathbf{48.68}\\\end{array}`

`\begin{array}{rcr}4.700 + 48.00(0.900 - x) + 49.00x & = & 48.68\\4.700 + 43.20 - 48.00x + 49.00x & = & 48.68\\47.90 +x & = & 48.68\\x & = & \mathbf{0.78}\\\end{array}`

The heaviest isotope has an abundance of 78 %.

Answer 2

Answer : The percent abundance of the heaviest isotope is, 78 %

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* \text{(Fractional abundance})_i`

As we are given that,

Average atomic mass = 48.68 amu

Mass of heaviest-weight isotope = 49.00 amu

Let the percentage abundance of heaviest-weight isotope = x %

Fractional abundance of heaviest-weight isotope =
`(x)/(100)`

Mass of lightest-weight isotope = 47.00 amu

Percentage abundance of lightest-weight isotope = 10 %

Fractional abundance of lightest-weight isotope =
`(10)/(100)`

Mass of middle-weight isotope = 48.00 amu

Percentage abundance of middle-weight isotope = [100 - (x + 10)] % = (90 - x) %

Fractional abundance of middle-weight isotope =
`((90-x))/(100)`

Now put all the given values in above formula, we get:

`48.68=[(47.0* (10)/(100))+(48.0* ((90-x))/(100))+(49.0* (x)/(100))]`

`x=78\%`

Therefore, the percent abundance of the heaviest isotope is, 78 %